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On Permutability and Submultiplicativity of Spectral Radius

Published online by Cambridge University Press:  20 November 2018

W. E. Longstaff  and
H. Radjavi
W. E. Longstaff
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia
H. Radjavi
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5
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Abstract

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Let r(T) denote the spectral radius of the operator T acting on a complex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say that r is permutable on 𝓢 if r(ABC) = r(BAC), for every A,B,C ∈ 𝓢. We say that r is submultiplicative on 𝓢 if r(AB) ≤ r(A)r(B), for every A, B ∈ 𝓢. It is known that, if r is permutable on 𝓢, then it is submultiplicative. We show that the converse holds in each of the following cases: (i) 𝓢 consists of compact operators (ii) 𝓢 consists of normal operators (iii) 𝓢 is generated by orthogonal projections.

Keywords

47A1547D0347A1020M2015A30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 5 , 01 October 1995 , pp. 1007 - 1022
Copyright
Copyright © Canadian Mathematical Society 1995

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