arXiv subjects

Subrata Shyam Roy

Publications and source records attributed to Subrata Shyam Roy.

12 recordsLinked to original sources

arXiv2025

Brown-Halmos type Theorems on the proper images of bounded symmetric domains

Let $Ω\subseteq\mathbb C^n$ be a bounded symmetric domain and $f :Ω\to Ω^\prime\subseteq \mathbb C^n$ be a proper holomorphic mapping which is factored by a finite complex reflection group $G.$ We identify a family of reproducing kernel Hilbert spaces on $Ω^\prime$ arising naturally from the isotypic decomposition of the regular representation of $G$ on the Hardy space $H^2(Ω).$ Each element of this family can be realized as a closed subspace of some $L^2$-space on the Šilov boundary of $Ω^\prime$. The reproducing kernel Hilbert space associated to the sign representation of $G$ is the Hardy space $H^2(Ω^\prime).$ We establish a Brown-Halmos type characterization for the Toeplitz operators on $H^2(Ω^\prime),$ where $Ω^\prime$ is the image of the open unit polydisc $\mathbb D^n$ in $\mathbb C^n$ under a proper holomorphic mapping factored by the finite complex reflection group $G(m,p,n).$ Moreover, we prove various multiplicative properties of Toeplitz operators on $H^2(Ω^\prime)$, where $Ω^\prime$ is a proper holomorphic image of a bounded symmetric domain.

math.CV↗

arXiv2024

Contractive Hilbert modules on quotient domains

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbolΘ_n$-contraction is a commuting tuple of operators on a Hilbert space having $$\overline{\boldsymbolΘ}_n:=\{\boldsymbolθ(z)=(θ_1(z),\ldots,θ_n(z)):z\in\overline{\mathbb D}^n\}$$ as a spectral set, where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $\boldsymbolΘ_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $\boldsymbolΘ_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $\boldsymbolΘ_n$-isometries is obtained.

math.FA↗

arXiv2022

On irreducibility of a certain class of homogeneous operators obtained from quotient modules

Let $ Ω\subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(Ω)$ of $Ω$ depending upon a subgroup $G$ of Aut$(Ω)$ whenever the tuple of multiplication operators on $\mathcal{M}$ is homogeneous with respect to $G$ and both $\mathcal{M}$ as well as $\mathcal{M}_q$ are in the Cowen-Douglas class. We show that these compression of multiplication operators might be reducible even if the tuple of multiplication operators on $\mathcal{M}$ is irreducible by exhibiting a concrete example. Moreover, the irreducible components of these reducible operators are identified as Generalized Wilkins' operators.

math.FA↗

arXiv2020

Reducing submodules of Hilbert Modules and Chevalley-Shephard-Todd Theorem

Let $G$ be a finite pseudoreflection group, $Ω\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(Ω)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbolθ}$ on $\mathcal H,$ where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbolθ= (θ_1, \ldots, θ_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(Ω)$ of holomorphic functions on $Ω$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbolθ}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbolθ(Ω)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $Ω$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(Ω).$

math.CV↗

arXiv2018

The Agler-Young Class

This note introduces a special class of tuples of bounded operators on a Hilbert space. It is called the Agler Young class. Major results about this class include a Wold decomposition and a dilation theorem. The structure of the dilation is completely spelt out. A characterization of this class using the hereditary functional calculus of Agler is obtained and examples are discussed. Toeplitz operators play a major role in this note. An Agler-Young pair arising from a truncated Toeplitz operator is characterized. Thus, we extend results obtained in the case of commuting operators by several authors over many decades to the non-commutative situation. The results for the commuting case can be recovered as special cases.

math.FA↗

arXiv2017

Reducing sub-modules of the Bergman module $\mathbb A^{(λ)}(\mathbb D^n)$ under the action of the symmetric group

The weighted Bergman spaces on the polydisc, $\mathbb A^{(λ)}(\mathbb D^n)$, $λ>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $χ_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $χ_{\boldsymbol p}(1) \ne χ_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.

math.FA↗

arXiv2015

Dilations of Γ-contractions by solving operator equations

For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation S-S*P = (I-P*P)^1/2 X(I-P*P) 1/2, where X is a bounded operator on Ran(I-P*P) 1/2 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domain \Gamme = {(z 1 +z 2, z 1z 2) : |z1|{\leq} 1, |z2| {\leq}1} {\subseteq} C2 as a spectral set, is called a \Gamme-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction (S,P). This allows us to construct an explicit Γ-isometric dilation of a Γ-contraction (S,P). We prove the other way too, i.e, for a commuting pair (S,P) with |P|| {\leq} 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S,P) is a Γ-contraction. We show that for a pure Γ-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S = C +C*P. Any Γ-isometry can be written in this form where P now is an isometry commuting with C and C*. Any Γ-unitary is of this form as well with P and C being commuting unitaries. Examples of Γ-contractions on reproducing kernel Hilbert spaces and their Γ-isometric dilations are discussed.

math.FA↗

arXiv2013

A Note on $Γ_n$-isometries

In this note we characterize the distinguished boundary of the symmetrized polydisc and thereby develop a model theory for $Γ_n$-isometries along the lines of \cite{AY}. We further prove that for invariant subspaces of $Γ_n$-isometries, similar to the case $n=2$ \cite{S}, Beurling-Lax-Halmos type representation holds.

math.FA↗

arXiv2011

Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.

math.FA↗

arXiv2009

Quantum symmetries of classical spaces

We give a general scheme for constructing faithful actions of genuine (noncommutative as $C^*$ algebra) compact quantum groups on classical topological spaces. Using this, we show that: (i) a compact connected classical space can have a faithful action by a genuine compact quantum group, and (ii) there exists a spectral triple on a classical connected compact space for which the quantum group of orientation and volume preserving isometries (in the sense of \cite{qorient}) is a genuine quantum group.

math.QA↗

arXiv2006

Curvature calculations for a class of homogeneous operators

For an operator $T$ in the class ${\mathrm B}_n(Ω)$, introduced in \cite{CD}, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle $E_T$ determine the unitary equivalence class of the operator $T$. In the paper \cite{CD2}, the authors ask if there exists some pair of inequivalent oprators $T_1$ and $T_2$ for which the simultaneous unitary equivalence class of the curvature along with all the covariant derivatives coincide except for the derivative of the highest order. Here we show that some of the covariant derivatives are necessary to determine the unitary equivalence class of the operators in ${\mathrm B}_n(Ω)$. Our examples consist of homogeneous operators. For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives are determined from the simultaneous unitary equivalence class of these at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples.

math.FA↗

arXiv2006

On the irreducibility of a class of homogeneous operators

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra $\mathcal{A}(\mathbb D)$ as quotients of certain natural modules over the function algebra $\mathcal{A}(\mathbb D^2)$. These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class ${\mathrm B}_k(\mathbb D)$ and that they are mutually inequivalent, irreducible and homogeneous.

math.FA↗