Spectral inclusion properties of the numerical range in a space with an indefinite metric
type: Item
instance of: scholarly article
title: Spectral inclusion properties of the numerical range in a space with an indefinite metric
zbMATH Open document ID: 1228.47006
author: Deyu Wu
author: Alatancang
published in: Linear Algebra and its Applications
publication date: 2011-07-15T00:00:00Z
review text: A classical result in operator theory is the spectral inclusion property for bounded operators in Hilbert spaces: all eigenvalues of a bounded operator are included in the numerical range and the spectrum is included in the closure of the numerical range. In a Krein space, there exists an adopted notion of a numerical range \(W_{\mathcal J}(A)\) with respect to the indefinite inner product \([\cdot, \cdot]\) for a (possibly unbounded) operator \(A\), i.e., \[ W_{\mathcal J}(A) := \{[Ax,x][x,x]^{-1} : x \in \text{dom}\,A,\quad [x,x]\neq 0\}. \] For non-negative operators (i.e., \([Ax,x]\geq 0\) for \(x\in \text{dom}\, A\) and \(\rho(A)\neq \emptyset\)), the spectral inclusion property for \(W_{\mathcal J}(A)\) is a consequence of the spectral function, see, e.g., \textit{H.\ Langer} [Lect.\ Notes Math.\ 948, 1--46 (1982; Zbl 0511.47023)]. Here, the authors give an elementary proof of this fact which is direct and does not make use of the spectral function.
reviewed by: Carsten Trunk
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47B50
zbMATH DE Number: 5924524
zbMATH Keywords: numerical range
zbMATH Keywords: \(\mathcal J\)-numerical range
zbMATH Keywords: indefinite inner product space
zbMATH Keywords: uniformly \(\mathcal J\)-positive operator
MaRDI profile type: Publication
full work available at URL: j.laa.2011.02.053
OpenAlex ID: W1975936763
cites work: Shapes and computer generation of numerical ranges of Krein space operators
cites work: Numerical ranges of an operator on an indefinite inner product space
cites work: Q4403356
cites work: On the geometry of numerical ranges in spaces with an indefinite inner product
cites work: On Generalized Numerical Ranges of Operators on an Indefinite Inner Product Space
cites work: Numerical range of linear operators in spaces with indefinite metric
cites work: Q4692935
DOI: 10.1016/J.LAA.2011.02.053
description: scientific article; zbMATH DE number 5924524