arXiv subjects

Bhaskar Paul

Publications and source records attributed to Bhaskar Paul.

9 recordsLinked to original sources

arXiv2026

Dilation and Functional Models for Pure $\mathbf{\Theta}_n$-Contractions and the von Neumann Inequality on Distinguished Varieties in $\mathbf{\Theta}_n$

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbf{\Theta}_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbf{\Theta}_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbf{\Theta}_n$-contractions. Finally, we show that for a $\mathbf{\Theta}_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbf{\Theta}_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbf{\Theta}_n$.

math.FA↗

arXiv2026

On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions

This paper studies the domain $\mathbf{\Theta}_n$ from the perspective of operator theory. We obtain several characterizations of $\mathbf{\Theta}_n$-contractions (respectively, $\mathbf{\Theta}_n$-unitaries and $\mathbf{\Theta}_n$-isometries) and establish their relationships with $\Gamma_n$-contractions (respectively, $\Gamma_n$-unitaries and $\Gamma_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\mathbf{\Theta}_{n+1}$-contractions (respectively, $\mathbf{\Theta}_{n+1}$-unitaries and $\mathbf{\Theta}_{n+1}$-isometries). We prove that every $\mathbf{\Theta}_n$-contraction admits a canonical decomposition into the direct sum of a $\mathbf{\Theta}_n$-unitary and a completely non-unitary $\mathbf{\Theta}_n$-contraction. We further develop a dilation theory for $\mathbf{\Theta}_n$-contractions by obtaining necessary and sufficient conditions for the existence of minimal $\mathbf{\Theta}_n$-isometric dilations. As an application, we show that the minimal $\Gamma_n$-isometric dilation arises as a special case of the minimal $\mathbf{\Theta}_n$-isometric dilation. Finally, we identify a class of $\mathbf{\Theta}_2$-contractions that always admit $\mathbf{\Theta}_2$-isometric extensions.

math.FA↗

arXiv2026

Rational $\mathbf{\Theta_n}$-Inner Function and its Application in Interpolation Problem

In this paper, we investigate several geometric and function-theoretic properties of the domain $\mathbf{\Theta}_n$. We obtain new characterizations of its distinguished boundary and introduce the notion of a \textit{$\mathbf{\Theta}_n$-inner function}, together with several illustrative examples. We establish connections between $\mathbf{\Theta}n$-inner functions and $\Gamma_n$-inner functions, tetra-inner functions, and $\mathbf{\Theta}_{n+1}$-inner functions. Furthermore, we derive an explicit characterization of rational $\mathbf{\Theta}_n$-inner functions. As an application, for any finite collection of distinct interpolation nodes in $\mathbb{D}$ and prescribed target points in $\mathbf{\Theta}_n$, we obtain an explicit formula for the rational $\mathbf{\Theta}_n$-inner function satisfying the given interpolation data.

math.FA↗

arXiv2025

Functional models for $\Gamma_n$-contractions

This article develops several functional models for a given $\Gamma_n$-contraction. The first model is motivated by the Douglas functional model for a contraction. We then establish factorization results that clarify the relationship between a minimal isometric dilation and an arbitrary isometric dilation of a contraction. Using these factorization results, we obtain a Sz.-Nagy-Foias type functional model for a completely non-unitary $\Gamma_n$-contraction, as well as Sch\"affer type functional model for $\Gamma_n$-contraction.

math.FA↗

arXiv2025

Admissible Fundamental Operators and Models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction

We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] = [\tilde{F}^*_j, \tilde{F}_{7-i}]$,$w(\tilde{F}^*_i + \tilde{F}_{7-i}z) \leqslant 1$ and these operators satisfy \[(\tilde{F}^*_i + \tilde{F}_{7-i}z)\Theta_{T_7}(z) = \Theta_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] for $1 \leqslant i, j \leqslant 6$ for some $(F_1, \dots, F_6) \in \mathcal{B}(\mathcal{D}_{T_7})$ with $w(F^*_i + F_{7-i}z) \leqslant 1$ for $1 \leqslant i \leqslant 6$, then there exists a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $(T_1, \dots, T_7)$ such that $F_1, \dots, F_6$ are the fundamental operators of $(T_1, \dots, T_7)$ and $\tilde{F}_1, \dots, \tilde{F}_6$ are the fundamental operators of $(T^*_1, \dots, T^*_7)$. We also prove similar type of result for pure $\Gamma_{E(3; 2; 1, 2)}$-contraction. We explicitly construct a $\Gamma_{E(3; 3; 1, 1, 1)}$-unitary (respectively, a $\Gamma_{E(3; 2; 1, 2)}$-unitary) starting from a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, a $\Gamma_{E(3; 2; 1, 2)}$-contraction). Further, we develop functional models for general $\Gamma_{E(3; 3; 1, 1, 1)}$-isometries (respectively, $\Gamma_{E(3; 2; 1, 2)}$-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contractions). Finally, we present a Schaffer-type model for the $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation (respectively, the $\Gamma_{E(3; 2; 1, 2)}$-isometric dilation).

math.FA↗

arXiv2025

Functional Models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction, $\Gamma_{E(3; 2; 1, 2)}$-contraction and Tetrablock contraction

We obtain various characterizations of the fundamental operators of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a $\Gamma_{E(3; 2; 1, 2)}$-contraction. We describe functional models for \textit{pure $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction} and \textit{pure $\Gamma_{E(3; 2; 1, 2)}$-contraction}. We give a complete set of unitary invariants for a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a pure $\Gamma_{E(3; 2; 1, 2)}$-contraction. We demonstrate the functional models for a certain class of completely non-unitary $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T} = (T_1, \dots, T_7)$ and completely non-unitary $\Gamma_{E(3; 2; 1, 2)}$-contraction $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ which satisfy the following conditions: \begin{equation}\label{Condition 1} \begin{aligned} &T^*_iT_7 = T_7T^*_i \,\, \text{for} \,\, 1 \leqslant i \leqslant 6 \end{aligned} \end{equation} and \begin{equation}\label{Condition 2} \begin{aligned} &S^*_iS_3 = S_3S^*_i, \tilde{S}^*_jS_3 = S_3\tilde{S}^*_j \,\, \text{for} \,\, 1 \leqslant i, j \leqslant 2, \end{aligned} \end{equation} respectively. We also describe a functional model for a completely non-unitary tetrablock contraction $\textbf{T} = (A_1,A_2,P)$ that satisfies \begin{equation}\label{Condition 3} \begin{aligned} A^*_iP = PA^*_i \,\, \text{for $1 \leqslant i \leqslant 2$}. \end{aligned} \end{equation} By exhibiting counter examples, we show that such abstract model of tetrablock contraction, $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction may not exist if we drop the hypothesis of the above equations, respectively..

math.FA↗

arXiv2025

Necessary Conditions for $\Gamma_{E(3; 3; 1, 1, 1)}$-Isometric Dilation, $\Gamma_{E(3; 2; 1, 2)}$-Isometric Dilation and $\mathcal{\bar{P}}$-Isometric Dilation

A fundamental theorem of Sz.-Nagy states that a contraction $T$ on a Hilbert space can be dilated to an isometry $V.$ A more multivariable context of recent significance for these concepts involves substituting the unit disk with $\Gamma_{E(3; 3; 1, 1, 1)}, \Gamma_{E(3; 2; 1, 2)},$ and pentablock. We demonstrate the necessary conditions for the existence of $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation, $\Gamma_{E(3; 2; 1, 2)}$-isometric dilation and pentablock-isometric dilation. We construct a class of $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions that are always dilate . We create an example of a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction that has a $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation such that $[F_{7-i}^*, F_j] \ne [F_{7-j}^*, F_i] $ for some $i,j$ with $1\leq i ,j\leq 6,$ where $F_i$ and $F_{7-i}, 1\leq i \leq 6$ are the fundamental operators of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T}=(T_1, \dots, T_7).$ We also produce an example of a $\Gamma_{E(3; 2; 1, 2)}$-contraction that has a $\Gamma_{E(3; 2; 1, 2)}$-isometric dilation by which $$[G^*_1, G_1] \neq [\tilde{G}^*_2, \tilde{G}_2]~{\rm{ and }}~[2G^*_2, 2G_2] \neq [2\tilde{G}^*_1, 2\tilde{G}_1],$$ where $G_1, 2G_2, 2\tilde{G}_1, \tilde{G}_2$ are the fundamental operators of $\textbf{S}$. As a result, the set of sufficient conditions for the existence of a $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation and $\Gamma_{E(3; 2; 1; 2)} $-isometric dilations presented in Theorem \ref{conddilation} and Theorem \ref{condilation1}, respectively, are not generally necessary. We construct explicit $\Gamma_{E(3; 3; 1, 1, 1)} $-isometric, $\Gamma_{E(3; 2; 1; 2)} $-isometric dilations and $\mathcal{\bar{P}}$-isometric dilation of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction, $\Gamma_{E(3; 2; 1; 2)}$-contraction and $\mathcal{\bar{P}}$-contraction, respectively.

math.FA↗

arXiv2025

Canonical Decompositions and Conditional Dilations of $\Gamma_{E(3; 3; 1, 1, 1)}$-Contraction and $\Gamma_{E(3; 2; 1, 2)}$-Contraction

A $7$-tuple of commuting bounded operators $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$\Gamma_{E(3; 3; 1, 1, 1)}$-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\mathbf{T}$. Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators on $\mathcal{H}$ satisfying $S_i \tilde{S}_j = \tilde{S}_j S_i$ for $1 \leq i \leq 3$ and $1 \leq j \leq 2$. The tuple $\mathbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is called a \textit{$\Gamma_{E(3; 2; 1, 2)}$-contraction} if $\Gamma_{E(3; 2; 1, 2)}$ is a spectral set for $\mathbf{S}$. In this paper, we establish the existence and uniqueness of the fundamental operators associated with $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a pure $\Gamma_{E(3; 2; 1, 2)}$-isometry. We also construct a conditional dilation for a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a $\Gamma_{E(3; 2; 1, 2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contraction) admits a unique decomposition as a direct sum of a $\Gamma_{E(3; 3; 1, 1, 1)}$-unitary (respectively, $\Gamma_{E(3; 2; 1, 2)}$-unitary) and a completely non-unitary $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contraction).

math.FA↗

arXiv2025

Operators on Hilbert Space having $\Gamma_{E(3; 3; 1, 1, 1)}$ and $\Gamma_{E(3; 2; 1, 2)}$ as Spectral Sets

A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$\Gamma_{E(3; 3; 1, 1, 1)} $-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $ Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators defined on a Hilbert space $\mathcal{H}$ with $S_i\tilde{S}_j = \tilde{S}_jS_i$ for $1 \leqslant i \leqslant 3$ and $1 \leqslant j \leqslant 2$. We say that $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is a $\Gamma_{E(3; 2; 1, 2)} $-contraction if $ \Gamma_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. We derive various properties of $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for $\Gamma_{E(3; 3; 1, 1,1 )}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions. We explore the structure of $\Gamma_{E(3; 3; 1, 1, 1)}$-unitaries and $\Gamma_{E(3; 2; 1, 2)}$-unitaries and elaborate on the relationship between them. We also study various properties of $\Gamma_{E(3; 3; 1, 1, 1)}$-isometries and $\Gamma_{E(3; 2; 1, 2)}$-isometries. We discuss the Wold Decomposition for a $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a $\Gamma_{E(3; 2; 1, 2)}$-isometry. We further outline the structure theorem for a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a pure $\Gamma_{E(3; 2; 1, 2)}$-isometry.

math.FA↗