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Poornendu Kumar

Publications and source records attributed to Poornendu Kumar.

17 recordsLinked to original sources
arXiv2026

Law of iterated logarithm for inner functions

In a recent work [\emph{Adv. Math.} 401 (2022), Paper No. 108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin. In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients. We also identify the full set of subsequential limit points at the LIL scale. Using the Aleksandrov--Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions. We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.

math.CV↗
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

Multiplicative linear functionals on reproducing kernel Hilbert spaces

The classical Gleason--Kahane--Zelazko theorem characterizes multiplicative linear functionals on a unital Banach algebra through the scalar identity $\Lambda(x^{2})=\Lambda(x)^{2}$. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in $\mathbb{C}^{d}$, replacing conditions on the whole space by tractable conditions involving only kernel functions. Our first main result shows that if $k$ is a diagonal holomorphic kernel on a domain $\Omega\subseteq\mathbb{C}^{d}$ containing the origin, and if $k_{w}^{2}\in\mathcal H(k)$ for every $w\in\Omega$, then a bounded linear functional $\Lambda$ on $\mathcal H(k)$ satisfying $\Lambda(1)=1$ is multiplicative if and only if $\Lambda(k_{w}^{2})=\Lambda(k_{w})^{2}$ for all $w\in\Omega$. Kernels satisfying $2$-point Pick property and their powers furnish a natural class of examples. When $k$ arises as a Schur product or a tensor product of complete diagonal Pick kernels, we obtain a further, more algebraic characterization of multiplicativity, expressed in terms of the values of $\Lambda$ on kernel functions and their reciprocals. This framework subsumes the weighted Bergman and Dirichlet-type spaces, as well as the Hardy space on the polydisc. We extend the analysis to Hilbert spaces associated with diagonal Dirichlet series kernels on half-planes, encompassing in particular the Hardy space of Dirichlet series and its Riemann zeta reproducing kernel. Explicit examples demonstrate that the boundedness hypothesis on $\Lambda$ cannot be omitted. Finally, our characterization of multiplicative linear functionals leads to characterizations of weighted composition operators on a reproducing kernel Hilbert space associated with a diagonal holomorphic kernel.

math.FA↗
arXiv2026

Cayley--Hamilton tuples: an interplay between algebraic varieties and joint spectra

We introduce the notion of Cayley--Hamilton tuples: these are commuting operator tuples that are annihilated by a non-zero polynomial and such that its Taylor joint spectrum coincides with the algebraic variety determined by its annihilating ideal. Commuting matrix tuples are Cayley--Hamilton tuples. We provide two families of Cayley--Hamilton tuples in the infinite dimensional setting with additional details. What arises as a by-product is a concrete characterization of distinguished varieties in the polydisk in terms of Taylor joint spectrum of commuting isometries. These varieties have been of interest in various fields of mathematics over the last two decades. The Taylor and Waelbroeck joint spectrum of a Cayley--Hamilton tuple are shown to be the same. It is also shown that the support of the annihilating ideal of a Cayley--Hamilton tuple is the same as its joint spectrum. As an application, we deduce an algebraic characterization of bi-variate polynomials whose zero set intersected with the closed bidisk is the joint spectrum of a commuting isometric pair.

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

Annihilating ideals and Agler--McCarthy spectral varieties in the bidisc

The closed unit bidisc $\overline{\mathbb{D}}^2$ is known to be a spectral set for any pair $(T_1,T_2)$ of commuting contractions. When each $T_i$ is pure and has finite defect, the pair admits a much smaller spectral set: the closure of a distinguished variety $V$ inside the bidisc $\mathbb{D}^2$. We find conditions on $(T_1,T_2)$ that guarantee that the closure of $V$ is a minimal spectral set. In addition, we examine the relationship between $V$ and the annihilating ideal $\text{Ann}(T_1,T_2)$ in $H^\infty(\mathbb{D}^2)$. While $V$ is typically strictly larger than the zero set of $\text{Ann}(T_1,T_2)$, we isolate a natural constrained isometric co-extension $(S_1,S_2)$ of $(T_1,T_2)$ whose Taylor spectrum is contained in $V$ and is closely linked to the so-called support of $\text{Ann}(T_1,T_2)$. We also characterize when $\text{Ann}(T_1,T_2)$ is the ideal of functions vanishing on the joint point spectrum of $(S_1^*,S_2^*)$.

math.FA↗
arXiv2024

Herglotz representation for operator-valued function on a set associated with test functions

The Herglotz representation theorem for holomorphic functions with non-negative real part is a fundamental result in the theory of holomorphic functions. In this paper, we reinterpret the Herglotz representation in the context of modern techniques, specifically realization formula. This reinterpretation is then extended to operator-valued functions on arbitrary sets, in association with a collection of test functions.

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 $\mu$-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

Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$

Douglas and Rudin proved that any unimodular function on the unit circle $\T$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\T$.

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 Poincar\'e 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

Herglotz's representation and Caratheodory's approximation

Herglotz's representation of holomorphic functions with positive real part and Carath\'eodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain $\Omega$, a version of Heglotz's representation is known. Carath\'eodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Koranyi and Pukanszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carath\'eodory's approximation theorem while proving this new form of Herglotz's representation. Carath\'eodory's approximation theorem is also proved for matrix-valued functions on a multi-connected domain.

math.FA↗
arXiv2022

Function theory on quotient domains related to the polydisc

Inner functions are the backbone of holomorphic function theory. This paper studies the inner functions on quotient domains of the open unit polydisc, $\bD^d$, arising from the group action of finite pseudo-reflection groups. Such quotient domains are known to be biholomorphic to the proper image $\theta(\bD^d)$ of $\bD^d$ under certain polynomial maps $\theta: \bD^d \to \theta(\bD^d)$. The main contributions of this paper are as follows: 1) We show that the closed algebra generated by inner functions on $\theta(\bD^d)$ forms a proper subalgebra of $H^\infty(\theta(\bD^d))$, the algebra of bounded holomorphic functions on $\theta(\bD^d)$. 2) The set of all rational inner functions on $\theta(\bD^d)$ is shown to be dense in the norm-unit ball of $H^\infty(\theta(\bD^d))$ with respect to the uniform compact-open topology, thereby proving the Carath\'eodory approximation result. 3) As an application of the Carath\'eodory approximation theorem, we approximate holomorphic functions on $\theta(\bD^d)$ that are continuous in the closure of ${\theta(\bD^d)}$ by convex combinations of rational inner functions in the $L^2 $-norm, thereby obtaining a version of the Fisher's theorem. 4) Given the two approximation results above, establishing a structure for rational inner functions is essential. We have identified the structure of rational inner functions on $\theta(\mathbb{D}^d)$. 5) The Carath\'eodory approximation for operator-valued functions is also discussed.

math.FA↗
arXiv2022

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 Carath\'eodory. 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↗
arXiv2021

Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk

Starting with a solvable Nevanlinna-Pick interpolation problem with the initial data coming from the symmetrized bidisk, this paper studies the corresponding uniqueness set, i.e., the largest set in the domain where all solutions to the problem coincide. It is shown that the uniqueness set coincides with an algebraic variety in the domain. The algebraic variety - canonically constructed from the interpolation data - is called the uniqueness variety. It was shown that the uniqueness variety contains a distinguished variety which by definition is the zero set of a two-variable polynomial that intersects the domain and exits through its distinguished boundary. A complete algebraic and geometric characterizations of distinguished varieties are obtained in this paper.

math.FA↗
arXiv2020

Distinguished Varieties Through the Berger--Coburn--Lebow Theorem

A distinguished algebraic variety in $\mathbb{C}^2$ has been the focus of much research in recent years because of good reasons. This note gives a different perspective. (1) We find a new characterization of an algebraic variety $\mathcal W$ which is distinguished with respect to the bidisc. It is in terms of the joint spectrum of a pair of commuting linear matrix pencils. (2) There is a characterization known of $\mathbb{D}^2\cap\mathcal{W}$ due to a seminal work of Agler and McCarthy. We show that Agler--McCarthy characterization can be obtained from the new one and vice versa. (3) En route, we develop a new realization formula for operator-valued contractive analytic functions on the unit disc. (4) There is a one-to-one correspondence between operator valued contractive holomorphic functions and {\em canonical model triples}. This pertains to the new realization formula mentioned above. (5) Pal and Shalit gave a characterization of an algebraic variety, which is distinguished with respect to the symmetrized bidisc, in terms of a matrix of numerical radius no larger than $1$. We refine their result by making the class of matrices strictly smaller. (6) In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc. At the root of our work is the Berger--Coburn--Lebow theorem characterizing a commuting tuple of isometries.

math.FA↗