arXiv subjects

Jaydeb Sarkar

Publications and source records attributed to Jaydeb Sarkar.

At least 37 records · Page 2Linked to original sources

arXiv2025

Inner and characteristic functions in polydiscs

Characteristic functions of linear operators are analytic functions that serve as complete unitary invariants. Such functions, as long as they are built in a natural and canonical manner, provide representations of inner functions on a suitable domain and make significant contributions to the development of various theories in Hilbert function spaces. In this paper, we solve this problem in polydiscs. In particular, we present a concrete description of the characteristic functions of tuples of commuting pure contractions and, consequently, provide a description of inner functions on polydiscs.

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arXiv2024

Invariant subspaces of perturbed backward shift

We represent closed subspaces of the Hardy space that are invariant under finite-rank perturbations of the backward shift. We apply this to classify almost invariant subspaces of the backward shift and represent a more refined version of nearly invariant subspaces. Kernels of certain perturbed Toeplitz operators are examples of the newly introduced nearly invariant subspaces.

math.FA↗

arXiv2024

A Bishop-Phelps-Bollobás theorem for bounded analytic functions

Let $H^\infty$ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by $\mathscr{B}(H^\infty)$ the Banach space of all bounded linear operators from $H^\infty$ to itself. We prove that the Bishop-Phelps-Bollobás property holds for $\mathscr{B}(H^\infty)$. As an application to our approach, we prove that the Bishop-Phelps-Bollobás property also holds for operator ideals of $\mathscr{B}(H^\infty)$.

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arXiv2023

Pairs of inner projections and two applications

Orthogonal projections onto closed subspaces of $H^2(\mathbb{D}^n)$ of the form $φH^2(\mathbb{D}^n)$ for inner functions $φ$ on $\mathbb{D}^n$ are referred to as inner projections, where $H^2(\mathbb{D}^n)$ denotes the Hardy space over the open unit polydisc $\mathbb{D}^n$. In this paper, we classify pairs of commuting inner projections. We also present two seemingly independent applications: the first is an answer to a question posed by R. G. Douglas, and the second is a complete classification of partially isometric truncated Toeplitz operators with inner symbols on the polydisc.

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arXiv2023

A Bishop-Phelps-Bollobas theorem for disc algebra

Let $\mathbb{D}$ represent the open unit disc in $\mathbb{C}$. Denote by $A(\mathbb{D})$ the disc algebra, and $\mathscr{B}(X, A(\mathbb{\mathbb{D}}))$ the Banach space of all bounded linear operators from a Banach space $X$ into $A(\mathbb{D})$. We prove that, under the assumption of equicontinuity at a point in $\partial \mathbb{D}$, the Bishop-Phelps-Bollobás property holds for $\mathscr{B}(X, A(\mathbb{\mathbb{D}}))$.

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arXiv2023

Automorphisms and generalized projections on spaces of analytic functions

We present complete classifications of automorphisms of two closed subalgebras of the bounded analytic functions on the open unit disc $\mathbb{D}$, namely, the subalgebra of functions vanishing at the origin, and the subalgebra of functions whose first derivative vanishes at the origin. The later subalgebra is known as the Neil algebra. We also characterize generalized tri-circular projections on $H^{p}(\mathbb{D})$ and $H^{p}(\mathbb{D}^2)$, $1\leq p \leq \infty$, $p\neq 2$.

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arXiv2022

Orthogonal decompositions and twisted isometries II

We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify all the known orthogonal decomposition related results in the literature.

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arXiv2022

Characterizations of complex symmetric Toeplitz operators

We present complete characterizations of Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is $S$-Toeplitz for some unilateral shift $S$ and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift $S$. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results answer the well known open question about characterizations of complex symmetric Toeplitz operators.

math.FA↗

arXiv2022

Orthogonal decompositions and twisted isometries

Let $n > 1$. Let $\{U_{ij}\}_{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i$'s are in the commutator $\{U_{st}: s \neq t\}'$, and $V_i^*V_j=U_{ij}^*V_jV_i^*$, $i \neq j$ We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-Wold type orthogonal decomposition, and prove that the universal $C^*$-algebra generated by $\mathcal{U}_n$-twisted isometry is nuclear. We exhibit concrete analytic models of $\mathcal{U}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^*$-algebras generated by $\mathcal{U}_n$-twisted isometries and the unitary equivalence classes of the non-zero irreducible representations of twisted noncommutative tori. Our motivation of $\mathcal{U}_n$-twisted isometries stems from the classical rotation $C^*$-algebras, Heisenberg group $C^*$-algebras, and a recent work of de Jeu and Pinto.

math.OA↗

arXiv2022

Invariant subspaces of idempotents on Hilbert spaces

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.

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arXiv2022

Pairs of projections and commuting isometries

It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\ & & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised by He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries.

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arXiv2022

Schur functions and inner functions on the bidisc

We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in terms of Agler kernels, two-variable Schur functions that admit one variable factor.

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arXiv2022

Partially isometric Toeplitz operators on the polydisc

A Toeplitz operator $T_φ$, $φ\in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $φ_1, φ_2 \in H^\infty(\mathbb{D}^n)$ such that $φ_1$ and $φ_2$ depends on different variables and $φ= \barφ_1 φ_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in $L^\infty(\mathbb{T}^n)$, $n > 1$, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that $T_φ$ is a shift whenever $φ$ is inner in $H^\infty(\mathbb{D}^n)$.

math.FA↗

arXiv2022

Tridiagonal shifts as compact + isometry

Let $\{a_n\}_{n\geq 0}$ and $\{b_n\}_{n\geq 0}$ be sequences of scalars. Suppose $a_n \neq 0$ for all $n \geq 0$. We consider the tridiagonal kernel (also known as band kernel with bandwidth one) as \[ k(z, w) = \sum_{n=0}^\infty ((a_n + b_n z)z^n) \overline{(({a}_n + {b}_n {w}) {w}^n)} \qquad (z, w \in \mathbb{D}), \] where $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Denote by $M_z$ the multiplication operator on the reproducing kernel Hilbert space corresponding to the kernel $k$. Assume that $M_z$ is left-invertible. We prove that $M_z =$ compact $+$ isometry if and only if $|\frac{b_n}{a_n}-\frac{b_{n+1}}{a_{n+1}}|\rightarrow 0$ and $|\frac{a_n}{a_{n+1}}| \rightarrow 1$.

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arXiv2022

Left-invertibility of rank-one perturbations

For each isometry $V$ acting on some Hilbert space and a pair of vectors $f$ and $g$ in the same Hilbert space, we associate a nonnegative number $c(V;f,g)$ defined by \[ c(V; f,g) = (\|f\|^2 - \|V^*f\|^2) \|g\|^2 + |1 + \langle V^*f , g\rangle|^2. \] We prove that the rank-one perturbation $V + f \otimes g$ is left-invertible if and only if \[ c(V;f,g) \neq 0. \] We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function $z$. Finally, we examine $D + f \otimes g$, where $D$ is a diagonal operator with nonzero diagonal entries and $f$ and $g$ are vectors with nonzero Fourier coefficients. We prove that $D + f\otimes g$ is left-invertible if and only if $D+f\otimes g$ is invertible.

math.FA↗

arXiv2021

Model spaces invariant under composition operators

Given a holomorphic self-map $φ$ of $\D$ (the open unit disc in $\mathbb{C}$), the composition operator $C_φ f = f \circ φ$, $f \in H^2(\mathbb{\D})$, defines a bounded linear operator on the Hardy space $H^2(\mathbb{\D})$. The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb{\D})$, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

math.FA↗

arXiv2021

Invariant subspaces of analytic perturbations

By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, we first isolate a natural class of finite rank operators for which the corresponding perturbations are analytic, and then we present a complete classification of invariant subspaces of those analytic perturbations. We also exhibit some instructive examples and point out several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations.

math.FA↗

arXiv2021

Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms

Given scalars $a_n (\neq 0)$ and $b_n$, $n \geq 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $\mathbb{D}$ defined by \[ k(z, w) = \sum_{n=0}^\infty \Big((a_n + b_n z)z^n\Big) \Big((\bar{a}_n + \bar{b}_n \bar{w}) \bar{w}^n \Big) \qquad (z, w \in \mathbb{D}). \] This defines a reproducing kernel Hilbert space $\mathcal{H}_k$ (known as tridiagonal space) of analytic functions on $\mathbb{D}$ with $\{(a_n + b_nz) z^n\}_{n=0}^\infty$ as an orthonormal basis. We consider shift operators $M_z$ on $\mathcal{H}_k$ and prove that $M_z$ is left-invertible if and only if $\{|{a_n}/{a_{n+1}}|\}_{n\geq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin's models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel $k$, as above, is preserved under Shimorin model if and only if $b_0=0$ or that $M_z$ is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fails to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.

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