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Abhay Jindal

Publications and source records attributed to Abhay Jindal.

11 recordsLinked to original sources
arXiv2026

Spectra of $1/k$-Contractions via Characteristic Functions

A unitarily invariant complete Nevanlinna--Pick (CNP) kernel $k$ on the Euclidean ball gives rise to a natural class of operator tuples on Hilbert spaces, known as $1/k$-contractions. We establish a lower estimate for the Taylor joint spectrum of $1/k$-contractions with finite defect in terms of the associated characteristic function. Under the additional assumption that $k$ satisfies the Corona property, this lower estimate coincides with an upper estimate due to Clou\^atre--Timko [Adv. Math., 2023], yielding an exact characterization of the Taylor joint spectrum in terms of the characteristic function. As an application, we determine the Taylor joint spectrum of quotient modules of CNP spaces. A key ingredient in the proof is a Beurling--Lax--Halmos theorem for these spaces, established in terms of characteristic functions.

math.FA↗
arXiv2026

Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence

We study pure unital completely positive maps on the finite Toeplitz operator system $ T_{d}$ of $d \times d$ Toeplitz matrices. Our first main result gives an explicit characterization of pure UCP maps from $T_{d}$ to $M_n$ in terms of positive $n\times n$ matrix-valued trigonometric polynomials of degree at most $d-1$. This characterization provides a checkable criterion for deciding when a given UCP map is pure. As a first application, we show that every pure UCP map from $ T_{d}$ to $M_n$ admits a unique UCP extension to the generated $C^*$-algebra. As a second application, we prove that, for each fixed $n$, the space of pure UCP maps from $T_{d}$ to $M_n$, equipped with the matricial Connes distance, converges in the Gromov--Hausdorff sense to the space of normalized positive $n\times n$ matrix-valued Borel measures on the unit circle, equipped with the matricial Monge--Kantorovich distance.

math.OA↗
arXiv2026

Operator-Valued Positivstellens\"atze on Matrix Convex Sets and Free Products of Finite Abelian Groups

We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let $p$ be an operator-valued polynomial in $B(H)\otimes C $ of degree at most $2d+1$, where $H$ is separable and infinite-dimensional. Let $L(x)=I+\sum_{j=1}^{g} A_j x_j$ be a monic linear operator pencil, and let $D_L=\{X: L(X) \geq 0\}$ be the associated matrix convex set. We show that $p$ is positive on $D_L$ if and only if $p=r^*r+q^*\pi(L)q$, where $q$ and $r$ have degree at most $d$, and $\pi$ is a unital completely positive map on the operator system generated by the coefficients of $L$. The proof combines a Hahn--Banach separation argument with a tailored GNS construction. The main challenge is that the separation occurs in the product ultraweak topology, so boundedness of the resulting GNS operators is not automatic. We first handle bounded matrix convex sets, using closedness of the cone of weighted squares in the product ultraweak topology as the key technical input, and then pass to the general unbounded case by an approximation argument. Finally, we apply this convex Positivstellensatz to prove an operator-valued noncommutative Fejer--Riesz theorem on free products of finite abelian groups. The key additional ingredients are the universal $*$-algebra povm(n) associated with POVMs, a perfect Positivstellensatz for povm(n), and Boca's theorem on free products of completely positive maps. As a consequence, every positive operator-valued trigonometric polynomial on a free product of finite abelian groups admits a sum-of-squares factorization with explicit complexity bounds.

math.FA↗
arXiv2025

Positive operator-valued noncommutative polynomials are squares

We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann.~Math., 2002) and one of the authors (Linear Alg.~Appl., 2001). Specifically, we show that every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$. An analogous statement holds for operator-valued noncommutative trigonometric polynomials. Our approach follows the now standard sum-of-squares (sos) paradigm but requires new results and constructions tailored to operator coefficients. Assuming a positive $p$ is not sos, Hahn--Banach separation yields a linear functional that is positive on the sos cone and negative on $p$; a Gelfand--Naimark--Segal (GNS) construction then produces a representing tuple $Y$ leading to contradiction since $p$ was assumed positive on $Y$. The main technical input is a canonical tuple $A$ of self-adjoint operators and, in the unitary case, a canonical tuple $U$ of unitaries, both constructed from the left-regular representation on Fock space. We prove that, up to a universal constant, the norms $\|p(A)\|$ and $\|p(U)\|$ bound the operator norm of any positive semidefinite Gram matrix $G$ representing the sos polynomial $p$. This uniform control is the key input in showing that the cone of (sums of) squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then produces a separating functional that is continuous for the weak operator topology (WOT). This two-step passage between the ultraweak and WOT topologies constitutes our separation argument and yields the required WOT closedness of the sos cone. With this in hand, the GNS construction associates to such a separating linear functional a finite-rank positive semidefinite noncommutative Hankel matrix and, on its range, produces the desired tuple $Y$.

math.FA↗
arXiv2024

Operator Theory on the Pentablock

The pentablock, denoted as $\cP,$ is defined as follows: $$\cP= \left\{ (a_{21}, {\rm tr}(A), {\rm det}(A)) : A = [a_{ij}]_{2 \times 2} \text{ with } \|A\|<1 \right\}.$$ It originated from the work of Agler--Lykova--Young in connection with a particular case of the $μ$-synthesis problem. It is a non-convex, polynomially convex, $\mathbb{C}$-convex, star-like about the origin, and inhomogeneous domain. This paper deals with operator theory on the pentablock. We study pentablock unitaries and isometries, providing an algebraic characterization of pentablock isometries. En route, we provide the Wold-type decomposition for pentablock isometries, which consists of three parts: the unitary part, the pure part, and a new component. We define this novel component as the quasi-pentablock unitary and provide a functional model for it. Additionally, a model for a class of pure pentablock isometries has been found, along with some examples. Furthermore, a representation resembling the Beurling-Lax-Halmos paradigm has been presented for the invariant subspaces of pentablock pure isometries.

math.FA↗
arXiv2024

Kernels with complete Nevanlinna-Pick factors and the characteristic function

The Sz.-Nagy Foias characteristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna-Pick kernels. Indeed, a unitarily invariant kernel on the unit ball {\em admits} a characteristic function if and only if it is a complete Nevanlinna-Pick kernel. However, what has captured our curiosity are the recent advancements in constructing characteristic functions for kernels that do not have complete Nevanlinna-Pick property. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector-valued Drury-Arveson space (thus the Hardy space in case of the unit disc). We present a unified framework for deriving characteristic functions for kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not only encapsulates all previously documented cases but also achieves a remarkable level of generalization, thereby expanding the concept of the characteristic function substantially. We also provide an explanation for the prominence of the Drury-Arveson kernel in all previously established results by showing that the Drury-Arveson kernel was the natural choice inherently suitable for those situations.

math.FA↗
arXiv2024

Complete Nevanlinna-Pick Kernels and the Curvature Invariant

We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by $s$ and a commuting $d$-tuple of bounded operators $T = (T_{1}, \dots, T_{d})$ satisfying a natural contractivity condition with respect to $s$. We associate with $T$ its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of $\bfT$. The instrument which makes this possible is the characteristic function developed in \cite{BJ}. \medskip We present an asymptotic formula for the curvature invariant. In the special case when $\bfT$ is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fibre dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of $\bfT$ specifically when its characteristic function is a polynomial.

math.FA↗
arXiv2023

Complete Nevanlinna-Pick kernels, the Schwarz lemma and the Schur algorithm

We investigate the Schwarz lemma and the Schur algorithm for elements in the unit ball of the multiplier algebra of a reproducing kernel Hilbert space on the open unit ball whose kernel satisfies the complete Nevanlinna-Pick property. This paper also explores the Poincaré contractivity for elements in the unit ball of the multiplier algebra of a reproducing kernel Hilbert space whose kernel satisfies the property mentioned above.

math.FA↗
arXiv2023

A dilation theoretic approach to approximation by inner functions

Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of $H^\infty$. We first prove a generalization of a result of Carathéodory. This generalization has many applications. A uniform approximation result for matrix-valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix-valued functions for whom a naturally associated kernel has finitely many negative squares. This uses the Krein-Langer factorization. Approximation results for $J$-contractive meromorphic functions where $J$ induces an indefinite metric on $\mathbb C^N$ are proved using the Potapov-Ginzburg Theorem. Moreover, approximation results for holomorphic functions on the unit disc with values in certain other domains of interest are also proved.

math.CV↗
arXiv2023

Complete Nevanlinna-Pick kernels And The Characteristic Function

This note finds a new characterization of complete Nevanlinna-Pick kernels on the Euclidean unit ball. The classical theory of Sz.-Nagy and Foias about the characteristic function is extended in this note to a commuting tuple $\bfT$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from $H_k \otimes \cE$ to $H_k \otimes \cF$, {\em factoring} a certain positive operator, for suitable Hilbert spaces $\cE$ and $\cF$ depending on $\bfT$. There is a converse, which roughly says that if a kernel $k$ {\em admits} a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain.

math.FA↗