The numerical range of positive operators on Banach lattices
type: Item
instance of: scholarly article
title: The numerical range of positive operators on Banach lattices
zbMATH Open document ID: 1342.47007
author: Agnes Radl
published in: Positivity
publication date: 2015-09-22T00:00:00Z
review text: Let \(A\) be an operator in a Banach space \(X\). The spatial numerical range of \(A\), which is an extension of the concept of numerical range of operators on a Hilbert space, is defined as \[ V(A)=\{\varphi(Ax): x\in X,\;\varphi\in X',\;\|\varphi\|=\|x\|=\varphi(x)=1\}, \] and let \(v(A)=\sup\{|\lambda|: \lambda\in V(A)\}\). For a positive operator \(A\) in a Banach lattice, the authors consider the following four main questions: {\parindent=6mm \begin{itemize} \item[(1)] Is \(v(A)\in V(A)\)? \item [(2)] For a regular operator \(B\) with \(|B|\leq A\), is \(v(B)\leq v(B)\)? \item [(3)] Is \(V(A)\) symmetric with respect to the real axis? \item [(4)] For an irreducible element \(A\) with \(e^{i\theta}v(A)\in V(A)\), is \(V(A)=e^{i\theta}V(A)\)? \end{itemize}} The authors investigate these questions and answer them for both the numerical radius and the block numerical radius.
reviewed by: Maryam Khosravi
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 6484288
zbMATH Keywords: numerical range
zbMATH Keywords: block numerical range
zbMATH Keywords: Banach lattices
zbMATH Keywords: positive operators
describes a project that uses: Eigtool
MaRDI profile type: Publication
full work available at URL: s11117-014-0318-0
OpenAlex ID: W1982376517
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DOI: 10.1007/S11117-014-0318-0
description: scientific article; zbMATH DE number 6484288