On intertwining operators
type: Item
instance of: scholarly article
title: On intertwining operators
zbMATH Open document ID: 0652.47019
DOI: 10.1007/BF01298834
published in: Monatshefte für Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178397
review text: Let B(H) denote the algebra of operators on the Hilbert space H into itself. Given \(A,B\in B(H)\), define C(A,B) and R(A,B):B(H)\(\to B(H)\) by \(C(A,B)X=AX-XB\) and \(R(A,B)X=AXB-X\). Our purpose in this note is a two fold one. We show firstly that if A and \(B^*\in B(H)\) are dominant operators such that the pure part of B has non-trivial kernel, then \(C^ n(A,B)X=0\), n some natural number, implies that \(C(A,B)X=C(A^*,B^*)X=0\). Secondly it is shown that if A and \(B^*\) are contractions with \(C_{.0}\) completely non-unitary parts, then \(R^ n(A,B)X=0\) for some natural number n implies that \(R(A,B)X=R(A^*,B^*)X=C(A,B^*)X=C(A^*,B)X=0\). In the particular case in which X is compact, it is shown that this result extends to all contractions A and B.
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4063852
zbMATH Keywords: dominant operators
zbMATH Keywords: contractions with \(C_{.0}\) completely non-unitary parts
author: B. P. Duggal
reviewed by: B. P. Duggal
MaRDI profile type: Publication
cites work: Q5566804
cites work: Contractions with a Unitary Part
cites work: On dominant operators
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cites work: Q5600360
cites work: The unitary part of paranormal operators
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cites work: Q4090924
cites work: Q3711168
OpenAlex ID: W1968986614
description: scientific article; zbMATH DE number 4063852