arXiv subjects

Jeet Sampat

Publications and source records attributed to Jeet Sampat.

8 recordsLinked to original sources

arXiv2026

Cyclicity of stable matrix free polynomials over non-commutative operator unit balls

We consider the algebra of square matrices of bounded non-commutative (NC) functions over NC operator unit balls (unit balls corresponding to finite-dimensional operator spaces) and characterize cyclic matrix free polynomials with respect to the canonical weak-* topology. More precisely, we show that a matrix free polynomial generates a weak-* dense left/right ideal if and only if it is stable, i.e., non-singular at every point in the NC operator unit ball. To this end, we establish a version of the Neuwirth--Ginsberg--Newman inequality for stable matrix free polynomials. We combine our techniques with the theory of realizations to establish cyclicity of stable NC rational functions that are uniformly continuous across the boundary, and we recover known results about cyclicity of NC rational functions in the matrix-valued free Hardy space over the NC unit row-ball. Lastly, we introduce the NC parallel sum function: a stable NC rational function that is contractive over the NC bidisk, which cannot be extended uniformly across the boundary, and determine its cyclicity using properties of accretive operators.

math.FA↗

arXiv2026

A non-commutative de Branges-Rovnyak model for row contractions

We extend the de Branges-Rovnyak model for completely non-coisometric (CNC) linear contractions on a Hilbert space to the non-commutative multivariate setting of CNC row contractions. Namely, we show that any CNC contraction from several copies of a Hilbert space into a single copy is unitarily equivalent to the adjoint of the restricted backward right shifts acting on the de Branges-Rovnyak space of a contractive left multiplier between vector-valued "free Hardy spaces" of square-summable power series in several non-commuting (NC) variables. This contractive, operator-valued left multiplier, the characteristic function of the CNC row contraction, is a complete unitary invariant and it is always column-extreme as a contractive left multiplier. Our construction builds a model reproducing kernel Hilbert space of NC functions using a "non-commutative resolvent" of the row contraction, $T$, which is the inverse of the monic, affine linear pencil of $T$ in a certain NC unit row-ball of the NC universe of all row tuples of square matrices of all finite sizes.

math.FA↗

arXiv2025

Zeros of Holomorphic Functions in Commuting and Non-commuting Variables as Spectral Data

We characterize the zero sets of functions in the Schur--Agler class over the unit polydisk as well as functions in the unit ball of the multiplier algebra of the Drury--Arveson space via operators associated with a unitary realization formula for these functions. To this end, new notions of `eigenvalues' for tuples of operators are introduced, where the eigenvalues depend on the operator space structure of the ambient domain. Several examples showcasing the properties of these eigenvalues and the zero sets of rational inner functions in the Schur--Agler class are also presented. We further generalize this result to a large class of non-commuting (NC) holomorphic functions whose ambient domain is given by the unit ball of a matrix of linear polynomials. This includes the NC counterparts of the unit polydisk and the Euclidean unit ball. We also show for functions in the Schur--Agler class over NC matrix unit balls that their zeros along the topological boundary are contained in an appropriately defined `approximate point spectrum' of the associated realization operator, and so are points along the Shilov boundary where the boundary values are not isometric/coisometric. This, in-turn, provides an identical result for the commutative case.

math.CV↗

arXiv2025

Shift-cyclicity in analytic function spaces

In this survey, we consider Banach spaces of analytic functions in one and several complex variables for which: (i) polynomials are dense, (ii) point-evaluations on the domain are bounded linear functionals, and (iii) the shift operator corresponding to each variable is a bounded linear map. We discuss the problem of determining the shift-cyclic functions in such a space, i.e., functions whose polynomial multiples form a dense subspace. This problem is known to be intimately connected to some deep problems in other areas of mathematics, such as the dilation completeness problem and even the Riemann hypothesis. What makes determining shift-cyclic functions so difficult is that often we need to employ techniques that are specific to the space in consideration. We therefore cover several different function spaces that have frequently appeared in the past such as the Hardy spaces, Dirichlet-type spaces, complete Pick spaces and Bergman spaces. We highlight the similarities and the differences between shift-cyclic functions among these spaces and list some important general properties that shift-cyclic functions in any given analytic function space must share. Throughout this discussion, we also motivate and provide a large list of open problems related to shift-cyclicity.

math.FA↗

arXiv2025

Jointly cyclic polynomials and maximal domains

For a (not necessarily locally convex) topological vector space $\mathcal{X}$ of holomorphic functions in one complex variable, we show that the shift invariant subspace generated by a set of polynomials is $\mathcal{X}$ if and only if their common vanishing set contains no point at which the evaluation functional is continuous. For two variables, we show that this problem can be reduced to determining the cyclicity of a single polynomial and obtain partial results for more than two variables. We proceed to examine the maximal domain, i.e., the set of all points for which the evaluation functional is continuous. When $\mathcal{X}$ is metrizable, we show that the maximal domain must be an $F_σ$ set, and then construct Hilbert function spaces on the unit disk whose maximal domain is the disk plus an arbitrary subset of the boundary that is both $F_σ$ and $G_δ$.

math.CV↗

arXiv2025

Weak-* and completely isometric structure of noncommutative function algebras

We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the following classification theorem: under very mild assumptions on the varieties, two such algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are completely isometrically and weak-* isomorphic if and only if there is a nc biholomorphism between the varieties. For matrix spanning homogeneous varieties in injective operator balls, we further sharpen this equivalence, showing that there exists a linear isomorphism between the respective balls that maps one variety onto the other; in general, we show, the homogeneity condition cannot be dropped. We highlight some difficulties and open problems, contrasting with the well studied case of row ball.

math.OA↗

arXiv2024

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

math.OA↗

arXiv2021

Cyclicity preserving operators on spaces of analytic functions in $\mathbb{C}^n$

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_α \, (α\in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-Żelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable.

math.CV↗