Norm estimates for functions of matrices with simple spectrum
type: Item
instance of: scholarly article
title: Norm estimates for functions of matrices with simple spectrum
zbMATH Open document ID: 1205.15038
author: Michael I. Gil'
publication date: 2010-10-14T00:00:00Z
review text: In 1958 \textit{I. M. Gelfand} and \textit{G. E. Schilow} [Some Questions of Theory of Differential Equations. Moscow: Nauka (1958); cf. Einige Fragen zur Theorie der Differentialgleichungen. Berlin: VEB Deutscher Verlag der Wissenschaften. 195 S. (1964; Zbl 0133.37503)] established an estimate for the norm of a~regular matrix-valued function in connection with their investigations of partial differential equations. However, the estimated obtained was not sharp. In the present paper the author establishes sharp estimates for the norm of matrix-valued functions of a~matrix having geometrically simple eigenvalues~\((\mu_n)\). The regularity of the function in question is not assumed. The main result is as follows: Let all the eigenvalues of an~\(n\times n\) complex matrix be geometrically simple, and let \(f\)~be a~function defined on~\(\sigma(A)\). Then \[ \|f(A)-f(\mu_m)I\|\leqslant \gamma_0 \sum^{m-1}_{k=1} |f(\mu_k)-f(\mu_{k+1})|, \] where \(\gamma_0= 1+\max_{j\leq m} l(L_j) p(A,\rho_j)\). If in addition the condition \(f(\mu_{k+1})\leqslant f(\mu_k)\), \(k=1,\dotsc,m-1)\), holds, then \(\|f(A)-f(\mu_m)I\|\leq \gamma_0|f(\mu_1) - f(\mu_m)|\). Here \(l(L_j)=(2\pi)^{-1} \int_{L_j}|dz|\), \(L_j\) is a~Jordan closed contour surrounding the eigenvalues \(\mu_1,\dots,\mu_j\), \(p(A,x)=\sum^{n-1}_{k=1} g^k(A)/[x^{k+1}\sqrt{k!}]\), \(x>0\), and \(\rho_j=\min\{|\lambda-\mu_k|\mid \lambda\in L_j\), \(k=1,\dotsc,j\}\). Also, applications of the estimates obtained to the periodic problem and two-point boundary value problem for vector differential equations are discussed.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 5800238
zbMATH Keywords: matrix-valued functions
zbMATH Keywords: simple spectrum
zbMATH Keywords: differential equations
zbMATH Keywords: periodic problem
zbMATH Keywords: boundary value problem
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: s12215-010-0016-0
OpenAlex ID: W2124063272
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DOI: 10.1007/S12215-010-0016-0
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5800238