Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set
type: Item
instance of: scholarly article
title: Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set
zbMATH Open document ID: 1475.47017
author: Gadadhar Misra
author: Avijit Pal
published in: Journal d'Analyse Mathématique
publication date: 2019-01-30T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{C}^m\). The Cowen-Douglas class consists of \(m\)-tuples of commuting bounded operators \(\mathbf{T}=(T_1,\dots,T_m)\) on a Hilbert space \(\mathcal{H}\) such that for \(w=(w_1,\dots,w_m)\in\Omega\), the dimension of the joint kernel \(\cap_{k=1}^m\ker(T_k-w_kI)\) is equal to \(1\); for \(w\in\Omega\) and \(h\in\mathcal{H}\), the operator \(D_{\mathbf{T}-wI}:\mathcal{H}\to\mathcal{H}\oplus\dots\oplus\mathcal{H}\) defined by \(D_{\mathbf{T}-wI}=((T_1-w_1I)h,\dots,(T_m-w_mI)h))\) has closed range; the closed linear span of \(\left\{\cap_{k=1}^m\ker (T_k-w_kI):w\in\Omega\right\}\) coincides with \(\mathcal{H}\). For an \(m\)-tuple \(\mathbf{T}\in B_1(\Omega)\), let \(N_{\mathbf{T}}(w)\) denote the restriction of \(\mathbf{T}\) to the subspace \(\mathcal{N}(w):=\cap_{i,j=1}^m\ker (T_i-w_iI)(T_j-w_jI)\). The authors study the contractivity and complete contractivity of homomorphisms of the polynomial ring \(P[z_1,\dots,z_m]\) defined by \(\rho_{N_{\mathbf{T}}(w)}(p)=p(N_{\mathbf{T}}(w))\) for \(p\in P[z_1,\dots,z_m]\). Some explicit examples arising from multiplication operators on the Bergman space of \(\Omega\) are considered in detail. For operators in \(B_1(\Omega)\), there exists a holomorphic choice \(\gamma(w)\) of eigenvalues at \(w\). The map \(\gamma\) defines a holomorphic Hermitian vector bundle \(E_{\mathbf{T}}\) on \(\Omega\). One can define the curvature \(\mathcal{K}_{\mathbf{T}}(w):=-\left(\left( \frac{\partial^2}{\partial w_i\partial\overline{w_j}}\log\|\gamma(w)\|^2 \right)\right)_{i,j=1}^m\). It is shown that contractive properties of \(\rho_{N_{\mathbf{T}}(w)}\) are equivalent to an inequality for the curvature of the Cowen-Dougles bundle \(E_{\mathbf{T}}\).
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B13
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 7008548
zbMATH Keywords: Cowen-Douglas class
zbMATH Keywords: localization operators
zbMATH Keywords: Bergman kernel
zbMATH Keywords: curvature inequalities
MaRDI profile type: Publication
full work available at URL: s11854-018-0054-7
OpenAlex ID: W2904680785
cites work: Homogeneous multiplication operators on bounded symmetric domains.
cites work: Contractive homomorphisms and tensor product norms
cites work: Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class
cites work: Q3425149
cites work: Complex geometry and operator theory
cites work: Q3709323
cites work: Generalized Bergman Kernels and the Cowen-Douglas Theory
cites work: Function spaces and reproducing kernels on bounded symmetric domains
cites work: Q5349018
cites work: Q4769499
cites work: Q4744553
cites work: Invariant distances and metrics in complex analysis
cites work: Q3924489
cites work: Analytic Invariants of Bounded Symmetric Domains
cites work: Q3333587
cites work: Completely contractive Hilbert modules and Parrott's example
cites work: Bounded modules, extremal problems, and a curvature inequality
cites work: Completely bounded modules and associated extremal problems
cites work: Representations of function algebras, abstract operator spaces, and Banach space geometry
cites work: Function theory in the unit ball of \({\mathbb{C}}^ n\)
DOI: 10.1007/S11854-018-0054-7
description: scientific article; zbMATH DE number 7008548