\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 \(g\) and a complex Hilbert space \(\mathcal H\). Let \(\mathcal A=\mathcal B(\mathcal H)\langle x_1,\dots,x_g\rangle\) be the algebra of noncommutative operator-valued polynomials in the free variables \(x_1,\dots,x_g\), and let \(\mathcal A_d\) denote the polynomials of degree at most \(d\). For \(f=\sum_w P_w w\in \mathcal A\) and a tuple \(Y=(Y_1,\dots,Y_g)\) of self-adjoint operators, define \(f(Y)=\sum_w P_w\otimes Y^w\). For \(f\in \mathcal A_{2d}\), which statement is equivalent to both of the following positivity conditions: (i) \(f(Y)\succeq 0\) for every Hilbert space \(\mathcal K\) and every self-adjoint tuple \(Y\in \mathcal B(\mathcal K)^g\); and (ii) \(f(Y)\succeq 0\) for every \(n\in\mathbb N\) and every self-adjoint tuple \(Y\in M_n(\mathbb C)^g\)?

There exist polynomials \(r_1,\dots,r_{N(d)}\in \mathcal A_d\), where \(N(d)=\sum_{i=0}^d g^i\), such that \(f=\sum_{i=1}^{N(d)} r_i^*r_i\). Moreover, if \(\mathcal H\) is infinite-dimensional, this is also equivalent to the existence of a single \(r\in \mathcal A_d\) such that \(f=r^*r\).

Choices:
B. There exists a constant \(\mu_d>0\), depending only on \(d\) and \(g\), such that for every positive semidefinite noncommutative Gram matrix \(\Sigma\) with \(f=V_d^*\Sigma V_d\), one has \(\|\Sigma\|\le \mu_d\,\|f(Y)\|\) for every self-adjoint tuple \(Y\). In particular, \(f\) is positive semidefinite on all self-adjoint tuples if and only if it admits such a Gram representation.
C. There exist finitely many polynomials \(r_1,\dots,r_m\in \mathcal A_d\) for some \(m\in\mathbb N\) such that \(f=\sum_{i=1}^m r_i^*r_i\).
D. There exists a finite-dimensional Hilbert space \(\mathcal E\), a self-adjoint tuple \(Y=(Y_1,\dots,Y_g)\in \mathcal B(\mathcal E)^g\), and a vector \(\gamma\in \mathcal H\otimes \mathcal E\) such that for every \(p\in \mathcal A_d\), one has \(\langle f(Y)\gamma,\gamma\rangle\ge 0\); equivalently, positivity on all matrix tuples is witnessed by a single finite-dimensional GNS model.
E. There exist polynomials \(r_1,\dots,r_{N(2d)}\in \mathcal A_{2d}\), where \(N(2d)=\sum_{i=0}^{2d} g^i\), such that \(f=\sum_{i=1}^{N(2d)} r_i^*r_i\). Moreover, if \(\mathcal H\) is infinite-dimensional, this is also equivalent to the existence of a single \(r\in \mathcal A_{2d}\) such that \(f=r^*r\).

uniform Gram-matrix bound misused as equivalence criterion

explicit bound N(d) on number of squares

global positivity replaced by existence of one witnessing GNS tuple/vector

degree and square-count bound shifted from d to 2d


[Structural invariance of GGD thresholds {[Theorem.~\ref{thm:main-threshold}]{}}]\label{thm:invariance}
		Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \(\mathcal{X}\), and let \(A\) be an ample line bundle on \(Y\) endowed with a smooth positively curved metric.
		Set \(L=\pi^*A\).
		Then there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\)$\colon$

		\begin{enumerate}[label=(\roman*)]
			\item all higher cohomology groups vanish,
			\[
			H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0,\quad i>0;
			\]
			\item consequently,
			\[
			h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\frac{1}{s}\!\int_Y\! \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)
			+O(m^{n-1}),
			\]
			where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\);
			\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kähler class \([\omega_A]\).
			Equivalently,
			\[
			\text{$Y$ is GGD-positive}
			\quad\Longleftrightarrow\quad
			\text{$\mathcal{X}$ is GGD-positive}.
			\]
		\end{enumerate}
		Thus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.

\label{sec:intro}

The Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\cite{GreenGriffiths1979} and developed extensively by Demailly~\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties.
 Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory.
 The minimal jet order \(k_0\) and weight slope \(\lambda_0\) for which
 \[
 H^0\!\big(X,E^{\mathrm{inv}}_{k,m}\otimes A^{-q}\big)\neq0,
 \qquad q\simeq \lambda_0 m,
 \]
 quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture.
 These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.

Orbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\u{a}un~\cite{CampanaPaun2016}, Borghesi–Tomassini~\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\cite{ToenVezzosi2008}.
 Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}.
 However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \(L^2\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds.
 This raises the structural question:
\begin{question}
Does the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?
\end{question}
 At first glance, the answer need not be obvious.
 The orbifold Riemann--Roch theorem of Satake~\cite{Satake1956} and Kawasaki~\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations.
 Thus, the asymptotic expansion of the Euler characteristic
 \[
 \chi(\mathcal{X},E_{k,m}\otimes \pi^*A^{-q})
 \]
 where $\pi\colon \mathcal{X}\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,
might, in principle, differ from its coarse counterpart on~$Y$.
 Furthermore, while the leading term of the Euler characteristic scales by \(1/s\), where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\) is the generic stabilizer order, the actual number of global sections \(h^0\) could still depend on higher cohomology groups \(H^i(\mathcal{X},E_{k,m}\otimes L^{-q})\).
 To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \(h^0\) and \(\chi\) coincide asymptotically.

\begin{theorem}
		\label{thm:main-threshold}
		Let $\pi\colon \X\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\omega_A$.
		Assume the curvature hypotheses \textup{(H1)}–\textup{(H4)} of \Cref{assump:curv}.
		Then$\colon $
		\begin{enumerate}[label=(\roman*), leftmargin=1.6em]
			\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\omega_A]$ and not on the presence of orbifold or stack structure.
			\item Equivalently,
			\[
			\text{$Y$ is GGD--positive at some jet order $k_0$}
			\quad\Longleftrightarrow\quad
			\text{$\X$ is GGD--positive at the same order.}
			\]
			\item The minimal jet order and asymptotic growth rate of $h^0(E^{\mathrm{GG}}_{k,m}\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.
		\end{enumerate}
	\end{theorem}

Orbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\u{a}un~\cite{CampanaPaun2016}, Borghesi–Tomassini~\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\cite{ToenVezzosi2008}.
 Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}.
 However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \(L^2\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds.
 This raises the structural question:
\begin{question}
Does the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?
\end{question}
 At first glance, the answer need not be obvious.
 The orbifold Riemann--Roch theorem of Satake~\cite{Satake1956} and Kawasaki~\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations.
 Thus, the asymptotic expansion of the Euler characteristic
 \[
 \chi(\X,E_{k,m}\otimes \pi^*A^{-q})
 \]
 where $\pi\colon \X\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,
might, in principle, differ from its coarse counterpart on~$Y$.
 Furthermore, while the leading term of the Euler characteristic scales by \(1/s\), where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\) is the generic stabilizer order, the actual number of global sections \(h^0\) could still depend on higher cohomology groups \(H^i(\X,E_{k,m}\otimes L^{-q})\).
 To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \(h^0\) and \(\chi\) coincide asymptotically.

The proof combines two analytic–cohomological mechanisms.
 First, the curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.
 Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as
 \[
 \chi(\X,E_{k,m}\otimes L^{-q})
 =\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
 +O(m^{n-1}),
 \]
 with all twisted-sector terms of order \(O(m^{n-1})\).
 Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide, yielding the stated invariance.

\begin{proposition}[Generic stabilizer and normalization of the untwisted integral]
 \label{prop:generic-stab-normalization}
 Let $\pi\colon \mathcal X\to Y$ be the coarse moduli map and let $n=\dim_{\C}\mathcal X$. Then$\colon $
 \begin{enumerate}[label=(\roman*)]
 \item (Existence and constancy) There exists a Zariski open dense sub-orbifold $\mathcal X^\circ\subset \mathcal X$ on which $|\Aut(x)|$ is constant. Its common value
 \[
 s \; \coloneq \; |\Stab_{\mathrm{gen}}|
 \]
 is the \emph{generic stabilizer order}. The function $x\mapsto |\Aut(x)|$ is upper semicontinuous, hence $|\Aut(x)|\ge s$ on $\mathcal X$.
 \item (Normalization) For any top-degree form $\alpha$ supported on the untwisted sector,
 \[
 \int_{\mathcal X}\alpha \;=\; \frac{1}{s}\int_Y \pi_*\alpha.
 \]
 Equivalently, on $\mathcal X^\circ$ the map $\pi$ is \'etale of degree $s$, and the identity-sector integral picks up the factor $1/s$.
 \end{enumerate}
 \end{proposition}

\begin{proposition}[Sectorwise asymptotics and slope control]
 \label{prop:slope-control}
 Let $\mathcal{X}$ be a connected compact complex orbifold of complex dimension $n$,
 and let $L=\pi^*A$ be the pullback of an ample line bundle on the coarse space $Y$.
 Then
 \[
 \chi(\mathcal{X},E\otimes L^{\otimes m})
 = \frac{1}{s}\int_Y \ch(E)\,e^{m\,c_1(A)}\,\Td(TY)
 + O(m^{n-1}),
 \qquad s=|\Stab_{\mathrm{gen}}|.
 \]
 Consequently, the sign of the asymptotic slope
 \[
 \mu(L) \coloneq \lim_{m\to\infty}\frac{\chi(\mathcal{X},E\otimes L^{\otimes m})}{m^n/n!} =\frac{1}{s}\int_Y \rk(E)\,c_1(A)^n
 \]
 is identical to that of the coarse manifold term.
 Twisted--sector corrections are of strictly lower order
 and therefore cannot alter the positivity or negativity of $\mu(L)$.
 \end{proposition}

\begin{lemma}[Untwisted control of cohomology dimensions]
 \label{lem:cohomology-control}
 Assume $(L,h)$ is a Hermitian line bundle with Nakano--positive curvature
 and that $L=\pi^*A$ descends from the coarse space.
 Then for all sufficiently large $m$,
 \[
 H^q(\mathcal{X},E\otimes L^{\otimes m})=0\quad\text{for all }q>0,
 \]
 and consequently
 \[
 h^0(\mathcal{X},E\otimes L^{\otimes m})
 =\chi(\mathcal{X},E\otimes L^{\otimes m})
 =\frac{1}{s}\int_Y \ch(E)\,e^{m\,c_1(A)}\,\Td(TY)
 +O(m^{n-1}).
 \]
 Thus the asymptotic growth of global sections is governed entirely by the untwisted (identity) sector.
 \end{lemma}

\begin{lemma}[Orbifold HRR asymptotics]
 \label{lem:eq-HRR}
 Let $\X$ be a compact complex orbifold with generic stabilizer of order $s$, and let $L=\pi^*A$ for an ample line bundle $A$ on the coarse space $Y$.
 Let $E^{\mathrm{inv}}_{k,m}$ denote the invariant jet (or DS) bundle.
 Then, as $m\to\infty$,
 \[
 \chi\!\bigl(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\bigr)
 =\frac{1}{s}\!\int_Y\!
 \ch(\underline{E}^{\mathrm{inv}}_{k,m})\,e^{-q\,c_1(A)}\,\Td(TY)
 +O(m^{n-1}),
 \]
 and all twisted-sector terms contribute only $O(m^{n-1})$ due to the fixed-locus dimension drop and the $m$–independence of the Kawasaki denominator.
 \end{lemma}

\begin{theorem}[Stack-theoretic GGD degeneracy]
 \label{thm:stack-ggd-final}
 Under Assumption~\ref{ass:equiv-setting3}, there exist integers $k\gg1$, $m\gg1$, and $q=q(k,m)>0$ such that$\colon $
 \begin{enumerate}[label=(\roman*), leftmargin=1.6em]
 \item $H^0(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q})\neq 0$;
 \item the common zero locus
 \[
 \mathcal{G}_k
 =\bigcap_{\substack{m\gg1\\0\le q\le q_0(k)}}
 Z\!\big(H^0(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q})\big)
 \]
 is a proper closed analytic substack $\mathcal{G}_k\subsetneq\X$;
 \item every nonconstant entire map $f\colon \C\to\X$ satisfies $f(\C)\subset\mathcal{G}_k$.
 \end{enumerate}
 \end{theorem}

\begin{theorem}
 \label{thm:main-threshold}
 Let $\pi\colon \X\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\omega_A$.
 Assume the curvature hypotheses \textup{(H1)}–\textup{(H4)} of \Cref{assump:curv}.
 Then$\colon $
 \begin{enumerate}[label=(\roman*), leftmargin=1.6em]
 \item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\omega_A]$ and not on the presence of orbifold or stack structure.
 \item Equivalently,
 \[
 \text{$Y$ is GGD--positive at some jet order $k_0$}
 \quad\Longleftrightarrow\quad
 \text{$\X$ is GGD--positive at the same order.}
 \]
 \item The minimal jet order and asymptotic growth rate of $h^0(E^{\mathrm{GG}}_{k,m}\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.
 \end{enumerate}
 \end{theorem}

The proof combines two analytic–cohomological mechanisms.
	First, the curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.
	Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as
	\[
	\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
	=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
	+O(m^{n-1}),
	\]
	with all twisted-sector terms of order \(O(m^{n-1})\).
	Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide, yielding the stated invariance.

	This conclusion situates the structural invariance of GGD thresholds at the intersection of analytic and stack-theoretic geometry.
	It refines Demailly’s curvature approach~\cite{Demailly1997,Demailly2011} and P\u{a}un’s vector-field method~\cite{Paun2008}, while connecting Satake–Kawasaki index theory~\cite{Kawasaki1979,Kawasaki1981} and modern stack Riemann--Roch theorems~\cite{Toen1999,Vistoli1989}.
	By aligning orbifold HRR asymptotics with curvature positivity, it confirms that orbifold structures preserve, rather than disturb, the hyperbolicity thresholds predicted by the Green--Griffiths--Lang conjecture.
	In particular, higher orbifold Betti numbers or stabilizers may influence lower-order corrections but have no effect on the leading asymptotic behavior that determines the GGD threshold.

	The remainder of the paper proceeds as follows.
	Section~\ref{sec:orbifold-hrr-ds} establishes the chartwise Riemann--Roch formula compatible with orbifold descent and applies it to jet bundles on $\X_k$.
	Section~\ref{sec:stack-ggd} develops the curvature–positivity package on the DS tower, derives \(L^2\)-vanishing and Bochner inequalities on orbifold charts, and proves the stack-theoretic GGD degeneracy theorem.
	Finally, Section~\ref{sec:ggd-structural-invariance} combines HRR asymptotics, Kodaira-type vanishing, and slope control to show that the GGD thresholds depend only on the coarse Kähler class and remain invariant under orbifold or stack structures, concluding with examples and applications.

	Throughout, compact analytic Deligne--Mumford stacks are identified with compact complex orbifolds after rigidification.
	Integration on $\mathcal{X}$ is normalized by the generic stabilizer order \(s=|\mathrm{Stab}_{\mathrm{gen}}|\), so that identity-sector integrals correspond to those on \(Y\) up to the factor \(1/s\).
	All Demailly--Semple and jet constructions are performed chartwise and descend by finite-group equivariance.

	\subsection*{Acknowledgements}
A substantial part of this work was carried out while the second author was visiting Texas State University to give a talk in the topology seminar.
The authors thank the host of the seminar Prof. Christine Lee, as well as the department chair and faculty members, for their warm hospitality and stimulating discussions.

To prove Theorem~\ref{thm:invariance}, the argument “combines two analytic–cohomological mechanisms.”

1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”

2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives
\[
\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
+O(m^{n-1}),
\]
“with all twisted-sector terms of order \(O(m^{n-1})\).”

3) **Vanishing to identify \(h^0\) with \(\chi\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).

To prove the main theorem, the argument “combines two analytic–cohomological mechanisms.”

1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”

2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives
\[
\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
+O(m^{n-1}),
\]
“with all twisted-sector terms of order \(O(m^{n-1})\).”

3) **Vanishing to identify \(h^0\) with \(\chi\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).

[Structural invariance of GGD thresholds {In establishing the main theorem,}{}]\label{thm:invariance}
		Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \(\mathcal{X}\), and let \(A\) be an ample line bundle on \(Y\) endowed with a smooth positively curved metric.
		Set \(L=\pi^*A\).
		Then there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\)$\colon$

		\begin{enumerate}[label=(\roman*)]
			\item all higher cohomology groups vanish,
			\[
			H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0,\quad i>0;
			\]
			\item consequently,
			\[
			h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\frac{1}{s}\!\int_Y\! \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)
			+O(m^{n-1}),
			\]
			where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\);
			\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kehler class \([\omega_A]\).
			Equivalently,
			\[
			\text{$Y$ is GGD-positive}
			\quad\Longleftrightarrow\quad
			\text{$\mathcal{X}$ is GGD-positive}.
			\]
		\end{enumerate}
		Thus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.

Existential–Universal
Biconditional or Equivalence

Question: Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold), let \(A\) be an ample line bundle on \(Y\) with a smooth positively curved metric, and set \(L=\pi^*A\). Say that a space is GGD-positive if some invariant jet differential bundle \(E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\) has a nonzero global section for suitable jet order \(k\) and integers \(m,q\). Which statement is equivalent to saying that the coarse space \(Y\) is GGD-positive?

The orbifold/stack \(\mathcal{X}\) is GGD-positive. Equivalently, the minimal jet order and the asymptotic slope at which invariant jet differentials exist depend only on the coarse K\"ahler class \([\omega_A]\), so orbifold compactification and rigidification do not alter or shift the GGD threshold.

Choices:
B. The orbifold/stack \(\mathcal{X}\) is GGD-positive after possibly replacing the jet order by some larger \(k'\ge k\) and rescaling the asymptotic slope by the generic stabilizer factor \(1/s\). Equivalently, orbifold compactification preserves existence of invariant jet differentials only up to this stabilizer-dependent shift of the GGD threshold.
C. The orbifold/stack \(\mathcal{X}\) is GGD-positive whenever the coarse space \(Y\) is GGD-positive.
D.
The orbifold/stack \(\mathcal{X}\) is GGD-positive if and only if the orbifold Euler characteristic satisfies
\[
\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)>0
\]
for all sufficiently large \(m\) and \(q\), since the twisted sectors contribute only the factor \(1/s\) and therefore automatically force nonzero global sections without any higher-cohomology vanishing hypothesis.
E.
The orbifold/stack \(\mathcal{X}\) is GGD-positive if and only if there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\),
\[
H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0\quad (i>0),
\]
and
\[
h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
=\frac{1}{s}\!\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)+O(m^{n-1}),
\]
so the GGD threshold is characterized exactly by this asymptotic formula.

dropped the converse equivalence and threshold invariance

need for higher-cohomology vanishing to identify h^0 with chi

equivalence between positivity and eventual vanishing/asymptotic formula for all large m,q