Computation of functions of certain operator matrices
type: Item
instance of: scholarly article
title: Computation of functions of certain operator matrices
zbMATH Open document ID: 0802.65069
DOI: 10.1016/0024-3795(93)90111-Z
author: Lawrence A. Harris
published in: Linear Algebra and its Applications
publication date: 1994-12-11T00:00:00Z
review text: The following result is obtained: Theorem 1. Let \(B_ 1\), \(C_ 1\in {\mathcal L}(H,K_ 1)\) and \(B_ 2\), \(C_ 2\in {\mathcal L}(H,K_ 2)\). Suppose \(f\) is a function which is holomorphic in an open set \(D\) containing the spectrum \((C^*_ 1 B_ 1+ C^*_ 2 B_ 2)\) and 0. Then \[ f\left(\left[{B_ 1 C^*_ 1\atop B_ 2 C^*_ 1} {B_ 1 C^*_ 2\atop B_ 2 C^*_ 2}\right]\right)= \left[ {f(0)I+ B_ 1 RC^*_ 1\atop B_ 2 RC^*_ 1} {B_ 1 R^*_ 2\atop f(0)I+ B_ 2 RC^*_ 2}\right], \] where \(R= g(C^*_ 1 B_ 1+ C^*_ 2 B_ 2)\) and \(g\) is the holomorphic extension to \(D\) of \(g(z)= [f(z)- f(0)]/z\). Here \(H\), \(K_ 1\), \(K_ 2\) are Hilbert spaces and \({\mathcal L}(H,K)\) denotes the space of bounded linear operators from \(H\) to \(K\). Some illustrations are given.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 540055
zbMATH Keywords: functions of operator matrices
zbMATH Keywords: functional calculus
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: bounded linear operators
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
cites work: On nth Roots of Positive Operators
cites work: Q5660202
cites work: Q4024768
cites work: Q3244654
cites work: Q3249198
description: scientific article; zbMATH DE number 540055