arXiv subjects

Srijan Sarkar

Publications and source records attributed to Srijan Sarkar.

12 recordsLinked to original sources

arXiv2026

Finite-rank commutators of projections onto shift-invariant subspaces

In this article, using Halmos' two projections theorem, we completely characterize finite-rank commutators of orthogonal projections onto shift-invariant subspaces $\phi_1 H^2(\mathbb{D})$ and $\phi_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ corresponding to inner functions $\phi_1, \phi_2$ on the unit disc $\mathbb{D}$. Using our methods, we connect the finite-rank commutator with the Fredholmness of the projections $(P_{\phi_1}, P_{\phi_2})$ as introduced by Avron, Seiler and Simon. We conclude with several characterizations on the polydisc.

math.FA↗

arXiv2025

On products and partial isometry of Toeplitz operators with operator-valued symbols

We solve the following problems associated with Toeplitz operators $T_Φ$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$. $(I)$ Given operator-valued bounded analytic functions $Γ, Ψ$ on $\mathbb{D}^n$, we completely characterize when the product $M_ΓM_Ψ^*$ becomes a Toeplitz operator by identifying tractable conditions on the functions. Furthermore, these conditions can be used to explicitly write the product into a sum of simple Toeplitz operators. $(II)$ We prove that partially isometric Toeplitz operators admit the following factorization: \[ T_Φ = M_Γ M_Ψ^*, \] where, $Γ, Ψ$ are operator-valued inner functions on $\mathbb{D}^n$. A few of the immediate consequences are: $(a)$ every partially isometric Toeplitz operator has a partially isometric symbol almost everywhere on $\mathbb{T}^n$ (distinguished boundary of $\mathbb{D}^n$), $(b)$ any partially isometric analytic Toeplitz operator is of the form $M_{ΓV^*}$, where $Γ$ is an operator-valued inner function and $V$ is an constant isometry. In connection with the result $(ii)$, we establish and use a crucial phenomenon: the range of partially isometric Toeplitz operators is always a Beurling-type invariant subspace of $H_{\mathcal{E}}^2(\mathbb{D}^n)$. Our results are new even in the case of Hardy spaces over the unit disc and extend the work of Brown--Douglas, Deepak--Pradhan--Sarkar on scalar-valued spaces.

math.FA↗

arXiv2024

Two problems on submodules of $H^2(\mathbb{D}^n)$

Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2(\mathbb{D}^n)$ (where $n \geq 2$), let $R_{z_j}:=M_{z_j}|_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ ev_{z_j}$, for each $j \in \{1,\ldots,n\}$. Here, $ev_{z_j}$ is the operator evaluating at $0$ in the $z_j$-th variable. In this article, we prove that given any subset $Λ\subseteq \{1,\ldots,n\}$, there exists a collection of one-variable inner functions $\{ϕ_λ(z_λ)\}_{λ\in Λ}$ on $\mathbb{D}^n$, such that \[ \mathcal{S} = \sum_{λ\in Λ} ϕ_λ(z_λ)H^2(\mathbb{D}^n), \] if and only if the conditions $ (I_{\mathcal{S}}-E_{z_k}E_{z_k}^*)(I_{\mathcal{S}}-R_{z_k}R_{z_k}^*)=0$ for all $k \in \{1,\dots,n\} \setminus Λ$, and $(I_{\mathcal{S}}-E_{z_{i}}E_{z_{i}}^*)(I_{\mathcal{S}}-R_{z_{i}}R_{z_{i}}^*)(I_{\mathcal{S}}-E_{z_{j}}E_{z_{j}}^*)(I_{\mathcal{S}}-R_{z_{j}}R_{z_{j}}^*)=0$ for all distinct $i,j \in Λ$ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules.

math.FA↗

arXiv2024

Unitary parts of Toeplitz operators with operator-valued symbols

Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is unitary. We show that for a Hilbert space $\mathcal{E}$ and operator-valued symbol $Φ\in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T})$, the Toeplitz operator $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ has such a unitary subspace if and only if there exists a Hilbert space $\mathcal{F}$, an inner function $Θ(z) \in H_{\mathcal{B}(\mathcal{F}, \mathcal{E})}^{\infty}(\mathbb{D})$, and a unitary $U:\mathcal{F} \rightarrow \mathcal{F}$ such that \[ Φ(e^{it}) Θ(e^{it}) = Θ(e^{it}) U \quad \text{and} \quad Φ(e^{it})^* Θ(e^{it}) = Θ(e^{it}) U^* \quad (\text{ a.e. on }\mathbb{T}). \] This result can be seen as a generalization of the corresponding result for Toeplitz operators on $H^2(\mathbb{D})$ by Goor in [13]. We provide finer characterizations for analytic Toeplitz operators by finding the correspondence between the unitary parts of $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $Φ(0)$ on $\mathcal{E}$.

math.FA↗

arXiv2023

A note on the column-row property

In this article, we study the following question asked by Michael Hartz in a recent paper \cite{Hartz}: \textit{which operator spaces satisfy the column-row property?} We provide a complete classification of the column-row property for non-commutative $L_{p}$-spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the column-row property and discuss their existence and non-existence for various natural examples of operator spaces.

math.FA↗

arXiv2021

Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operator topology. In this article, we prove that for contractions $T$, which commute with certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued multiplier $Φ$ of a vector-valued reproducing kernel Hilbert space (rkHs) is said to be pure contractive if the associated multiplication operator $M_Φ$ is a pure contraction. Using the above result, we find that operator-valued mulitpliers $Φ(\textbf{z})$ of several vector-valued rkHs's on the polydisc $\mathbb{D}^n$ as well as the unit ball $\mathbb{B}_n$ in $\mathbb{C}^n$ are pure contractive if and only if $Φ(0)$ is a pure contraction on the underlying Hilbert space. The list includes Hardy, Bergman and Drury-Arveson spaces. Finally, we present some applications of our characterization of pure contractive multipliers associated with the polydisc.

math.FA↗

arXiv2021

Characterization of C-Symmetric Toeplitz operators for a Class of Conjugations in Hardy Spaces

In this article, we introduce a new class of conjugations in the scalar valued Hardy space $H^2_{\mathbb{C}}(\mathbb{D})$ and provide a characterization of a complex symmetric Toeplitz operator $T_ϕ$ with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator $T_Φ$ on the vector valued Hardy space $\hdct$ with respect to certain conjugations introduced in \cite{CamaGP,KangKoLee,LeeKo2019}.

math.FA↗

arXiv2021

Pairs of Commuting Pure Contractions and Isometric dilation

A particular case of the fundamental Sz.-Nagy--Foias functional model for a contraction states that a pure contraction always dilates to a pure isometry. We are interested in the similar question for pairs, more precisely: does a pair of commuting pure contractions always dilate to a pair of commuting pure isometries? The purpose of this article is to identify pairs of commuting pure contractions for which the above question has an affirmative answer. Our method is based on an explicit structure of isometric dilation for pure pairs of commuting contractions obtained in a recent work by Das, Sarkar and the author.

math.FA↗

arXiv2019

Multiplicities, invariant subspaces and an additive formula

Let $T = (T_1, \ldots, T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots, n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb{C}$. If $\mathcal{Q}_i^{\bot}$, $i = 1, \ldots, n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\boldsymbol{z}} = (M_{z_1}, \ldots, M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^\perp$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb{C}^n$ is given by \[ \mbox{mult}_{M_{\boldsymbol z}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^\perp}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^\perp = \sum_{i=1}^n (\mbox{mult}_{M_z|_{\mathcal{Q}_i^\perp}} (\mathcal{Q}_i^{\bot})) = n. \] A similar result holds for the Bergman space over the unit polydisc.

math.FA↗

arXiv2017

Factorizations of Contractions

The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exists a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_Φ, M_Ψ)$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where \[ Φ(z)=(P+zP^{\perp})U^*\;\text{and}\; Ψ(z)=U(P^{\perp}+zP) \;;(z \in \mathbb{D}). \] Here we prove a similar factorization result for pure contractions. More particularly, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$. Then $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exists $\mathcal{B}(\mathcal{D})$-valued polynomials $φ$ and $ψ$ of degree $ \leq 1$ such that $\mathcal{Q}$ is a joint $(M_φ^*, M_ψ^*)$-invariant subspace, \[P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{φψ}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{ψφ}|_{\mathcal{Q}} \; \mbox{and} \;(T_1, T_2) \cong (P_{\mathcal{Q}} M_φ|_{\mathcal{Q}}, P_{\mathcal{Q}} M_ψ|_{\mathcal{Q}}).\]

math.FA↗

arXiv2017

Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$

We initiate a study of asymptotic Toeplitz operators on the Hardy space $H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$). We also study the Toeplitz operators in the polydisc setting. Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let $T_{z_i}$ denote the multiplication operator on $H^2(\mathbb{D}^n)$ by the $i^{th}$ coordinate function $z_i$, $i =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then the following hold: (i) $T$ is a Toeplitz operator (that is, $T = P_{H^2(\mathbb{D}^n)} M_φ|_{H^2(\mathbb{D}^n)}$, where $M_φ$ is the Laurent operator on $L^{2}(\mathbb{T}^n)$ for some $φ\in L^\infty(\mathbb{T}^n)$) if and only if $T_{z_i}^* T T_{z_i} = T$ for all $i = 1, \ldots, n$. (ii) $T$ is an asymptotic Toeplitz operator if and only if $T = \mbox{~Toeplitz} + \mbox{~compact}$. The case $n = 1$ is the well known results of Brown and Halmos, and Feintuch, respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.

math.FA↗

arXiv2016

Factorizations of Kernels and Reproducing Kernel Hilbert Spaces

The paper discusses a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, it is proved that for a large class of spaces isometric multipliers are trivial. One also gives for certain spaces conditions for obtaining a particular type of dilation, as well as a classification of Brehmer type submodules.

math.FA↗