Numerical range of two operators in semi-inner product spaces
type: Item
instance of: scholarly article
title: Numerical range of two operators in semi-inner product spaces
zbMATH Open document ID: 1258.47010
DOI: 10.1155/2012/846396
published in: Abstract and Applied Analysis
publication date: 2012-05-14T00:00:00Z
review text: The authors define, for any two operators \(T\) and \(A\) on a complex semi-inner product space \((X,[.,.])\) with domains \(D(T)\) and \(D(A)\), their numerical range \(W(T,A)\equiv \{[Tx, Ax]:\| Ax\|= 1\), \(x\in D(T)\cap D(A)\}\), the spectrum \(\sigma(T,A)\equiv \{\lambda\in\mathbb{C}: T-\lambda\) is not invertible\}, and other familiar spectral subsets of the latter. Then they show that \(\sigma(T,A)\) is contained in the closure of \(W(T,A)\), thus generalizing the classical result for a single operator \(T\). The authors proceed to define such sets associated with two nonlinear operators \(T\) and \(A\), and prove an analogous containment relation between the approximate point spectrum and the closure of the numerical range of \(T\) and \(A\). Finally, they establish an approximation method for solving the operator equation \((A-T)y= Ax\) for nonlinear \(T\) and \(A\) on a complex Banach space.
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 6034251
zbMATH Keywords: semi-inner product space
zbMATH Keywords: numerical range
zbMATH Keywords: spectrum
Wikidata QID: Q58696929
author: N. K. Sahu
author: Chandal Nahak
author: Sudarsan Nanda
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/846396
OpenAlex ID: W2153399013
cites work: Semi-Inner-Product Spaces
cites work: Classes of Semi-Inner-Product Spaces
cites work: A numerical range for two linear operators
cites work: The closure of the numerical range contains the spectrum
cites work: Q5608364
cites work: Q5678038
cites work: The numerical range of a tridiagonal operator
cites work: The numerical radius of a weighted shift operator with geometric weights
cites work: On geometric properties of the numerical range
cites work: Spectra of products and numerical ranges
cites work: Q3530481
cites work: The numerical range of nonlinear Banach space operators
cites work: A note on numerical ranges of operators
description: scientific article; zbMATH DE number 6034251