\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 $\mathcal L(G)$.


A classical result of Almgren \cite{A}, also known as Calabi-Almgren theorem, asserts that any minimal 2-sphere in $\mathbb{S}^3$ must be a totally geodesic equator; see also \cite{C}. Almgren's proof relies on the use of the Hopf differential, a holomorphic quadratic differential associated to any constant mean curvature immersion in a space form, which vanishes precisely when the surface is totally umbilical.

The problem of understanding to what extent Almgren's uniqueness theorem extends to more general classes of surfaces was recently studied by Gálvez, Mira and Tassi \cite{GMT}. More specifically, they proved that if $\Sigma \subset \mathbb{S}^3$ is an analytic {\em saddle} sphere in $\mathbb{S}^3$, then $\Sigma$ must be a totally geodesic equator. The {\em saddle} condition here means that the product $\kappa_1\kappa_2$ of the principal curvatures of $\Sigma$ is non-positive. They also constructed $\mathcal{C}^\infty$ saddle spheres in $\mathbb{S}^3$ different from equators, showing that the analyticity hypothesis is necessary to ensure uniqueness. Other examples of non-analytic saddle spheres were previously constructed by Martinez-Maure \cite{MM} and Panina \cite{P}.

In the context of surfaces with boundary, the natural analogue of Almgren's theorem is due to Nitsche \cite{N}, who proved in 1985 that any free boundary minimal disk in the Euclidean unit ball $\mathbb{B}^3$ must be equatorial; see Figure \ref{fig:FBdisk}. We recall that a compact orientable surface $\Sigma$ with boundary $\partial \Sigma$ immersed in $\mathbb{B}^3$ is said to be \emph{free boundary} in the unit ball if $\partial \Sigma \subset \partial \mathbb{B}^3$ and $\Sigma$ meets $\partial \mathbb{B}^3$ orthogonally along $\partial \Sigma$. This condition appears naturally when one considers critical points for the area functional among all surfaces $\Sigma$ with boundary in $\partial \mathbb{B}^3$.

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{FB_disk.png}
\caption{Equatorial disk in $\mathbb{B}^3$.}
\label{fig:FBdisk}
\end{figure}

The idea behind Nitsche's theorem can be viewed as the free boundary version of Almgren’s uniqueness result. Indeed, assume by contradiction that $\Sigma$ is a free boundary minimal disk which is not totally geodesic, and consider the pair of line fields $L_1,L_2$ associated to the set of principal directions of the surface. These line fields are defined on the set of non-umbilical points of $\Sigma$. Since the Hopf differential of $\Sigma$ is holomorphic, it follows that the set of umbilical points of $\Sigma$ is discrete, and the index of the line fields is negative at each such point. On the other hand, the free boundary condition implies that at each non-umbilical point of $\partial \Sigma$, one of the line fields $L_1,L_2$ is tangent to $\partial \Sigma$ and the other one is orthogonal. We may then reflect $L_1,L_2$ by symmetry along the boundary. This implies the existence of a pair of line fields on a 2-sphere with a discrete set of singularities of negative index. This contradicts Poincaré-Hopf theorem \cite{Hop}. Hence, any free boundary disk $\Sigma$ in $\mathbb{B}^3$ must be totally geodesic.

It is possible to extend Nitsche's uniqueness result to broader classes of surfaces. Indeed, one can show that any free boundary disk $\Sigma$ modelled by a geometric PDE of the type
\begin{equation}\label{eq:Weingarten}
    W(\kappa_1,\kappa_2) = 0, \; \; \; \text{ with }  \; \; \; W(0,0) = 0  \; \; \;  \text{ and }  \; \; \;  W_{\kappa_1}W_{\kappa_2} > 0
\end{equation}
for some differentiable symmetric function $W(x,y)$ (i.e., $W(x,y) = W(y,x)$) must be totally geodesic. The surfaces satisfying \eqref{eq:Weingarten} are also known as {\em special elliptic Weingarten surfaces of minimal type}; see for example \cite{FGM,FM,RSa,SaT}. We note that minimal surfaces are special Weingarten surfaces in the particular case $W(\kappa_1,\kappa_2) = \kappa_1 + \kappa_2$. We also remark that every surface satisfying \eqref{eq:Weingarten} must be saddle.

In view of these results, it is natural to ponder whether a free boundary analogue of the result by Gálvez-Mira-Tassi \cite{GMT} holds. Specifically, we ask the following:

\begin{enumerate}[(1)]
\item Is every free boundary analytic saddle disk in $\mathbb{B}^3$ equatorial?
\item Are there non-equatorial free boundary saddle disks in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$?
\end{enumerate}

An affirmative answer to both these problems would provide a sharp geometric extension of Nitsche's uniqueness theorem. In other words, we could view the class of free boundary analytic saddle disks as the {\em largest} family of surfaces in which Nitsche's result holds.

Regarding the uniqueness in the analytic case, the results obtained in \cite{GMT} could allow us to control the index of the line fields associated to the principal directions at the interior umbilical points of a free boundary saddle disk. However, the analysis of the index on the boundary is substantially more involved, and new techniques would be needed to deal with this situation.

On the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\Sigma$ in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$ presents several difficulties. The examples constructed in \cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\Sigma$; see \cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:

The problem of understanding to what extent Almgren's uniqueness theorem extends to more general classes of surfaces was recently studied by Gálvez, Mira and Tassi \cite{GMT}. More specifically, they proved that if $\Sigma \subset \mathbb{S}^3$ is an analytic {\em saddle} sphere in $\mathbb{S}^3$, then $\Sigma$ must be a totally geodesic equator. The {\em saddle} condition here means that the product $\kappa_1\kappa_2$ of the principal curvatures of $\Sigma$ is non-positive. They also constructed $\mathcal{C}^\infty$ saddle spheres in $\mathbb{S}^3$ different from equators, showing that the analyticity hypothesis is necessary to ensure uniqueness. Other examples of non-analytic saddle spheres were previously constructed by Martinez-Maure \cite{MM} and Panina \cite{P}.

In the context of surfaces with boundary, the natural analogue of Almgren's theorem is due to Nitsche \cite{N}, who proved in 1985 that any free boundary minimal disk in the Euclidean unit ball $\mathbb{B}^3$ must be equatorial; see Figure \ref{fig:FBdisk}. We recall that a compact orientable surface $\Sigma$ with boundary $\partial \Sigma$ immersed in $\mathbb{B}^3$ is said to be \emph{free boundary} in the unit ball if $\partial \Sigma \subset \partial \mathbb{B}^3$ and $\Sigma$ meets $\partial \mathbb{B}^3$ orthogonally along $\partial \Sigma$. This condition appears naturally when one considers critical points for the area functional among all surfaces $\Sigma$ with boundary in $\partial \mathbb{B}^3$.

The idea behind Nitsche's theorem can be viewed as the free boundary version of Almgren’s uniqueness result. Indeed, assume by contradiction that $\Sigma$ is a free boundary minimal disk which is not totally geodesic, and consider the pair of line fields $L_1,L_2$ associated to the set of principal directions of the surface. These line fields are defined on the set of non-umbilical points of $\Sigma$. Since the Hopf differential of $\Sigma$ is holomorphic, it follows that the set of umbilical points of $\Sigma$ is discrete, and the index of the line fields is negative at each such point. On the other hand, the free boundary condition implies that at each non-umbilical point of $\partial \Sigma$, one of the line fields $L_1,L_2$ is tangent to $\partial \Sigma$ and the other one is orthogonal. We may then reflect $L_1,L_2$ by symmetry along the boundary. This implies the existence of a pair of line fields on a 2-sphere with a discrete set of singularities of negative index. This contradicts Poincaré-Hopf theorem \cite{Hop}. Hence, any free boundary disk $\Sigma$ in $\mathbb{B}^3$ must be totally geodesic.

\begin{enumerate}[(1)]
\item Is every free boundary analytic saddle disk in $\mathbb{B}^3$ equatorial?
\item Are there non-equatorial free boundary saddle disks in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$?
\end{enumerate}

Regarding the uniqueness in the analytic case, the results obtained in \cite{GMT} could allow us to control the index of the line fields associated to the principal directions at the interior umbilical points of a free boundary saddle disk. However, the analysis of the index on the boundary is substantially more involved, and new techniques would be needed to deal with this situation.

On the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\Sigma$ in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$ presents several difficulties. The examples constructed in \cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\Sigma$; see \cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{FB_disk.png}
\caption{Equatorial disk in $\mathbb{B}^3$.}
\label{fig:FBdisk}
\end{figure}

\begin{enumerate}[(1)]
\item Is every free boundary analytic saddle disk in $\mathbb{B}^3$ equatorial?
\item Are there non-equatorial free boundary saddle disks in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$?
\end{enumerate}

On the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\Sigma$ in $\mathbb{B}^3$ of class $\mathcal{C}^\infty$ presents several difficulties. The examples constructed in \cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\Sigma$; see \cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{psigamma.png}
\caption{One of the free boundary saddle disks in Theorem \ref{thm:main}.}
\label{fig:psigamma}
\end{figure}

We emphasize that the constructed examples are never analytic. We do not know whether analytic free boundary saddle disks other than the equatorial ones exist. In light of the results by Gálvez, Mira and Tassi \cite{GMT}, it is natural to conjecture that uniqueness holds when we restrict to the family of analytic examples. We also note that all the disks obtained in Theorem \ref{thm:main} possess self-intersections; see Figure \ref{fig:psigamma}. It remains an open question whether embedded free boundary saddle disks in $\mathbb{B}^3$ must necessarily be equatorial, regardless of their regularity. Finally, since the examples constructed here contain planar pieces, we also ask whether every free boundary saddle disk in $\mathbb{B}^3$ must have one such portion.

\section{Construction of free boundary disks in \texorpdfstring{$\mathbb{B}^3$}{}}\label{sec:2}
We will introduce a method to construct a free boundary immersion $\psi(u,v)$ of a disk in the unit ball $\mathbb{B}^3 \subset \mathbb{R}^3$ in terms of a certain ribbon-shaped planar curve $\gamma(u)$, which we describe next. In what follows, let $\mathbb{D}^2 = \mathbb{B}^3 \cap \{x_3 = 0\}$ denote the horizontal unit disk in $\mathbb{B}^3$.

\begin{remark}
 There exist infinitely many $\mathcal{C}^\infty$ curves in $\mathbb{D}^2$ satisfying the properties listed above. However, none of these curves is analytic.
\end{remark}

\begin{proposition}\label{pro:psiFB}
 Let $\gamma(u)$ be a ribbon and $\psi_\gamma(u,v)$ be the corresponding immersion in Definition \ref{def:psigamma}. Then, $\Sigma_\gamma$ is a free boundary disk in $\mathbb{B}^3$.
\end{proposition}
\begin{proof}
 We may decompose $\Sigma_\gamma$ into three parts, namely
 \begin{equation}\label{eq:Sigma123}
 \begin{aligned}
 \Sigma_{\gamma}^{(1)} &:= \psi_\gamma\left(\mathcal{D} \cap \{u \leq -u_2\}\right), \\
 \Sigma_{\gamma}^{(2)} &:= \psi_\gamma\left(\mathcal{D} \cap \{-u_2 < u < u_2\}\right), \\
 \Sigma_{\gamma}^{(3)} &:= \psi_\gamma\left(\mathcal{D} \cap \{u \geq u_2\}\right),
 \end{aligned}
 \end{equation}
 respectively. It is clear that $\Sigma_{\gamma}^{(1)}$ and $\Sigma_{\gamma}^{(3)}$ are pieces of equatorial disks, so they meet $\partial \mathbb{B}^3$ orthogonally; see Definition \ref{def:ribbon}. We also note that the piece $\Sigma_{\gamma}^{(2)}$ intersects $\partial \mathbb{B}^3$ along the curves $u \mapsto \psi_{\gamma}(u,\pm v_{\gamma(u)})$, $u \in (-u_2,u_2)$. For every $u_0 \in (-u_2,u_2)$, the circle $v \mapsto \psi_\gamma(u_0,v) = c_{\gamma(u_0)}(v)$ meets $\partial \mathbb{B}^3$ orthogonally; see Definition \ref{def:cpv}. Thus, $\Sigma_{\gamma}^{(2)}$ meets $\partial \mathbb{B}^3$ orthogonally, as we wanted to show.
\end{proof}

\subsection*{Proof of Theorem \ref{thm:main}}
Let $\gamma(u)$ be any ribbon and consider the family of ribbons $\gamma_t$ in Definition \ref{def:gammat}. According to Propositions \ref{pro:curvaturapositiva} and \ref{pro:curvaturanegativa}, for sufficiently small values of $t$, the corresponding disks $\Sigma_{\gamma_t} \subset \mathbb{B}^3$ are saddle. Moreover, by Proposition \ref{pro:psiFB}, they are free boundary in $\mathbb{B}^3$. This completes the proof of Theorem \ref{thm:main}.

\begin{definition}\label{def:ribbon}
    Let $\gamma$ be a planar regular curve in $\mathbb{D}^2$. We say that $\gamma$ is a {\em ribbon} if, up to an isometry, it admits a parametrization $\gamma(u):[-u_3,u_3] \to \mathbb{D}^2$, where $u$ is the arc-length parameter, satisfying the following properties:
    \begin{enumerate}
    \item $\Psi(\gamma(u)) = \gamma(-u)$, where $\Psi$ denotes the symmetry with respect to the line $\{x = y \}$. In particular, $\gamma(0)$ belongs to this line.
    \item There exist $a \in (0,1)$ and $u_2 \geq 0$ such that, on the interval $[u_2,u_3]$, $\gamma(u)$ parametrizes the segment $\{0\}\times [-1,a] \times \{0\}$, with $\gamma(u_3) = (0,-1,0)$. In particular, $u_2 = u_3 - (1+a) > 0.$ Consequently, on the interval $[-u_3,-u_2]$, $\gamma(u)$ parametrizes the segment $[-1,a]\times \{0\} \times \{0\}$, and $\gamma(-u_3) = (-1,0,0)$.
    \item On the interval $(-u_2,u_2)$, $\gamma(u)$ is an embedded, strictly convex curve contained in the open planar quadrant $\left((0,1)\times(0,1)\times \{0\}\right) \cap \mathbb{D}^2$; see Figure \ref{fig:gamma}. The rotation index of the curve is thus $3/4$.
    \item The curvature $\kappa(u)$ of $\gamma(u)$ is strictly decreasing on the interval $(u_1,u_2)$ for some $u_1 \in (0,u_2)$. By symmetry, $\kappa(u)$ is increasing for $u \in (-u_2,-u_1)$.
    \end{enumerate}
\end{definition}

\begin{theorem}\label{thm:main}
There exist infinitely many $\mathcal{C}^\infty$ free boundary saddle disks in the unit ball $\mathbb{B}^3$ which are not equatorial.
\end{theorem}

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{psigamma.png}
\caption{One of the free boundary saddle disks in Theorem \ref{thm:main}.}
\label{fig:psigamma}
\end{figure}

We emphasize that the constructed examples are never analytic. We do not know whether analytic free boundary saddle disks other than the equatorial ones exist. In light of the results by Gálvez, Mira and Tassi \cite{GMT}, it is natural to conjecture that uniqueness holds when we restrict to the family of analytic examples. We also note that all the disks obtained in Theorem \ref{thm:main} possess self-intersections; see Figure \ref{fig:psigamma}. It remains an open question whether embedded free boundary saddle disks in $\mathbb{B}^3$ must necessarily be equatorial, regardless of their regularity. Finally, since the examples constructed here contain planar pieces, we also ask whether every free boundary saddle disk in $\mathbb{B}^3$ must have one such portion.

We briefly outline the structure of the paper. In Section \ref{sec:2}, we will construct a family of free boundary disks in $\mathbb{B}^3$. These surfaces are foliated by circles, and all of them contain a ribbon-shaped planar curve $\gamma$; see Definition \ref{def:ribbon}. Later on, in Section \ref{sec:saddledisks}, we will show that, under additional hypotheses on the ribbon curve $\gamma$, the associated free boundary disks are saddle.

\subsection*{Acknowledgements}
This research has been financially supported by Grants PID2020-118137GB-I00 and PID2024-160586NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by ESF+.

We briefly outline the structure of the paper. Next, we will construct a family of free boundary disks in $\mathbb{B}^3$. These surfaces are foliated by circles, and all of them contain a ribbon-shaped planar curve $\gamma$; specifically:

\begin{definition}\label{def:ribbon}
    Let $\gamma$ be a planar regular curve in $\mathbb{D}^2$. We say that $\gamma$ is a {\em ribbon} if, up to an isometry, it admits a parametrization $\gamma(u):[-u_3,u_3] \to \mathbb{D}^2$, where $u$ is the arc-length parameter, satisfying the following properties:
    \begin{enumerate}
    \item $\Psi(\gamma(u)) = \gamma(-u)$, where $\Psi$ denotes the symmetry with respect to the line $\{x = y \}$. In particular, $\gamma(0)$ belongs to this line.
    \item There exist $a \in (0,1)$ and $u_2 \geq 0$ such that, on the interval $[u_2,u_3]$, $\gamma(u)$ parametrizes the segment $\{0\}\times [-1,a] \times \{0\}$, with $\gamma(u_3) = (0,-1,0)$. In particular, $u_2 = u_3 - (1+a) > 0.$ Consequently, on the interval $[-u_3,-u_2]$, $\gamma(u)$ parametrizes the segment $[-1,a]\times \{0\} \times \{0\}$, and $\gamma(-u_3) = (-1,0,0)$.
    \item On the interval $(-u_2,u_2)$, $\gamma(u)$ is an embedded, strictly convex curve contained in the open planar quadrant $\left((0,1)\times(0,1)\times \{0\}\right) \cap \mathbb{D}^2$; see Figure \ref{fig:gamma}. The rotation index of the curve is thus $3/4$.
    \item The curvature $\kappa(u)$ of $\gamma(u)$ is strictly decreasing on the interval $(u_1,u_2)$ for some $u_1 \in (0,u_2)$. By symmetry, $\kappa(u)$ is increasing for $u \in (-u_2,-u_1)$.
    \end{enumerate}
\end{definition}

Later on, we will show that, under additional hypotheses on the ribbon curve $\gamma$, the associated free boundary disks are saddle.

\label{thm:main}
There exist infinitely many $\mathcal{C}^\infty$ free boundary saddle disks in the unit ball $\mathbb{B}^3$ which are not equatorial.,

Question: In the Euclidean unit ball $\mathbb{B}^3$, call a compact $\mathcal{C}^\infty$ immersed disk $\Sigma$ a free boundary disk if $\partial \Sigma \subset \partial \mathbb{B}^3$ and $\Sigma$ meets $\partial \mathbb{B}^3$ orthogonally along $\partial \Sigma$. Call it a saddle disk if the product $\kappa_1\kappa_2$ of its principal curvatures is everywhere non-positive. An equatorial disk means the intersection of $\mathbb{B}^3$ with a plane through the origin. Which existence statement holds?

There exist infinitely many $\mathcal{C}^\infty$ disks $\Sigma \subset \mathbb{B}^3$ that are free boundary and saddle, and that are not equatorial.

Choices:
B. There exists a unique $\mathcal{C}^\infty$ disk $\Sigma \subset \mathbb{B}^3$ that is free boundary and saddle, and every such disk is non-equatorial.
C. There exists at least one $\mathcal{C}^\infty$ disk $\Sigma \subset \mathbb{B}^3$ that is free boundary and saddle, and that is not equatorial.
D. There exist infinitely many real-analytic disks $\Sigma \subset \mathbb{B}^3$ that are free boundary and saddle, and that are not equatorial.
E. There exist infinitely many $\mathcal{C}^\infty$ disks $\Sigma \subset \mathbb{B}^3$ that are saddle and not equatorial, and whose boundaries lie in $\partial \mathbb{B}^3$.