\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\mathcal{B}(\mathcal{K})^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\mathbb N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\mathbb C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
		\end{enumerate}

		\noindent If $\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
			\item \label{i:sos:iv}
			There exists $r\in\cA_d$ s.t.
			\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}

		\end{enumerate}

\begin{remark}[What's new?]\rm
		\label{rem:whatsnew}
		The passage to operator coefficients necessitates several novel
		results and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\ref{thm:sos} and Theorem~\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).
		Namely, the Hahn-Banach theorem produces a separating linear functional $\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction
		based on $\varphi$ ultimately produces a tuple $Y$.
		Here we
		roughly follow the outline of \cite{MP05}.

		A key construction is that of a tuple of self-adjoint operators $A$
		based upon the left regular representation on Fock space; see Section~\ref{sec:fock}.
		We then show that, up to a universal constant, for a sum of squares polynomial $p,$
		the norm
		of $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\ref{prop:bounded}.
		This uniform bound is the main input in Section~\ref{sec:top} for proving that the cone $\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.

		On the GNS side, we introduce a new argument that exploits the WOT to associate to a
		separating linear functional a finite-rank psd noncommutative representing Hankel matrix and,
		on its range, construct the desired tuple~$Y$; see Section~\ref{sec:gns}. For the unitary result,
		Theorem~\ref{thm:usos}, we additionally modify the construction of Section~\ref{sec:fock} to produce a canonical
		tuple of unitary operators from the left-regular representation and adapt the GNS procedure
		to obtain a unitary tuple together with a representing vector that realizes the separating
		functional as a vector state; see Subsection~\ref{ssec:uGNS}.\qed
	\end{remark}

	\subsection{Reader's guide}\label{ssec:readersguide}

	The paper is structured as follows. The convex cone of sums of squares (making an appearance in \eqref{eq:sosthm} of Theorem~\ref{thm:sos}) is introduced and characterized in the next Section~\ref{sec:sos}. In Section~\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\cF_\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\ref{sec:top} we
	introduce a suitable topology on $\cA_\ad$ and
	collect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\ref{sec:gns}; see Proposition~\ref{prop:GNS}.
	Then Theorem~\ref{thm:sos} is proved in Section~\ref{sec:proof} and
	Theorem~\ref{thm:usos} is proved in Section~\ref{sec:uproof}.

	\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\ref{prop:closedcone}.

	\section{Convex Cone of (Sums of) Squares}\label{sec:sos}

	In this section a key player in the proof of Theorem~\ref{thm:sos}, the convex cone of sums of squares of polynomials, is introduced and
	studied. The main result in this section is Proposition~\ref{prop:closed under addition} (see also Remark \ref{rem:soscone}), which  gives a bound on the number of sums of squares needed to write a polynomial as a sum of squares.

	\begin{lemma}\label{lem:psd}
		If $T: \cH^{n} \to \cH^{n}$ be a psd linear map, then there exist linear maps $R_{i}: \cH \to \cH^{n},$ $i = 1, \dots, n,$ such that $T = \sum_{i=1}^{n} R_{i}R_{i}^{*}.$ Moreover, if $\cH$ is infinite-dimensional, then $T = R R^{*}$ for some $R : \cH \to \cH^{n}.$
	\end{lemma}
	\begin{proof}

		Since $T$ is psd, there exists a linear map $\tilde{R}: \cH^{n} \to \cH^{n}$ such that $T = \tilde{R} \tilde{R}^{*}.$ Write
		\[
		\tilde{R} \ =\  \begin{bmatrix}
			R_{1}, \dots, R_{n}
		\end{bmatrix}
		\]
		with respect to the orthogonal decomposition $\cH^{n} = \cH \oplus \dots \oplus \cH.$ The first part of the lemma follows by noting that each $R_{i}$ is a map from $\cH$ into $\cH^{n}.$ For the moreover part, let $U: \cH \to \cH^{n}$ be any unitary, and set $R = \tilde{R} U.$
	\end{proof}

	Index \df{$\mathbb{\cH}^{N(d)}$} and \df{$\cA_{d}^{N(d)}$} (the algebraic direct sum of $\cA_{d}$ with itself $N(d)$ times) by $\la x \ra_{d}.$
	Let $V_{d} \in \cA_{d}^{N}$ denote the \df{Veronese column vector} whose $w\in \la x \ra_{d}$ entry is $w$
	(adopting the  usual convention of viewing $w$ as the the $\cB(\cH)$-valued polynomial $I_\cH \, w$). For instance,
	if $\vg=2$ and $d=2$, then  \index{$V_d$}
	\[
	V_2\ =\  \text{col} \begin{pmatrix}
		1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2
	\end{pmatrix}.
	\]
	Let \df{$\cC_d$} denote the \df{cone of sums of squares} of polynomials of degree at most $d,$
	\begin{equation}\label{eq:sosdef}
		\cC_{d} \ :=\  \left\{ \sum\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \quad r_{i}\in \cA_{d},  \quad  i=1, \dots, N(d) \right\}
		\subseteq \cA_{2d}.
	\end{equation}
	Given $r\in \cA_d,$ the column vector  $R$  with $w$ entry $R_{w}^{*}$
	is called the \df{coefficient vector} of $r$ since  $r = R^{*} V_{d}.$ In particular,
	\[
	r^* r \ =\  V_d^* RR^* V_d
	\]
	so that  $r^*r$ has a representation  as  $V_d^* \SG V_d$ for a psd matrix $\SG.$

	\begin{prop} \label{prop:closed under addition}
		A polynomial $p\in \cA_{2d}$ is in $\cC_d$ if and only if there is a psd block matrix $\SG$ such that
		\begin{equation}
			\label{e:gram-rep}
			p \ =\  V_d^* \SG V_d.
		\end{equation}
		In fact, if $p=V_d^* \SG V_d,$ then factoring $\SG=\sum_{j=1}^{N(d)} R_jR_j^*$  with $R_j:\cH \to \oplus_{w\in \la x\ra_d}\cH$
		as in Lemma~\ref{lem:psd},
		setting $r_j = R_{j}^{*} V_{d} $
		gives,
		\[
		p \ =\ \sum_{j=1}^{N(d)} r_j^* r_j.
		\]

		In particular,  the set $\cC_d$  is a (convex) cone. 	\end{prop}

	We call any psd block matrix $\SG$ satisfying equation~\eqref{e:gram-rep} a \df{Gram representation} for $p.$

	\begin{proof}
		Given a sum of squares $p =  \sum_{i=1}^{N(d)} r_{i}^{*}r_{i},$ writing $r_j=R_j^{*} V_d$ gives
		$p = \sum_{i=1}^{N(d)} V_{d}^{*} R_{i} R_{i}^{*} V_{d},$ where $R_{i}$ is the coefficient vector corresponding to the polynomial $r_{i}.$  It follows that  $p = V_{d}^{*} \SG V_{d},$ where $\SG = \sum_{i=1}^{N(d)} R_{i} R_{i}^{*}.$ In particular, $\SG : \cH^{N(d)} \to \cH^{N(d)}$ is a psd linear map.

		Conversely, suppose there is a psd linear map $\SG:\oplus_{w\in \la x\ra_d} \cH \to \oplus_{w\in \la x \ra_d}\cH$ such that $p = V_d^* \SG V_d.$
		By Lemma~\ref{lem:psd}, there exist $R_j: \cH\to \oplus_{w\in \la x \ra_d}\cH$ such  that
		$\SG=\sum_{j=1}^{N(d)} R_j R_j^*.$ Setting
		$r_j =R_j^* V_d$, one obtains $p=\sum_{j=1}^{N(d)} r_j^*r_j.$

		By what has already been proved,
		if  $p,q \in \cC_{d},$ then there exist (psd) Gram representations  $p = V^{*} \SG_{p} V$ and $ q = V^{*} \SG_{q} V.$
		Now $p + q = V_{d}^{*} (\SG_{p} + \SG _{q}) V_{d} .$ Since $\SG_{p} + \SG_{q} : \cH^{N(d)}  \to \cH^{N(d)}$ is a psd linear map,
		what has already been proved shows
		$p + q \in \cC_{d}.$
	\end{proof}

	\begin{corollary}\label{cor:soscone}
		Letting   $V_{d} \in \cA_{d}^{N(d)}$ denote  the Veronese column vector,
		the convex cone of sums of squares of degree at most $2d$ is
		\[
		\cC_{d} = \left\{ V_{d}^{*} \SG V_{d}: \quad \SG =  [\SG_{v,w}]_{v,w\in \la x \ra_{d}} \in \cB(\cH)^{N(d) \times N(d)}, \quad \SG \succeq 0 \right\}.
		\]
	\end{corollary}

	\begin{remark}\rm\label{rem:soscone}
		It follows from the preceding discussion that the convex cone $\cC_{d}$ takes the form
		\[
		\cC_{d} \ =\ \{r^{*}r : \quad r \in \cA_{d}\}
		\]
		when the Hilbert space $\cH$ is infinite-dimensional.
		\qed
	\end{remark}

\label{ssec:uGNS}
	The only other point that needs attention is the proof of a suitable GNS construction as in Proposition~\ref{prop:GNS}.
	Since we cannot rely on non-cancellation of the highest orde

\begin{prop}\label{prop:closedcone}
		The convex cone $\cC_{d}$ is closed in $\cA_{\ad}$.
	\end{prop}

\begin{proof}
		Recall that we have endowed $\la x \ra$ with graded lexicographic order. If $\ad\ge |v|$ and  $v>w$, then
		\[
		\langle  A^{w} \Omega, e_{v} \rangle \ =\  0
		\]
		by Lemma~\ref{lem:faithful}. Hence, $\ME_\ad$ is upper triangular.
		Moreover, each diagonal entry is $1$ by Lemma~\ref{lem:faithful}. Thus, $\ME_{\ad}$ is invertible.
	\end{proof}

	\subsection{Extraction formula for coefficients}\label{ssec:coeff}
	Let $q = \sum Q_{w} w \in \cA_{\ad}.$ For $v\in \la x\ra_{\ad}$ 	define the linear functional
	$\Omega_{v} : \cB(\Ftgd) \to \mathbb{C}$ 	by
	\[
	\Omega_{{v}}(T) = \langle  T \Omega, e_{v} \rangle.
	\]
	The operator coefficients \(Q_v\) are obtained from $q(A)$  by solving the linear system
	\begin{align*}
		Z_{v}(q) \ :=&\ (\mathrm{id}_{\cB(\cH)} \otimes \Omega_{v}) q(A) \ =\   \sum\limits_{w} Q_{w} \otimes \Omega_{v} (A^{w}) \\
		\ =&\   \sum\limits_{w} \langle  A^{w} \Omega , e_{v} \rangle \, Q_{w} \ =\   \sum\limits_{w} [\ME_{\ad}]_{v,w} Q_{w},
	\end{align*}
	where $[\ME_{\ad}]_{v,w}$ is the $(v,w)$ entry of the matrix $\ME_{\ad}.$ In short, \index{$[\ME_d]$}
	\begin{equation}\label{eq:coeff2}
		Z(q) \ =\  \ME_{\ad} Q,
	\end{equation}
	where $Z(q)$ and $Q$ are column vectors with $Z_{v}(q)$ and $Q_{v}$  as the $v^{\rm th}$ entry of $Z$ and $Q,$ respectively.
	Since,  by Lemma~\ref{lem:invertible M}, $\ME_d$ is invertible,
	\begin{equation} \label{eq:coeff}
		Q \ =\  \ME_{\ad}^{-1} Z(q).
	\end{equation}
	{We refer to $\ME$ as the \df{extraction matrix}, and equation~\eqref{eq:coeff} as the \df{extraction formula} for the coefficients of $q.$ Note
		that this formula depends only upon $q(A);$ that is, the coefficients of $q$ are determined uniquely
		by $q(A).$}

	It follows from equation~\eqref{eq:coeff}  that  there exists a positive constant $\lambda_{\ad}$ (independent of $q$) such that
	\begin{equation} \label{eq:Coeff bound}
		\|Q_{w}\| \ \leq\  \lambda_{\ad} \, \|q(A)\| \quad \text{ for all $w \in \la x\ra_{\ad}$.}
	\end{equation}

	\begin{prop} \label{prop:bounded}
		If  $p \in \cC_{d},$ then the set
		\[
		\Gamma_p \ =\  \{\, \SG \in \cB(\cH)^{N(d)\times N(d)} \ : \quad \SG \succeq 0, \quad   V_d^* \SG V_d = p \,\}
		\]
		is norm bounded $($with respect to the operator norm on $\cB(\cH^{N(d)})$$)$. More precisely,
		there exists a constant $\mu_{d}$ $($depending only on $d$ and $\vg$ and not on $p$$)$ such that,
		for all $\SG\in\Gamma_p,$
		\[
		\|\SG\| \ \leq\  \mu_{d} \; \|p(A)\| ,
		\]
		where the tuple $A=A^{(\ad)}$ is defined in \eqref{eq:symcreate}
		for any $\ad\geq 2d$.
	\end{prop}

	\begin{proof}
		Fix $p \in \cC_{d}$ and $\SG \in \Gamma_p.$ Thus $p=V_d^* \SG V_d.$ By
		Proposition~\ref{prop:closed under addition}, there exists $Q_j: \cH\to \oplus_{w\in\la x\ra_d} \cH$ such that
		\begin{equation}
			\label{e:bound:1}
			p \ =\  \sum\limits_{j=1}^{N(d)} q_{j}^{*}q_{j} \ =\  V_d^* \left [ \sum\limits_{j=1}^{N(d)} Q_{j} Q_{j}^* \right ] V_d,
		\end{equation}
		where
		\[
		q_j \ =\   Q_j^* V_d = \sum_{w\in\la x\ra_d} Q_{j,w} w.
		\]
		By equation~\eqref{eq:Coeff bound}, for $v\in\la x \ra_d,$
		\[
		\| Q_{j,v}\| \ \le\  \lambda_d\; \|q_j(A)\|.
		\]
		From equation~\eqref{e:bound:1},
		\[
		\|q_j(A)\|^2 \ =\  \|q_j(A)^* q_j(A)\| \ \le\  \|p(A)\|.
		\]
		Thus, again using equation~\eqref{e:bound:1},
		\[
		\sum_{u,v\in \la x \ra_d} \|\SG_{u,v}\| \ \le\   \sum_{u,v\in \la x \ra_d} \, \sum_{j=1}^{N(d)} \|Q_{j,u}Q_{j,v}^*\|
		\ \le\  N(d)^3 \lambda_d^{2} \; \|p(A)\|.
		\]
		It follows that
		$\|\SG\|\le \mu_d \, \|p(A)\|$ for $\mu_d = N(d)^3 \lambda_d^{2}.$
	\end{proof}

\begin{equation}\label{eq:poly}
		p \ =\  \sum_{w\in\la x\ra}^{\rm finite} P_{w} w,
	\end{equation}

\begin{equation}
		\label{d:evalPunitary}
		\sum_{u\in\freeg}^{\rm finite} P_u u \ \mapsto\  \sum_u P_u^* u^{-1}.
	\end{equation}
	If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
	\[
	p(X)^* \ =\   \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*}   \ =\  p^*(X)
	\]
	for all $p\in\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\ccA$ and we let,
	for positive integers $d,$ \index{$\ccA$} \index{$\ccA_d$}
	\[
	\ccA_d \ =\ \left\{
	\sum_{\substack{u\in\freeg\\ |u|\le d}} P_u u \, :\, P_u \in \cB(\cH)
	\right\}.
	\]
	The number of words in $\freeg$ of length $\le d$, $(\freeg)_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
	\[
	N_{\rm red}(d)
	\ =\
	1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
	\ =\  \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
	\]

	\subsection{Main results}\label{ssec:mainresults}
	We are now ready to state our main results. The first is an operator-valued version of the classical 	sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
	is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
	results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

	\begin{thm}\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\cB(\cK)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}

\label{ssec:notation}
	Fix a positive integer $\vg$. Let
	\df{$\la x \ra$}  denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
	Its multiplicative identity

Positivity and factorization lie at the heart of real algebraic geometry and operator theory. In the commutative setting, positivity certificates via sums of squares (sos) trace back to
	Hilbert's $17^{\rm th}$ problem in 1900; for classical results and modern treatments see \cite{BCR98,Mar08,Sc24}.\looseness=-1

	In the 21st century, motivated by developments in linear systems theory \cite{SIG98,dOHMP09}, quantum physics \cite{brunner}, and free probability \cite{MS17}, the free (noncommutative) counterpart has evolved into a broad program within noncommutative function theory \cite{KVV14,MS11,AM15,BMV16,PTD22}. This framework encompasses noncommutative factorizations and noncommutative Positivstellens\"atze. Early landmarks include Helton’s theorem that (scalar) positive noncommutative polynomials are sums of squares \cite{Hel02} and McCullough’s factorization theory for noncommutative polynomials \cite{McC01}; see also \cite{HM04,HMP04,Po95,JM12,JMS21} and the references therein for further developments.

	This paper establishes operator-valued analogs of these factorization theorems: every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$, with an analogous result for operator-valued noncommutative trigonometric polynomials (elements of the free group algebra).

	Beyond the noncommutative positivity literature, our results resonate with classical and modern operator factorization themes, including canonical/state-space factorizations of Bart--Gohberg--Kaashoek and collaborators \cite{BGK79,BGKR10}, and the operator Fejér--Riesz and multivariable outer factorization lines \cite{DR10,DW05,GW05}. While our focus is the free (noncommutative) polynomial and free group contexts, the methods developed, such as the WOT-closure mechanism via Fock-space evaluations and the finite-rank Hankel realization, are of independent interest and may be useful in adjacent problems within free analysis and operator theory.

	\subsection*{Guide to the introduction.}
	Notation is introduced in Subsection~\ref{ssec:notation}. The main results are stated and their proofs outlined in Subsection~\ref{ssec:mainresults}, while Subsection~\ref{ssec:readersguide} provides a roadmap for the remainder of the paper.

	\subsection{Notation}\label{ssec:notation}
	Fix a positive integer $\vg$. Let
	\df{$\langle x \rangle$}  denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
	Its multiplicative identity is the empty word
	\df{$\varnothing$}.
	We endow $\langle x\rangle$ with the \df{graded lexicographic order}.
	The length of a word $w \in \langle x \rangle$ is denoted by $|w|.$ The
	set of all elements (words) of $\langle x \rangle$ of length (or degree) at most $d$
	is denoted \df{$\langle x \ra_{d}$}.
	The cardinality of $\langle x\ra_d$  is \index{$N(d)$}
	\[
	N(d) \ =\  \sum_{i=0}^d \vg^i \ =\  \frac{\vg^{d+1}-1}{\vg-1}.
	\]

	Let \df{$\mathcal{H}$} be a fixed complex Hilbert space.
	Let  \df{$\mathcal{B}(\mathcal{H})$} be the space of all bounded linear operators on $\mathcal{H}$, and
	let
	\df{$\mathcal{A} $} to be the free semigroup $\mathcal{B}(\mathcal{H})$-algebra on $x,$
	i.e., $\mathcal{A}=\mathcal{B}(\mathcal{H})\langle x\rangle$. An element $p$ of $\mathcal{A}$ takes the form,
	\begin{equation}\label{eq:poly}
		p \ =\  \sum_{w\in\langle x\rangle}^{\rm finite} P_{w} w,
	\end{equation}
	where
	$P_{w} \in \mathcal{B}(\mathcal{H}),$ and is referred to as an (operator-valued)
	\df{polynomial} in $x.$
	Let \df{$\cA_{d}$} denote the elements from $\mathcal{A}$ of degree at most $d$.

	Equip $\mathcal{A}$ with the involution \df{$^*$}: on letters, $x_{j}^{*} = x_{j},$ on a word $w = x_{i_1} \cdots x_{i_n} \in \langle x \rangle,$
	\[
	w^{*} \ =\  x_{i_n} \cdots x_{i_1};
	\]
	and, on a polynomial $p$ as in \eqref{eq:poly}, \index{$p^*$}
	\[
	p^{*} \ =\  \sum P_{w}^{*} w^{*},
	\]
	where $P_{w}^{*}$ is the adjoint of the operator $P_{w}$ in $\mathcal{B}(\mathcal{H}).$

	Let $\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \dots, X_{\vg})$ be a tuple of operators from $\mathcal{B}(\mathcal{K}).$  The
	\df{evaluation} of $p$ at $X$ is defined as
	\[
	p(X) \ =\  \sum P_{w} \otimes X^{w},
	\]
	where $X^{w} = X_{i_1} \dots X_{i_n}$ for $w = x_{i_1} \dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.

	\subsubsection{Trigonometric polynomials}

	We will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\mathbb F_{\vg}$ on the $\vg$  letters  $x_i$, $i=1,\ldots,\vg$.
	Elements of $\mathbb F_{\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.

	Let $\mathscr{A}$ be the algebra $\mathcal{B}(\mathcal{H})[\mathbb F_{\vg}] = \mathcal{B}(\mathcal{H})\otimes\mathbb C[\mathbb F_{\vg}]$. Its elements are called \df{trigonometric polynomials}, and $\mathscr{A}$
	is
	endowed with the involution
	\begin{equation}
		\label{d:evalPunitary}
		\sum_{u\in\mathbb F_{\vg}}^{\rm finite} P_u u \ \mapsto\  \sum_u P_u^* u^{-1}.
	\end{equation}
	If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
	\[
	p(X)^* \ =\   \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*}   \ =\  p^*(X)
	\]
	for all $p\in\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\mathscr{A}$ and we let,
	for positive integers $d,$ \index{$\mathscr{A}$} \index{$\ccA_d$}
	\[
	\ccA_d \ =\ \left\{
	\sum_{\substack{u\in\mathbb F_{\vg}\\ |u|\le d}} P_u u \, :\, P_u \in \mathcal{B}(\mathcal{H})
	\right\}.
	\]
	The number of words in $\mathbb F_{\vg}$ of length $\le d$, $(\mathbb F_{\vg})_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
	\[
	N_{\rm red}(d)
	\ =\
	1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
	\ =\  \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
	\]

	\subsection{Main results}\label{ssec:mainresults}
	We are now ready to state our main results. The first is an operator-valued version of the classical 	sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
	is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
	results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\subsection{Notation}\label{ssec:notation}
 Fix a positive integer $\vg$. Let
 \df{$\langle x \rangle$} denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
 Its multiplicative identity is the empty word
 \df{$\varnothing$}.
 We endow $\langle x\rangle$ with the \df{graded lexicographic order}.
 The length of a word $w \in \langle x \rangle$ is denoted by $|w|.$ The
 set of all elements (words) of $\langle x \rangle$ of length (or degree) at most $d$
 is denoted \df{$\langle x \ra_{d}$}.
 The cardinality of $\langle x\ra_d$ is \index{$N(d)$}
 \[
 N(d) \ =\ \sum_{i=0}^d \vg^i \ =\ \frac{\vg^{d+1}-1}{\vg-1}.
 \]

Let \df{$\mathcal{H}$} be a fixed complex Hilbert space.
 Let \df{$\mathcal{B}(\mathcal{H})$} be the space of all bounded linear operators on $\mathcal{H}$, and
 let
 \df{$\mathcal{A} $} to be the free semigroup $\mathcal{B}(\mathcal{H})$-algebra on $x,$
 i.e., $\mathcal{A}=\mathcal{B}(\mathcal{H})\langle x\rangle$. An element $p$ of $\mathcal{A}$ takes the form,
 \begin{equation}\label{eq:poly}
 p \ =\ \sum_{w\in\langle x\rangle}^{\rm finite} P_{w} w,
 \end{equation}
 where
 $P_{w} \in \mathcal{B}(\mathcal{H}),$ and is referred to as an (operator-valued)
 \df{polynomial} in $x.$
 Let \df{$\cA_{d}$} denote the elements from $\mathcal{A}$ of degree at most $d$.

Let $\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \dots, X_{\vg})$ be a tuple of operators from $\mathcal{B}(\mathcal{K}).$ The
 \df{evaluation} of $p$ at $X$ is defined as
 \[
 p(X) \ =\ \sum P_{w} \otimes X^{w},
 \]
 where $X^{w} = X_{i_1} \dots X_{i_n}$ for $w = x_{i_1} \dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.

We will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\mathbb F_{\vg}$ on the $\vg$ letters $x_i$, $i=1,\ldots,\vg$.
 Elements of $\mathbb F_{\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.

Let $\mathscr{A}$ be the algebra $\mathcal{B}(\mathcal{H})[\mathbb F_{\vg}] = \mathcal{B}(\mathcal{H})\otimes\mathbb C[\mathbb F_{\vg}]$. Its elements are called \df{trigonometric polynomials}, and $\mathscr{A}$
 is
 endowed with the involution
 \begin{equation}
 \label{d:evalPunitary}
 \sum_{u\in\mathbb F_{\vg}}^{\rm finite} P_u u \ \mapsto\ \sum_u P_u^* u^{-1}.
 \end{equation}
 If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
 \[
 p(X)^* \ =\ \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*} \ =\ p^*(X)
 \]
 for all $p\in\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\mathscr{A}$ and we let,
 for positive integers $d,$ \index{$\mathscr{A}$} \index{$\ccA_d$}
 \[
 \ccA_d \ =\ \left\{
 \sum_{\substack{u\in\mathbb F_{\vg}\\ |u|\le d}} P_u u \, :\, P_u \in \mathcal{B}(\mathcal{H})
 \right\}.
 \]
 The number of words in $\mathbb F_{\vg}$ of length $\le d$, $(\mathbb F_{\vg})_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
 \[
 N_{\rm red}(d)
 \ =\
 1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
 \ =\ \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
 \]

\subsection{Main results}\label{ssec:mainresults}
 We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
 is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
 results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\begin{thm}\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\cB(\cK)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
		\end{enumerate}

		\noindent If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
			\item \label{i:sos:iv}
			There exists $r\in\cA_d$ s.t.
			\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}

		\end{enumerate}
	\end{thm}

\begin{thm}\label{thm:usos}
		For $f\in\ccA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:usos:i}
			For any Hilbert space $\cK$ and any tuple of unitary operators  $U = (U_{1}, \dots, U_{\vg})\in\cB(\cK)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:ii}
			For any $n\in\N$ and any tuple of unitary matrices
			$U = (U_{1}, \dots, U_{\vg})\in M_n(\C)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:iii}
			There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
			\begin{equation*}\label{eq:usosthm}
				f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
			\end{equation*}
		\end{enumerate}
		If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item[\rm (iv)] \label{i:usos:iv}
			There exists $r\in\ccA_d$ s.t.
			\begin{equation*}
				f\ =\ r^*r.
			\end{equation*}
		\end{enumerate}
	\end{thm}

Let $\ccA$ be the algebra $\cB(\cH)[\freeg] = \cB(\cH)\otimes\C[\freeg]$. Its elements are called \df{trigonometric polynomials}, and $\ccA$
 is
 endowed with the involution
 \begin{equation}
 \label{d:evalPunitary}
 \sum_{u\in\freeg}^{\rm finite} P_u u \ \mapsto\ \sum_u P_u^* u^{-1}.
 \end{equation}
 If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
 \[
 p(X)^* \ =\ \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*} \ =\ p^*(X)
 \]
 for all $p\in\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\ccA$ and we let,
 for positive integers $d,$ \index{$\ccA$} \index{$\ccA_d$}
 \[
 \ccA_d \ =\ \left\{
 \sum_{\substack{u\in\freeg\\ |u|\le d}} P_u u \, :\, P_u \in \cB(\cH)
 \right\}.
 \]
 The number of words in $\freeg$ of length $\le d$, $(\freeg)_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
 \[
 N_{\rm red}(d)
 \ =\
 1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
 \ =\ \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
 \]

\subsection{Main results}\label{ssec:mainresults}
 We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
 is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
 results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\noindent If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
 \begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
 \item \label{i:sos:iv}
 There exists $r\in\cA_d$ s.t.
 \begin{equation}\label{eq:1sosthm}
 f \ =\ r^*r.
 \end{equation}

\begin{remark} \rm
 Several remarks related to Theorem~\ref{thm:sos} are in order. \begin{enumerate}[\rm(a)]\itemsep=5pt
 \item Item~\ref{i:sos:iii} can also be phrased as a factorization result.
 Letting $r=\text{col} \begin{pmatrix} r_1 & \cdots & r_{N(d)}\end{pmatrix} \in
 \cA^{N(d)}=
 \cB(\cH,\cH^{N(d)})\la x\ra$, \eqref{eq:sosthm} simply states
 \[
 f\ =\ r^*r.
 \]
 We refer to \cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.
 \item
 That item~\ref{i:sos:iii} implies
 item~\ref{i:sos:i} implies item~\ref{i:sos:ii} is trivial. The main content of Theorem~\ref{thm:sos} is that item~\ref{i:sos:ii}
 implies item~\ref{i:sos:iii}. A routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\ref{rem:soscone}.
 \item
 Our proof yields no bound on the size $n$ of matrices needed in item~\ref{i:sos:ii}.
 \item
 From Theorem~\ref{thm:sos} one can easily
 deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification
 $z_j\mapsto \real{z_j}= \frac{z_j+z_j^*}2$ and hence
 $z_j^*\mapsto \imag{z_j}= \frac{z_j-z_j^*}{2i}$.
 \qed
 \end{enumerate}
 \end{remark} The following result is the unitary version of Theorem~\ref{thm:sos}.

\begin{thm}\label{thm:usos}
 For $f\in\ccA_{2d}$ the following are equivalent:
 \begin{enumerate}[\rm(i)]\itemsep=5pt
 \item \label{i:usos:i}
 For any Hilbert space $\cK$ and any tuple of unitary operators $U = (U_{1}, \dots, U_{\vg})\in\cB(\cK)^\vg$, $f(U)\succeq0$;
 \item \label{i:usos:ii}
 For any $n\in\N$ and any tuple of unitary matrices
 $U = (U_{1}, \dots, U_{\vg})\in M_n(\C)^\vg$, $f(U)\succeq0$;
 \item \label{i:usos:iii}
 There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
 \begin{equation*}\label{eq:usosthm}
 f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
 \end{equation*}
 \end{enumerate}
 If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
 \begin{enumerate}[\rm(i)]\itemsep=5pt
 \item[\rm (iv)] \label{i:usos:iv}
 There exists $r\in\ccA_d$ s.t.
 \begin{equation*}
 f\ =\ r^*r.
 \end{equation*}
 \end{enumerate}
 \end{thm}

Index \df{$\mathbb{\cH}^{N(d)}$} and \df{$\cA_{d}^{N(d)}$} (the algebraic direct sum of $\cA_{d}$ with itself $N(d)$ times) by $\la x \ra_{d}.$
 Let $V_{d} \in \cA_{d}^{N}$ denote the \df{Veronese column vector} whose $w\in \la x \ra_{d}$ entry is $w$
 (adopting the usual convention of viewing $w$ as the the $\cB(\cH)$-valued polynomial $I_\cH \, w$). For instance,
 if $\vg=2$ and $d=2$, then \index{$V_d$}
 \[
 V_2\ =\ \text{col} \begin{pmatrix}
 1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2
 \end{pmatrix}.
 \]
 Let \df{$\cC_d$} denote the \df{cone of sums of squares} of polynomials of degree at most $d,$
 \begin{equation}\label{eq:sosdef}
 \cC_{d} \ :=\ \left\{ \sum\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \quad r_{i}\in \cA_{d}, \quad i=1, \dots, N(d) \right\}
 \subseteq \cA_{2d}.
 \end{equation}
 Given $r\in \cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$
 is called the \df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular,
 \[
 r^* r \ =\ V_d^* RR^* V_d
 \]
 so that $r^*r$ has a representation as $V_d^* \SG V_d$ for a psd matrix $\SG.$

\begin{lemma}\label{lem:uweak2WOT}
 If $\varphi:\mathcal A_{2d}\to\mathbb C$ be an ultraweak continuous linear functional that
 separates the cone $\cC_{d}$ from a fixed polynomial $p$ in $\cA_{2d},$ that is,
 \[
 \varphi(r^{*}r) \geq 0 \quad \text{for all $r \in \cA_{d}$} \quad \text{and} \quad \varphi(p+p^{*}) <0,
 \]
 then there exists a WOT continuous linear functionl $\tilde{\varphi} : \cA_{2d} \to \mathbb{C}$ such that
 \[
 \tilde{\varphi}(r^{*}r) \geq 0 \quad \text{for all $r \in \cA_{d}$} \quad \text{and} \quad \tilde{\varphi}(p+p^{*}) <0.
 \]
 \end{lemma}
 \begin{proof}
 Since $\varphi$ is ultraweak continuous, there exist trace class operators $S_{w}$ $(w \in \la x\ra_{2d})$ in $\cB(\cH)$ such that
 \[
 \varphi(q) = \sum\limits_{w \in \la x\ra_{2d}} {\rm Tr}\, (S_{w} Q_{w}),
 \]
 where $q = \sum_{w \in \la x \ra_{2d}} Q_{w} w.$ For $r,r^\prime \in \cA_{d}$, \[
 \varphi(r^{*}r^\prime) \ =\ \sum\limits_{u,v\in \la x \ra_{d}} {\rm Tr}\, (S_{u^{*}v} R_{u}^{*}R^\prime_{v}),
 \]
 where $r = \sum_{u \in \la x \ra_{d}} R_{u} u$
 and $r^\prime = \sum_{v \in \la x \ra_{d}} R^\prime_{v} v.$ Denote by $S$ the $N(d) \times N(d)$ block operator matrix whose $(u,v)$ entry is $S_{v^{*}u}.$

\begin{prop}\label{prop:GNS}
 If $\varphi:\mathcal A_{2d+2}\to\mathbb C$ is a continuous linear functional such that
 \[
 \varphi(p^ *p)\ \ge\ 0 \]
 for all $p\in \cA_{d+1},$
 then there exist a finite-dimensional Hilbert space $\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\dots,Y_\vg)$ on $\cE$, and a vector $\gamma\in \cH\otimes \cE$ such that
 \[
 \varphi(q^ *p)\ =\ \big\langle p(Y)\gamma,\ q(Y)\gamma\big\rangle_{\cH\otimes \cE} \qquad\text{for all }p \in \cA_{d+1},\ q\in\mathcal A_{d}.
 \]
 Therefore, for all $p\in\mathcal A_{2d+1}$,
 \[
 \varphi(p)\ =\ \langle p(Y)\gamma,\ \gamma\rangle.
 \]
 \end{prop}

\begin{remark} \rm
		Several remarks related to Theorem~\ref{thm:sos} are in order. 		\begin{enumerate}[\rm(a)]\itemsep=5pt
			\item Item~\ref{i:sos:iii} can also be phrased as a factorization result.
			Letting $r=\text{col} \begin{pmatrix} r_1 & \cdots & r_{N(d)}\end{pmatrix} \in
			\mathcal{A}^{N(d)}=
			\mathcal{B}(\mathcal{H},\mathcal{H}^{N(d)})\langle x\rangle$, \eqref{eq:sosthm} simply states
			\[
			f\ =\ r^*r.
			\]
			We refer to \cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.
			\item
			That item~\ref{i:sos:iii} implies
			item~\ref{i:sos:i}  implies item~\ref{i:sos:ii}  is trivial. The main content of Theorem~\ref{thm:sos} is that item~\ref{i:sos:ii}
			implies item~\ref{i:sos:iii}. A routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm}  in the infinite-dimensional case, see Remark~\ref{rem:soscone}.
			\item
			Our proof yields no bound on the size $n$ of matrices needed in item~\ref{i:sos:ii}.
			\item
			From Theorem~\ref{thm:sos} one can easily
			deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification
			$z_j\mapsto \operatorname{real}{z_j}= \frac{z_j+z_j^*}2$ and hence
			$z_j^*\mapsto \operatorname{imag}{z_j}= \frac{z_j-z_j^*}{2i}$.
			\qed
		\end{enumerate}
	\end{remark} The following result is the unitary version of Theorem~\ref{thm:sos}.

	\begin{thm}\label{thm:usos}
		For $f\in\ccA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:usos:i}
			For any Hilbert space $\mathcal{K}$ and any tuple of unitary operators  $U = (U_{1}, \dots, U_{\vg})\in\mathcal{B}(\mathcal{K})^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:ii}
			For any $n\in\mathbb N$ and any tuple of unitary matrices
			$U = (U_{1}, \dots, U_{\vg})\in M_n(\mathbb C)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:iii}
			There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
			\begin{equation*}\label{eq:usosthm}
				f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
			\end{equation*}
		\end{enumerate}
		If $\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item[\rm (iv)] \label{i:usos:iv}
			There exists $r\in\ccA_d$ s.t.
			\begin{equation*}
				f\ =\ r^*r.
			\end{equation*}
		\end{enumerate}
	\end{thm}

	\begin{remark}[What's new?]\rm
		\label{rem:whatsnew}
		The passage to operator coefficients necessitates several novel
		results and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\ref{thm:sos} and Theorem~\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).
		Namely, the Hahn-Banach theorem produces a separating linear functional $\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction
		based on $\varphi$ ultimately produces a tuple $Y$.
		Here we
		roughly follow the outline of \cite{MP05}.

		A key construction is that of a tuple of self-adjoint operators $A$
		based upon the left regular representation on Fock space; see Section~\ref{sec:fock}.
		We then show that, up to a universal constant, for a sum of squares polynomial $p,$
		the norm
		of $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\ref{prop:bounded}.
		This uniform bound is the main input in Section~\ref{sec:top} for proving that the cone $\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.

		On the GNS side, we introduce a new argument that exploits the WOT to associate to a
		separating linear functional a finite-rank psd noncommutative representing Hankel matrix and,
		on its range, construct the desired tuple~$Y$; see Section~\ref{sec:gns}. For the unitary result,
		Theorem~\ref{thm:usos}, we additionally modify the construction of Section~\ref{sec:fock} to produce a canonical
		tuple of unitary operators from the left-regular representation and adapt the GNS procedure
		to obtain a unitary tuple together with a representing vector that realizes the separating
		functional as a vector state; see Subsection~\ref{ssec:uGNS}.\qed
	\end{remark}

	\subsection{Reader's guide}\label{ssec:readersguide}

	The paper is structured as follows. The convex cone of sums of squares (making an appearance in \eqref{eq:sosthm} of Theorem~\ref{thm:sos}) is introduced and characterized in the next Section~\ref{sec:sos}. In Section~\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\cF_\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\ref{sec:top} we
	introduce a suitable topology on $\cA_\lcal{d}$ and
	collect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\ref{sec:gns}; see Proposition~\ref{prop:GNS}.
	Then Theorem~\ref{thm:sos} is proved in Section~\ref{sec:proof} and
	Theorem~\ref{thm:usos} is proved in Section~\ref{sec:uproof}.

	\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\ref{prop:closedcone}.

The post-theorem discussion says the nontrivial direction in Theorem~\ref{thm:sos} is item~\ref{i:sos:ii} $\Rightarrow$ item~\ref{i:sos:iii} (while item~\ref{i:sos:iii} $\Rightarrow$ item~\ref{i:sos:i} $\Rightarrow$ item~\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\varphi$ to produce a tuple $Y$.

Key intermediate steps highlighted are:
\begin{itemize}
\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\ref{sec:fock}).
\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\|p(A)\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$ (Proposition~\ref{prop:bounded}).
\item Use this bound (Section~\ref{sec:top}) to show the sos cone $\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.
\item On the GNS side (Section~\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”
\end{itemize}
It also notes that in the infinite-dimensional case “a routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm}.”

The post-theorem discussion says the nontrivial direction in the main theorem is item~\ref{i:sos:ii} $\Rightarrow$ item~\ref{i:sos:iii} (while item~\ref{i:sos:iii} $\Rightarrow$ item~\ref{i:sos:i} $\Rightarrow$ item~\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\varphi$ to produce a tuple $Y$.

Key intermediate steps highlighted are:
\begin{itemize}
\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\ref{sec:fock}).
\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\|p(A)\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$. We first record the needed proposition.

\begin{prop} \label{prop:bounded}
		If  $p \in \cC_{d},$ then the set
		\[
		\Gamma_p \ =\  \{\, \SG \in \cB(\cH)^{N(d)\times N(d)} \ : \quad \SG \succeq 0, \quad   V_d^* \SG V_d = p \,\}
		\]
		is norm bounded $($with respect to the operator norm on $\cB(\cH^{N(d)})$$)$. More precisely,
		there exists a constant $\mu_{d}$ $($depending only on $d$ and $\vg$ and not on $p$$)$ such that,
		for all $\SG\in\Gamma_p,$
		\[
		\|\SG\| \ \leq\  \mu_{d} \; \|p(A)\| ,
		\]
		where the tuple $A=A^{(\ad)}$ is defined in \eqref{eq:symcreate}
		for any $\ad\geq 2d$.
	\end{prop}

\item Use this bound (Section~\ref{sec:top}) to show the sos cone $\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.
\item On the GNS side (Section~\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”
\end{itemize}
It also notes that in the infinite-dimensional case “a routine argument shows the equivalence between
\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
and
\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}
.”

Question: Fix a positive integer $\vg$ and a complex Hilbert space $\mathcal H$. Let $\mathcal A=\mathcal B(\mathcal H)\langle x_1,\dots,x_{\vg}\rangle$ be the algebra of operator-valued noncommutative polynomials in free variables $x_1,\dots,x_{\vg}$, and let $\mathcal A_d$ denote the polynomials of degree at most $d$. For a Hilbert space $\mathcal K$ and a self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in \mathcal B(\mathcal K)^{\vg}$, $f(Y)$ denotes the evaluation of $f$ at $Y$. Also set $N(d)=\sum_{i=0}^d \vg^i$. Given $f\in \mathcal A_{2d}$, which existence statement holds precisely under the positivity assumption that $f(Y)\succeq 0$ for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y\in\mathcal B(\mathcal K)^{\vg}$ (equivalently, for every $n\in\mathbb N$ and every self-adjoint tuple $Y\in M_n(\mathbb C)^{\vg}$)?

For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.

Choices:
B. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(2d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(2d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.
C. If there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$, then for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$. Moreover, if $\mathcal H$ is infinite-dimensional and $f=r^*r$ for some $r\in\mathcal A_d$, then the same positivity conclusion holds.
D. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.
E. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\le N(d)$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.

dropped converse/equivalence, keeping only sos-implies-positivity

self-adjointness hypothesis in evaluation


\label{thm:main}
    There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.

Consider a graph $G=(V,E)$ with an even number of vertices. A \emph{perfect matching} is a set of edges $M\subseteq E$ incident with each vertex exactly once. $G$ is \emph{matching covered} if every edge of $G$ belongs to some perfect matching.
The theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\'asz. An object of particular interest is the \emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \begin{equation*}
 \mathcal L(G): = \lat(\mathcal{M})= \{\sum_{M \in \mathcal{M}}k_M \mathbf{1}_M: k_M \in \mathbb{Z}\},
\end{equation*} where $\mathcal{M}$ denotes the set of all perfect matchings of $G$.
Lov\'asz \cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \emph{bricks} and \emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings.
A natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:

\section{Introduction}

The main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \emph{bipartite relaxation}:\begin{equation}\label{eq:pm-bipartite}
 P(G):=\left\{x\in \R^A_{+}: x(\delta(v))= 1 \forall v\in V \right\}.
\end{equation}
Matching covered graphs for which $PM(G)=P(G)$ are called \emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\zL(G)$ either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.

Since we are not interested in edges that do not belong to any perfect matching, it is convenient to delete from $G$ all edges that are not in any perfect matching, thus making the graph matching covered.
The number of bricks in a tight cut decomposition of a matching covered graph affects the dimension of its perfect matching polytope. In particular, the dimension of the perfect matching polytope is \begin{equation}\label{eq:pm-dimension}
 \dim(PM(G)) = |E'|-|V|+1-b,
\end{equation} where $b$ denotes the number of bricks in a tight cut decomposition of $G$ and $E'$ is the set of edges that belong to a perfect matching \cite{edmonds1982brick}, with $E=E'$ for matching covered $G$. Furthermore, the dimension of the perfect matching lattice and its linear hull is \begin{equation}\label{eq:pm-lat-dimension}
 \dim (\lat(G)) = |E'|-|V|+2-b.
\end{equation}
We will call \emph{near-bricks} all matching covered graphs $G$ with exactly one brick.
The following fact due to Lov\'asz connects the linear hull of perfect matchings with $\zL(G)$: \begin{theorem}[Theorem $6.3$ in \cite{lovaszPM}]\label{thm:lovasz}
 In a matching covered graph $G=(V,E)$, let $\mathcal{M}$ be the set of perfect matchings. Then any $x \in \lin(\mathcal{M})\cap \Z^E$ satisfies $2x\in \zL(G)$. Furthermore, if $G$ has no Petersen bricks then $\lin(\mathcal{M})\cap \Z^E=\zL(G)$.
\end{theorem}

\begin{theorem}\label{thm:algo-correct}
 The result of Algorithm \ref{algorithm:basis-bvn} applied to a matching covered BvN graph is a set $\zB$ of linearly independent indicator vectors of perfect matchings that forms an integral basis for $\zL(G)$.
\end{theorem}
\begin{proof}
 Because $G$ is BvN it implies that any face of $P(G)$ has integral vertices. Therefore, all $y(e)$ found as a corner solution to \ref{LP2_match} will indeed be incidence vectors of some perfect matchings of $G$ (in fact, of $G[E_t]$).
 It is clear that the final set $\zB$ will contain exactly $\dim(PM(G))$ perfect matchings with the property that each one uses an edge not used in the previous ones. This means that the elements of $\zB$ are linearly independent and moreover, create an integral basis for $\lin(\zB)$.
\end{proof}

If, given any brick $G$, the above process terminates in $|E|-|V|+1$ steps and all the output vectors are integral, then they correspond to a basis for $\zL(G)$.
Therefore, the only algorithm obstruction in the non-BvN case is \ref{LP2_match} having a fractional solution $x^*$. This means that $x^*$ is a fractional vertex of $P(G)$ itself, which gives a certificate that $G$ is non-BvN.
As $x^*$ has to be half-integral, its support consists of vertex-disjoint edges of weight $1$ and odd cycles of weight $1/2$. Picking any of the odd cycles, we can consider the odd cut $C'$ whose shore is exactly the vertices of this cycle. Inevitably, we have $x^*(C')=0$.
In \Cref{sec:sep-find}, we describe a combinatorial algorithm to find an odd cut $C$ such that the inequality $x(C)\geq 1$ is violated by $x^*$ with more specific properties:
\begin{theorem}\label{thm:find-robust}
 Let $G$ be a brick that is non-BvN. Let $x^*$ be a fractional vertex of $P(G)$. There is a polynomial time algorithm that finds a separating cut $C$ of $G$ such that $x^*(G)<1$ and both $C$-contractions of $G$ are near-bricks whose brick is not the Petersen brick.
\end{theorem}

As in Section \ref{subsec:non-bvn-detect}, consider a non-BvN brick $G$ for which the BvN algorithm fails. Theorem \ref{thm:find-robust} guarantees the existence of a separating cut $C$ both of whose contractions, call them $G_1$ and $G_2$, have exactly one brick that is not the Petersen brick.
Since neither $G_1$ nor $G_2$ contain the Petersen graph, we can construct lattice bases $\zB_1$ and $\zB_2$ for the corresponding graphs. Since both contractions do not contain Petersen bricks, $\lin(\zB_i)\cap \Z^{E_i} = \zL(G_i)$ and hence by \Cref{thm:basis-comb} the set $\zB:=\zB_1 \odot \zB_2$ will be an integral basis for its linear hull. However, all elements of $\zB$ satisfy $x^T \mathbf{1}_C=1$, so to make it a basis for $\zL(G)$ we need to add at least one perfect matching $M$ with $|M\cap C|>1$.
We use the following fact shown in \cite{camposlucchesi}.
\begin{theorem}\label{thm:char-three}
 In a non-BvN brick that is not Petersen, every separating cut has a perfect matching intersecting it three times.
\end{theorem}
Such a perfect matching $M^*$ can be found in polynomial time by considering all possible intersections $M^*\cap C$ of size three. Finally, we can use $M^*$ to increment the basis. We can now prove that such a set $\zB \cup \{\mathbf{1}_{M^*}\}$ will be a basis for $\zL(G)$.
\begin{theorem}
 Let $G=(V,E)$ be a matching covered graph and let $C$ be a non-tight separating cut such that both $C$-contractions are near-bricks whose brick is not the Petersen brick. Let $M^*$ be a perfect matching with $|M^*\cap C|=3$. Then $\zB \cup \{\mathbf{1}_{M^*}\}$ is an integral basis for $\zL(G)$ consisting of perfect matchings.
\end{theorem}
\begin{proof}
 By Theorem \ref{thm:fdi-charact}, such $C$ defines a facet of $PM(G)$. Hence, using part $(iii)$ of Theorem \ref{thm:basis-comb}, $|\zB|=\dim(\zL(G))-1$. Thus, adding $M^*$ to $\zB$ makes it a basis for the linear hull of $\zL(G)$, so it suffices to prove that it will be an integral basis.
 Indeed, consider any $y \in \zL(G)$. We also know it is in the linear hull of $\zB\cup\{\mathbf{1}_{M^*}\}$. Therefore, we can write it as follows \begin{equation}\label{eq:linear-comb}
 y = \sum_{M\in \mathcal{M}} \alpha_M \mathbf{1}_M = \sum_{z^i\in \zB} \beta_i z^i+\beta_* \mathbf{1}_{M^*},
 \end{equation} where all $\alpha_M$ are integers and the goal is to prove that $\beta_i$ are all integer.
 The first equality of (\ref{eq:linear-comb}) when multiplied by $\textbf{1}_{E}$ gives \begin{equation*}
 \frac{1}{n/2} \textbf{1}^T_{E} y = \frac{1}{n/2}\sum_{M\in \mathcal{M}} \alpha_M \textbf{1}^T_{E}\mathbf{1}_M=\sum_{M\in \mathcal{M}} \alpha_M \in \Z.
 \end{equation*} Similarly, multiplying it by $\textbf{1}_C$ gives \begin{equation*}
 \textbf{1}^T_{C} y = \sum_{M\in \mathcal{M}} \alpha_M \textbf{1}^T_{C}\mathbf{1}_M\equiv \sum_{M\in \mathcal{M}} \alpha_M \mod 2.
 \end{equation*} Applying the same idea to the last expression of equality (\ref{eq:linear-comb}) we obtain \begin{equation*}
 \frac{1}{n/2} \textbf{1}^T_{E} y =\sum_{\mathbf{z}^i \in \zB} \beta_i + \beta_*, \textbf{1}^T_{C} y = \sum_{\mathbf{z}^i \in \zB} \beta_i + 3\beta_*.
 \end{equation*} Therefore, $\beta_*=\frac{1}{2} (\textbf{1}^T_{C} y-\frac{1}{n/2} \textbf{1}^T_{E} y)$, which is integral by the above.
 Finally, consider $y-\beta_* \mathbf{1}_{M^*} \in \lin(\zB)$. Since we already showed that $\zB$ is an integral basis for its linear hull, all the remaining coefficients $\beta_i$ are also integer.
\end{proof}

The main contribution of this paper is a new approach to basis construction for the perfect matching lattice based on polyhedral theory. In addition to the algorithmic result, we identify a class of graphs where a lattice basis can be obtained by a simple inductive process. Moreover, we connect the combinatorial and polyhedral properties of the perfect matchings, bridging our approach and structural methods of de Carvalho, Lucchesi, and Murty. More broadly, our polyhedral techniques offer new tools that may prove valuable in tackling longstanding open problems in matching theory.
To begin with, the \emph{perfect matching polytope} $PM(G)$ is the convex hull of its perfect matching incidence vectors and is described by the following set of inequalities (Edmonds, \cite{edmonds1965maximum}):
\begin{equation}\label{eq:pm-general}
    PM(G)=\left\{x\in \mathbb{R}^A_{+}: \begin{array}{cc}
    x(\delta(v))= 1 &  \forall v\in V\\
     x(C)\geq 1 &  \forall C \textmd{ odd cut}
\end{array}\right\}
\end{equation}
Here, a \emph{cut} $C = \delta(U)$ for some $U\subseteq V$ is the set of edges with exactly one endpoint in $U$; the sets $U$ and its complement $V\backslash U$ are the \emph{shores} of $C$. A cut is \emph{odd} if both of its shores have an odd cardinality.

The main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \emph{bipartite relaxation}:\begin{equation}\label{eq:pm-bipartite}
    P(G):=\left\{x\in \mathbb{R}^A_{+}: x(\delta(v))= 1  \forall v\in V \right\}.
\end{equation}
Matching covered graphs for which $PM(G)=P(G)$ are called \emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\mathcal L(G)$ either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.

The paper is organized as follows. In Section \ref{sec:prelim} we discuss a way to decompose the graph into simple pieces and show how to reduce the basis construction for $\mathcal L(G)$ to these pieces. In Section \ref{sec:bvn} we describe a basis construction algorithm for \emph{Birkhoff von Neumann (BvN)} graphs based on polyhedral techniques and discuss what happens when applying this algorithm to a non-BvN graph. In Section \ref{sec:put-together} we show how to decompose non-BvN graphs into BvN ones using a specific type of odd cuts in a process similar to the tight cut decomposition. In Section \ref{sec:sep-find} we provide an algorithm to find an appropriate cut to be used in the above mentioned decomposition. Finally, in Appendix \ref{sec:classical} we draw connections between our polyhedral view of the basis and the classical approach based on \emph{ear decompositions}.

To prove Theorem~\ref{thm:main}, the paper’s approach is “based on polyhedral theory” via the perfect matching polytope $PM(G)$ (Edmonds’ description \eqref{eq:pm-general}) and its “bipartite relaxation” $P(G)$ in \eqref{eq:pm-bipartite}. The “main idea … is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its bipartite relaxation,” i.e., to reduce to Birkhoff--von Neumann (BvN) graphs where $PM(G)=P(G)$. The algorithm for a basis of $\mathcal L(G)$ then “either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$ induced by an odd cut constraint $x(C)\ge 1$”; in the latter case, it “use[s] this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.” The overall proof/argument is organized by: reducing basis construction to decomposed “simple pieces” (Section~\ref{sec:prelim}); giving a basis construction algorithm for BvN graphs and analyzing applying it to non-BvN graphs (Section~\ref{sec:bvn}); decomposing non-BvN graphs into BvN ones “using a specific type of odd cuts” (Section~\ref{sec:put-together}); and providing an algorithm to find the needed cut (Section~\ref{sec:sep-find}).

Question:
Let $G=(V,E)$ be a matching covered graph, meaning that $G$ has a perfect matching and every edge of $G$ belongs to some perfect matching. For a perfect matching $M$, let $\mathbf{1}_M\in\{0,1\}^E$ denote its incidence vector, and let the perfect matching lattice be
\[
\mathcal L(G)=\left\{\sum_{M} k_M\mathbf{1}_M : k_M\in\mathbb Z,\ M\text{ a perfect matching of }G\right\}.
\]
A lattice basis for $\mathcal L(G)$ means a set of vectors in $\mathcal L(G)$ that is integer-linearly independent and whose integer span is exactly $\mathcal L(G)$. Which existence statement holds for such graphs?

There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.

Choices:
B. There exists a polynomial-time algorithm that, given any graph $G$ with a perfect matching, outputs a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
C. For every matching covered graph $G$, there exists a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
D. There exists a single polynomial-time algorithm that, given any matching covered graph $G$, outputs a basis of the real linear span of $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
E. There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs all incidence vectors of perfect matchings of $G$, and these vectors form a lattice basis for $\mathcal L(G)$.


\label{thm:main}
For $s >1 $ and $\delta>0$ there exist $c_s>0$ and $\ep_0>0$ with the following. For any $0< \epsilon \leq \ep_0$ and any zero-mean initial data $\om_{in} \in H^s(\mathbb{T} \times \mathbb{R})$ such that
\begin{equation}\label{eq thm main 1}
\| \om_{in}\|_{H^{s} }  \leq \epsilon \quad \text{and} \quad \| y \om_{in}\|_{L^{2} }  \leq \epsilon ,
\end{equation}
define the lifespan
\begin{equation}\label{eq thm main 2}
T_\epsilon: = c_s \epsilon^{-\delta_s }  \quad \text{with}  \quad  \delta_s =
    \begin{cases}
  \frac{1}{4-s}  \quad\text{if $1<s<2 $  } &\\
\frac{1}{2+\delta}    \quad\text{ if $s=2 $  } &\\
\frac12 \quad\text{if $s > 2 $  }. &
\end{cases}
\end{equation}

 Then the (unique) solution to \eqref{eq:eu_eq} satisfies the following.

\begin{itemize}

	\item (Regularity):
	The re-normalized vorticity $W( t, X,Y): = \omega(t,X + Yt, Y)$ satisfies
	\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\epsilon]; H^s)} \leq 3 \epsilon .
	\end{eqnarray}

	\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
	\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \epsilon \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \epsilon \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\epsilon] $ }
	\end{equation}
	where $P_{\neq 0}f : = f -\int_{\mathbb{T} } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\end{itemize}

\label{sec:intro}

In this note, we consider the 2D Euler equation on $\mathbb{T} \times \mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\omega : \mathbb{T} \times \mathbb{R} \to \mathbb{R}$ satisfies
\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \omega + y\p_x \omega + u \cdot \nabla \omega = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \omega  &\\
	\omega|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
where    $ \nabla^\perp = (- \p_y , \p_x)$ and $\om_{in}$ is the initial data.

We are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.

For the Couette flow, the linearized dynamics are well understood: phase mixing transfers energy to small scales in the vorticity, leading to algebraic decay of the velocity.

 At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \cite{MR4740211,MR4628607}.

In the studies \cite{MR3415068,MR4076093,MR4740211,MR4628607}, inviscid damping follows from asymptotic stability, which requires Gevrey-2  regularity. This assumption appears to be sharp-- below Gevrey-2, instability can occur~\cite{MR4630602}.

 For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\cite{MR2796139} and traveling waves~\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.

\subsection{Main result}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

In this note, we consider the 2D Euler equation on $\mathbb{T} \times \mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\omega : \mathbb{T} \times \mathbb{R} \to \mathbb{R}$ satisfies
\begin{equation}\label{eq:eu_eq}
\begin{cases}
 \p_t \omega + y\p_x \omega + u \cdot \nabla \omega = 0 &\\
 u = \nabla^\perp \psi, \quad \Delta \psi = \omega &\\
 \omega|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
where $ \nabla^\perp = (- \p_y , \p_x)$ and $\om_{in}$ is the initial data.

We are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.

At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \cite{MR4740211,MR4628607}.

For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\cite{MR2796139} and traveling waves~\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.

\subsection{Main result}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
 \begin{equation}\label{eq:damping intro}
 \begin{aligned}
 \|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
 \| u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
 \end{aligned}
 \quad \text{for all $t \in [0, T_\ep] $ }
 \end{equation}
 where $P_{\neq 0}f : = f -\int_{\T } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\begin{remark}
\hfill
\begin{enumerate}
 \item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler. There is no contradiction with steady states\footnote{Note that the examples in \cite{MR2796139} are on the periodic channel $\T \times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\T \times [-1 ,1]$ when restricting away from the boundary. }~\cite{MR2796139} or traveling waves~\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \eqref{eq:damping intro} at $t=T_\ep$.

\begin{lemma}\label{lemma:Hs a priori}
Let $s>1$. The system \eqref{eq:eu_XY} satisfies the a priori estimates:
\begin{equation}
\frac{ d \| W(t) \|_{H^s} }{dt} \lesssim \| \nabla U \|_{L^\infty } \| W \|_{H^s} +
\begin{cases}
\| J^{ 2} U \|_{ L^2 } \| W \|_{H^s} & \quad \text{1<s<2}\\
\| J^{ 2+\delta} U \|_{ L^2 } \| W \|_{H^s} &\quad \text{ s=2} \\
\| J^{ s} U \|_{ L^2 } \| W \|_{H^s} &\quad \text{ s>2}.
\end{cases}
\end{equation}

If $\int_{\T \times \R} \om_{in} = 0 $ and $ y \om_{in} \in L^2 $, then the system \eqref{eq:eu_XY} admits the following a priori estimates:
\begin{equation}\label{eq prop:apriori_yw}
\frac{ d \| W (t) \|_{ \bar{H}^s } }{dt} \lesssim \langle t\rangle^{\beta_s} \|W \|_{\bar{H}^s}^2 ,
\end{equation}
where the exponent $ \beta_s =3-s $ if $1<s<2 $, $ \beta_s =1+\delta $ for arbitrary $\delta>0$ if $s=2 $, and $ \beta_s =1 $ if $s > 2 $.

Combining \eqref{eq aux apriori 1b}, \eqref{eq aux apriori 5} and \eqref{eq aux apriori 6} with Lemma \ref{lemma:Hs a priori}, we have:
\begin{equation*}
\frac{ d \| W(t) \|_{\bar H^s} }{dt}
\begin{cases}
\lesssim \langle t\rangle^{3- s} \| W(t) \|_{\bar H^s}^2 \quad & \text{if $1<s<2$}\\
\lesssim_\delta \langle t\rangle^{1+\delta} \| W(t) \|_{\bar H^s}^2 \quad & \text{if $ s =2$}.
\end{cases}
\end{equation*}

We conclude that the decay estimate \eqref{eq:damping intro} follows from Lemma \ref{lemma:damping}; for $u^x$,
\begin{align*}
\| P_{\neq 0 } u^x (t)\|_{L^2} \lesssim \| P_{\neq 0 } \om \|_{\dot H^{-1}}
 \lesssim \langle t\rangle^{-1} \|W \|_{L^\infty_t([0,T_\ep]; H^1)}
\end{align*}
while for $u^y$ (applying Lemma \ref{lemma:damping} for $\p_x \om $ and $\p_X W$),
\begin{align*}
\| u^y (t)\|_{L^2} = \| P_{\neq 0 } \Delta^{-1}_t ( \p_X W) \|_{L^2} \lesssim \langle t \rangle^{- \min \{ 2,s\}} \|W \|_{L^\infty_t([0,T_\ep];H^s) }
\end{align*}
for any $s>1$ since the extra $\p_X$ can be absorbed in $H^s$. We have thus proven Theorem \ref{thm:main}

By standard duality, it suffices to bound
\begin{equation}
 \| P_{ \neq 0 } \om \|_{\dot H^{-1} } = \sup_{
 |\varphi \|_{\dot H^1} \leq 1 ; P_{=0} \varphi = 0} \int \om \varphi \, dx dy.
\end{equation}
So we fix such a smooth test function $\varphi \in C^\infty_c(\T \times \R)$ with zero $x$-mean and consider
\begin{equation}\label{eq:om_int}
 \int \om \varphi \, dx dy =\int W (X, Y )\varphi( X+ t Y, Y) \,dX dY.
\end{equation}
Because $\varphi$ has zero x-mean and periodic in $x$, there exists a unique zero x-mean function $ \phi= \p_x^{-1} \varphi$.
Then we use the identity
\begin{equation}\label{eq:om_int 2}
 \varphi(\cdot) = \frac{1}{t}\p_Y \left( \phi(\cdot) \right) - \frac{1}{t}\p_y \phi(\cdot)
\end{equation}
where the argument $(\cdot) = (X+ t Y, Y )$ is omitted.
Inserting this into \eqref{eq:om_int} gives
\begin{align}
 \int \om \varphi \, dx dy & =-\frac{1}{t} \int \p_Y W \phi(\cdot) \,dX dY \nonumber -\frac{1}{t} \int W \p_y \phi(\cdot) \,dX dY \nonumber \\
 & \leq t^{-1} \| \p_Y W \|_{L^2 } \| \phi \|_{L^2} + t^{-1} \| W \|_{L^2 } \| \p_y \phi \|_{L^2 } . \label{eq:aux10}
\end{align}
Note that $\left| \widehat{\phi}(k,\xi) \right|=\left| \frac{\widehat{\varphi}(k,\xi)}{k} \right|$, we have $\| \phi \|_{H^1} \lesssim \| \varphi \|_{\dot H^1} $, which gives
\begin{align}
 \Big|\int \om \varphi \, dx dy \Big| & \lesssim \frac{1}{t} \| W \|_{L^2_X H^1_Y} \| \varphi \|_{\dot H^1 } \nonumber .
\end{align}

\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\ep] $ }
	\end{equation}

\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}

\begin{remark}
\hfill
\begin{enumerate}
    \item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler.  There is no contradiction with steady states\footnote{Note that the examples in \cite{MR2796139} are on the periodic channel $\mathbb{T} \times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\mathbb{T} \times [-1 ,1]$  when restricting away from the boundary.   }~\cite{MR2796139} or traveling waves~\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \eqref{eq:damping intro} at $t=T_\epsilon$.

    \item While for  $s> 2$ the decaying rates match with those of the linearized problem, it is not clear to us what would be the optimal life span $T_\epsilon =\epsilon^{-\delta_s}$.

    \item The estimates are most effective for initial data of the form $\omega_{in} = \epsilon \omega_0$ for some fixed profile $ \omega_0$. In the limit $\epsilon \to 0+$, one sees that initially $ u  \sim  \omega  \sim \epsilon $ and  $ u $ exhibits the algebraic decay over the time interval $[0,T_\epsilon]$.

    \item The damping estimate \eqref{eq:damping intro} follows from the regularity \eqref{eq:reg intro}. Whether nonlinear inviscid damping can persist globally without asymptotic stability remains open, see~\cite{MR4302767} for results in the linear case.

    \item The instability construction in \cite{MR4630602} also satisfies our assumptions and estimates, with an eventual breakdown of \eqref{eq:reg intro} occurring long after $t=T_\epsilon$. A definite counterexample to inviscid damping for low Gevrey regularity is also open.
\end{enumerate}

\end{remark}

\subsection{Discussion}
The literature on stability/inviscid damping is vast; see, for example, the works  \cite{MR3710645,MR3772399,MR3987441,MR4400903} on general shears/vortices as well as \cite{MR3974608,MR4121130,MR4412070,MR4848788,MR4951440,2510.16378}  on related viscous problems and boundary effects.

In contrast, the study of nonlinear inviscid damping of 2D Euler equations in the Sobolev regime remains largely open.   The present work demonstrates that the inviscid damping remains effective on a long, but finite, time interval, which is much shorter than the nonlinear resonance breakdown time found in recent instability constructions~\cite{MR4630602}.

Our proof is short and elementary. We work in simplified shear-back coordinates of Bedrossian-Masmoudi~\cite{MR3415068,MR4076093,MR4740211,MR4628607} and exploit the fact that the principle of ``\emph{decay costs regularity}'' can, in some sense, be reversed. In these coordinates, the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors.

 It is desired to find the optimal exponent $T_\epsilon\sim  \epsilon^{-\delta_s}$. However, whether the lifespan obtained here can be extended, with or without additional assumptions, is unclear to us. Another important open question is whether nonlinear inviscid damping persists without the regularity of re-normalized vorticity \eqref{eq:reg intro}. See \cite{MR4302767} for the study of a linear case.

Our proof is described as "short and elementary." To prove Theorem~\ref{thm:main}, the authors "work in simplified shear-back coordinates of Bedrossian-Masmoudi" and "exploit the fact that the principle of `\emph{decay costs regularity}' can, in some sense, be reversed." In these coordinates, they use that "the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors," which yields the damping estimates (indeed, they also remark that "The damping estimate \eqref{eq:damping intro} follows from the regularity \eqref{eq:reg intro}.")

Our proof is described as "short and elementary." To prove the main theorem, the authors "work in simplified shear-back coordinates of Bedrossian-Masmoudi" and "exploit the fact that the principle of `\emph{decay costs regularity}' can, in some sense, be reversed." In these coordinates, they use that "the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors," which yields the damping estimates
\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\ep] $ }
	\end{equation}
(indeed, they also remark that the damping estimate above follows from the regularity
\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\ep]; H^s)} \leq 3 \ep .
	\end{eqnarray}
).

\label{thm:main}
For $s >1 $ and $\delta>0$ there exist $c_s>0$ and $\ep_0>0$ with the following. For any $0< \epsilon \leq \ep_0$ and any zero-mean initial data $\om_{in} \in H^s(\mathbb{T} \times \mathbb{R})$ such that
\begin{equation}\label{eq thm main 1}
\| \om_{in}\|_{H^{s} }  \leq \epsilon \quad \text{and} \quad \| y \om_{in}\|_{L^{2} }  \leq \epsilon ,
\end{equation}
define the lifespan
\begin{equation}\label{eq thm main 2}
T_\epsilon: = c_s \epsilon^{-\delta_s }  \quad \text{with}  \quad  \delta_s =
    \begin{cases}
  \frac{1}{4-s}  \quad\text{if $1<s<2 $  } &\\
\frac{1}{2+\delta}    \quad\text{ if $s=2 $  } &\\
\frac12 \quad\text{if $s > 2 $  }. &
\end{cases}
\end{equation}

 Then the (unique) solution to
\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
satisfies the following.

\begin{itemize}

	\item (Regularity):
	The re-normalized vorticity $W( t, X,Y): = \omega(t,X + Yt, Y)$ satisfies
	\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\epsilon]; H^s)} \leq 3 \epsilon .
	\end{eqnarray}

	\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
	\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \epsilon \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \epsilon \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\epsilon] $ }
	\end{equation}
	where $P_{\neq 0}f : = f -\int_{\mathbb{T} } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\end{itemize}

Existential–Universal
Inequality or Bound

Question:
Let \(s>1\) and \(\delta>0\). Consider the 2D Euler vorticity equation on \(\mathbb T\times\mathbb R\):
\[
\partial_t\omega+y\partial_x\omega+u\cdot\nabla\omega=0,\qquad u=\nabla^\perp\psi,\qquad \Delta\psi=\omega,\qquad \omega|_{t=0}=\omega_{in},
\]
where \(\nabla^\perp=(-\partial_y,\partial_x)\). Define the renormalized vorticity by \(W(t,X,Y):=\omega(t,X+Yt,Y)\), the projection removing the \(x\)-mean by \(P_{\neq 0}f:=f-\int_{\mathbb T}f(x,y)\,dx\), and \(\langle t\rangle=(1+t^2)^{1/2}\). Which explicit uniform small-data statement is valid for this problem?

There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]

Choices:
B.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\tilde\delta_s},\qquad
\tilde\delta_s=
\begin{cases}
\dfrac{1}{3-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{1+\delta}, & \text{if } s=2,\\[1mm]
1, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]
C.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1}.
\]
D.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]
E.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^{s+1})}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s+1\}}.
\]

lifespan exponent derived from time-growth in shear coordinates

dropped the \(u^y\) damping conclusion while keeping the same hypotheses and lifespan

weighted moment assumption \(\|y\omega_{in}\|_{L^2}\le \epsilon\)

one-derivative smoothing used for velocity upgraded to an \(H^{s+1}\) bound for \(W\)