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Avijit Pal

Publications and source records attributed to Avijit Pal.

17 recordsLinked to original sources
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

Few maps in the rich structure for the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$

The primary goal of a rich structure for some naturally occurring domains $\mathcal X$ is to connect four naturally occurring objects of analysis in the context of $3\times 3$ analytic matrix functions on $\mathbb D$. Combining this rich structure with the classical realisation formula and Hilbert space models in the sense of Agler, one can effectively construct functions in the space $\mathcal O(\mathbb D,\mathcal X)$ of analytic maps from $\mathbb D$ to $\mathcal X$. This allows one to obtain solvability criteria for two cases of the $\mu$-synthesis problem. We describe few maps in the rich structure. We define $SE$ map between $\mathcal S_{1}(\mathbb C^3,\mathbb C^3)$ and $\mathcal S_{3}(\mathbb C,\mathbb C)$ and establish the relation between $\mathcal{S}_{1}(\mathbb C^3,\mathbb C^3)$ and the set of analytic kernels on $\mathbb{D}^{3}$. We obtain the $UW$ procedure and using the $UW$ procedure we construct the $Upper \,\,W$ and $Upper\,\ E$ maps. We also construct $Right~S$ and $SE$ maps. We show how the interpolation problems for $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$ can be reduced to a standard matricial Nevanlinna-Pick problem.

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↗
arXiv2025

Function Theory and necessary conditions for a Schwarz lemma related to $\mu$-Synthesis Domains

A subset of $\mathbb{C}^7$ (respectively, of $\mathbb{C}^5$) associated with the structured singular value $\mu_E$, defined on $3 \times 3$ matrices, is denoted by $G_{E(3;3;1,1,1)}$ (respectively, by $G_{E(3;2;1,2)}$). In control engineering, the structured singular value $\mu_E$ plays a crucial role in analyzing the robustness and performance of linear feedback systems. We characterize the domain $G_{E(3;3;1,1,1)}$ and its closure $\Gamma_{E(3;3;1,1,1)}$, and employ realization formulas to describe both. The domain $G_{E(3;3;1,1,1)}$ and its closure are neither circular nor convex; however, they are simply connected. We provide an alternative proof of the polynomial and linear convexity of $\Gamma_{E(3;3;1,1,1)}$. Furthermore, we establish necessary conditions for a Schwarz lemma on the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, and describe the relationships between these two domains as well as between their closed boundaries.

math.FA↗
arXiv2021

Isometric Embeddability of $S_q^m$ into $S_p^n$

In this paper, we study existence of isometric embedding of $S_q^m$ into $S_p^n,$ where $1\leq p\neq q\leq \infty$ and $n\geq m\geq 2.$ We show that for all $n\geq m\geq 2$ if there exists a linear isometry from $S_q^m$ into $S_p^n$, where $(q,p)\in(1,\infty]\times(1,\infty) \cup(1,\infty)\setminus\{3\}\times\{1,\infty\}$ and $p\neq q,$ then we must have $q=2.$ This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever $S_q$ embeds isometrically into $S_p$ for $(q,p)\in \left(1,\infty\right)\times\left[2,\infty \right)\cup[4,\infty)\times\{1\} \cup\{\infty\}\times\left( 1,\infty\right)\cup[2,\infty)\times\{\infty\}$ with $p\neq q,$ we must have $q=2.$ Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative $L_p$-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for $m\geq 2$ and $1<q<2,$ $S_q^m$ embeds isometrically into $S_\infty^n$, was left open in \textit{Bull. London Math. Soc.} 52 (2020) 437-447.

math.FA↗
arXiv2021

Necessary conditions for existence of $Γ_n$-contractions and examples of $Γ_3$-contractions

The fundamental result of B. Sz. Nazy states that every contraction has a coisometric extension and a unitary dilation. The isometric dilation of a contraction on a Hilbert space motivated whether this theory can be extended sensibly to families of operators. It is natural to ask whether this idea can be generalized, where the contraction $T$ is substituted by a commuting $n$-tuples of operators $(S_1,\cdots, S_n)$ acting on some Hilbert space having $Γ_n$ as a spectral set. We derive the necessary conditions for the existence of a $Γ_n$-isometric dilation for $Γ_n$-contractions. Also we discuss an example of a $Γ_3$-contraction $(S_1, S_2, S_3)$ acting on some Hilbert space $\mathcal H,$ which has a $Γ_3$-isometric dilation, but it fails to satisfy the following condition: $$E_1^*E_1-E_1E_1^*= E_2^*E_2-E_2E_2^*,$$ where $E_1$ and $E_2$ are the fundamental operators of $(S_1, S_2, S_3),$ $(S_1,S_2)$ is a pair of commuting contractions and $S_3$ is a partial isometry. Thus, the set of sufficient conditions for the existence of a $Γ_3$-isometric dilation breaks down, in general, to be necessary, even when the $Γ_3$-contraction $(S_1, S_2, S_3)$ has the special structure as described above.

math.FA↗
arXiv2018

On $Γ_n$-contractions and their Conditional Dilations

We prove some estimates for elementary symmetric polynomials on $\mathbb D^n.$ We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc $Γ_n.$ Furthermore, we show the existence and uniqueness of solutions to the operator equations $$S_i-S_{n-i}^*S_n=D_{S_n}X_iD_{S_n}~~{\rm{and}}~~S_{n-i}-S_{i}^*S_n=D_{S_n}X_{n-i}D_{S_n},$$ where $X_i,X_{n-i}\in \mathcal B(\mathcal D_{S_n}), ~{\rm{for ~all~}} i=1,\ldots,(n-1),$ with numerical radius not greater than $1,$ for a $Γ_n$-contraction $(S_1,\ldots, S_n).$ We construct a conditional dilation of various classes of $Γ_n$-contractions. Various properties of a $Γ_n$-contraction and its explicit dilation allow us to construct a concrete functional model for a $Γ_n$-contraction. We describe the structure and additional characterization of $Γ_n$-unitaries and $Γ_n$-isometries in detail.

math.FA↗
arXiv2016

Infinite-dimensional features of matrices and pseudospectra

Given a Hilbert space operator $T$, the level sets of function $Ψ_T(z)=\|(T-z)^{-1}\|^{-1}$ determine the so-called pseudospectra of $T$. We set $Ψ_T$ to be zero on the spectrum of $T$. After giving some elementary properties of $Ψ_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator $T$, there is a sequence $\{T_n\}$ of finite matrices such that $Ψ_{T_n}(z)$ tends to $Ψ_{T}(z)$ uniformly on $\C$. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices. One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large order. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen--Douglas class operators. We use these results to show that, to the opposite to the function $Ψ_{S}$, the function $\|\sqrt{S-z}\,\|$ for certain finite matrices $S$ may oscillate arbitrarily fast even far away from the spectrum.

math.FA↗
arXiv2016

Contractivity and complete contractivity for finite dimensional Banach Spaces

Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$ and let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contractive linear maps are always completely contractive thus remains open. We answer this question for balls of the form $Ω_\mathbf A$ in $\mathbb C^2.$

math.FA↗
arXiv2015

Contractivity, complete contractivity and curvature inequalities

For any bounded domain $Ω$ in $\mathbb C^m,$ let ${\mathrm B}_1(Ω)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $\boldsymbol T$ in the Cowen-Douglas class ${\mathrm B}_1(Ω),$ let $N_{\boldsymbol T}(w)$ denote the restriction of $\boldsymbol T$ to the subspace ${\cap_{i,j=1}^m\ker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{\boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $ρ_{_{\!N_{\boldsymbol T}(w)}}(p) = p\big (N_{\boldsymbol T}(w) \big),\, p\in P[z_1, ..., z_m].$ We study the contractivity and complete contractivity of the homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}.$ Starting from the homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}},$ we construct a natural class of homomorphism $ρ_{_{\!N^{(λ)}(w)}}, λ>0,$ and relate the properties of $ρ_{_{\!N^{(λ)}(w)}}$ to that of $ρ_{_{\!N_{\boldsymbol T}(w)}}.$ Explicit examples arising from the multiplication operators on the Bergman space of $Ω$ are investigated in detail. Finally, it is shown that contractive properties of $ρ_{_{\!N_{\boldsymbol T}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{\boldsymbol T}$.

math.FA↗
arXiv2014

Contractivity, Complete Contractivity and Curvature inequalities

Let $\|\cdot\|_{\mathbf A}$ be a norm on $\mathbb C^m$ given by the formula $\|(z_1,\ldots,z_m)\|_{\mathbf A}=\|z_1A_1+\cdots+z_mA_m\|_{\rm op}$ for some choice of an $m$-tuple of $n\times n$ linearly independent matrices $\mathbf A=(A_1, \ldots, A_m).$ Let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A}.$ %For a holomorphic function $f$ on $Ω_\mathbf A,$ let %$ρ_{V}(f):=\left ( %\begin{smallmatrix} %f(w)I_p& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ %0 & f(w)I_q %\end{smallmatrix}\right ),$ where $V_1, \ldots, V_m$ are $p\times q$ %matrices. Given $p\times q$ matrices $V_1, \ldots, V_m$ and a function $f \in \mathcal O(Ω_\mathbf A),$ the algebra of function holomorphic on an open set $U$ containing the closed unit ball $\barΩ_\mathbf A$ define $$ρ_{V}(f):=\left ( \begin{smallmatrix} f(w)I_p& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ 0 & f(w)I_q \end{smallmatrix}\right ),$$ $w\in Ω_\mathbf A.$ Clearly, $ρ_{V}$ defines an algebra homomorphism. We study contractivity (resp. complete contractivity) of such homomorphisms. The characterization of those balls in $\mathbb C^2$ for which contractive linear maps are always completely contractive remained open. We answer this question for balls of the form $Ω_\mathbf A$ in $\mathbb C^2.$ The class of homomorphisms of the form $ρ_V$ arise from localization of operators in the Cowen-Douglas class of $Ω.$ The (complete) contractivity of a homomorphism in this class naturally produces inequalities for the curvature of the corresponding Cowen-Douglas bundle. This connection and some of its very interesting consequences are discussed.

math.FA↗