On numerical ranges and roots.
type: Item
instance of: scholarly article
title: On numerical ranges and roots.
zbMATH Open document ID: 1039.47003
DOI: 10.1016/S0022-247X(03)00158-6
author: Chi-Kwong Li
author: Leiba Rodman
author: Ilya M. Spitkovsky
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \({\mathfrak a}\) be a complex Banach algebra with unit \(e\). The algebraic numerical range \(V(a)\) of an element \(a\) in \({\mathfrak a}\) is the set \(\{f(a): f\) is a bounded linear functional on \({\mathfrak a}\) such that \(f(e)= \| f\|=1\}\). It is shown that if \(V(a)\) contains no negative real number, then for any \(w\), \(0< w< 1\), there exists a \(b_w\), the \(w\)th power of \(a\), in the closed unital subalgebra of \({\mathfrak a}\) generated by \(a\) whose spectrum is contained in the wedge \(\{re^{i\theta}\in \mathbb{C}: r\geq 0\), \(|\theta|< w\pi\}\) such that \(b_{w_1} b_{w_2}= a\) if \(w_1+w_2=1\). It is unique if \(a\) is invertible and \(w\) is the reciprocal of an integer. For a Hermitian Banach *-algebra \({\mathfrak a}\) (a Banach algebra with a conjugate linear involution * satisfying \((xy)^*= y^*x^*\) for all \(x\) and \(y\) such that every Hermitian element has real spectrum), the numerical range \(V(a)\) should be replaced by the *-numerical range \(V_*(a)= \{f(a): f\) as before with \((fx^*x)\geq 0\) for all \(x\) in \({\mathfrak a}\}\). If \({\mathfrak a}\) further satisfies the condition that \(V_*(a)\) contains \(\sigma(a)\) for every \(a\) in \({\mathfrak a}\), then it is proved that under the previous assumptions on \(a\) and \(w\), there exists \(b_w\) such that \(V_*(b_w)\) is itself contained in the wedge \(\{re^{i\theta}: r\geq 0\), \(|\theta|\leq w\pi\}\). These results generalize the corresponding ones for operators on a Hilbert space.
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 1956920
zbMATH Keywords: root
zbMATH Keywords: Hermitian Banach algebra
zbMATH Keywords: numerical range
MaRDI profile type: Publication
cites work: Q5608364
cites work: Q5678038
cites work: Convolution operators and factorization of almost periodic matrix functions
cites work: Q3999065
cites work: Q5341026
cites work: Square roots with positive definite Hermitian part
cites work: Uniqueness of matrix square roots under a numerical range condition
cites work: Uniqueness of matrix square roots and an application
cites work: Perturbation theory for linear operators.
cites work: Über die Wurzeln eines maximalen dissipativen Operators
cites work: Induced norms, states, and numerical ranges
cites work: Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States
cites work: Q5597817
cites work: On the square roots of strictly quasiaccretive complex matrices
cites work: Q4293544
cites work: Banach algebras and the general theory of \(^*\)-algebras. II: \(^*\)-algebras
cites work: Q5690019
cites work: On Banach algebra elements of thin numerical range
cites work: Growth conitions and the numerical range in a Banach algebra
full work available at URL: s0022-247x(03)00158-6
OpenAlex ID: W1990307743
Recommended article: Numerical range and Poncelet property.
description: scientific article; zbMATH DE number 1956920