Estimates for norms of resolvents of operators on tensor products of Hilbert spaces
type: Item
instance of: scholarly article
title: Estimates for norms of resolvents of operators on tensor products of Hilbert spaces
zbMATH Open document ID: 1082.47006
published in: Periodica Mathematica Hungarica
publication date: 2005-07-05T00:00:00Z
review text: Let \(H= E_{1}\otimes E_{2}\) be the tensor product of the Hilbert spaces \(E_{1}\), \(E_{2}\) and let \(M_{1}\), \(M_{2}\) be operators acting on \(E_{1}\), \(E_{2}\), respectively. Under the conditions that the domains of \(M_{j}\) and \(M_{j}^\ast\) coincide and that the difference \(M_{j} -M_{j}^\ast \in \tilde{C}_{2p_{j}}\) for some Neumann--Schatten class ideal \(\tilde{C}_{2p_{j}}\), \(p_{j}\in {\mathbb N}\), the author shows for the operator \(A= M_{1} \otimes I_{2} + I_{1}\otimes M_{2}\) the estimate \[ \| (A-\lambda)^{-1}\| E \leq \sum_{n=0}^{\infty} {b_{n}(M_{1},M_{2}) \over \varrho(A,\lambda)^{n+1}}, \] where \(\varrho(A,\lambda) = \inf_{t\in\sigma(A)}| t-\lambda| \). The general result is improved in the special cases where \(H\) is finite-dimensional or \(M_{j}-M_{j}^\ast\), \(j=1,2\), are Hilbert--Schmidt operators. As an application, estimates for the spectral variation are given in the situation \(B= A +Z\), where \(Z\) is bounded and \(A\) of the form given above. Finally, an integro-differential operator is discussed.
reviewed by: Eckhard Giere
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47G20
Mathematics Subject Classification ID: 47A80
zbMATH DE Number: 2186367
zbMATH Keywords: linear operators
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: tensor products
zbMATH Keywords: spectrum
zbMATH Keywords: integro-differential operators
author: Michael I. Gil'
MaRDI profile type: Publication
DOI: 10.1023/B:MAHU.0000040537.06348.2D
description: scientific article; zbMATH DE number 2186367