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On prespectrality of generalised derivations*

Published online by Cambridge University Press:  14 November 2011

Milan Hladnik
Milan Hladnik
Affiliation:
E. K. University of Ljubljana, Institute of Mathematics, Physics and Mechanics, Jadranska 19, 61000 Ljubljana, Yugoslavia
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Synopsis

In this paper it is proved that, for scalar-type operators a and b on an infinite dimensional separable complex Hilbert space H, the generalised derivation Δa,b, defined for bounded linear operators x onℋ by the equation Δa,bx = ax − xb, is a (scalar-type) prespectral operator of the class (the trace class operators on ℋ) if and only if at least one of the spectra σ(a)or σ(b)is finite. It is shown also that the same condition is necessary and sufficient for Δa,b restricted to any one of the von Neumann-Schatten classes(p≠2) to be a spectral operator (of scalar type). Our results may be compared with those of J. Anderson and C. Foiaş, who established in [1] that, for scalar-type a, b, Δa,b is a (scalar-type) spectral operator if and only if both spectra, σ(a) and σ(b), are finite. However, we use different and more direct methods to show the existence or nonexistence of the spectral resolution of identity for Δa,b.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 93 - 106
Copyright
Copyright © Royal Society of Edinburgh 1986

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