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Range inclusion for normal derivations

Published online by Cambridge University Press:  18 May 2009

C. K. Fong
C. K. Fong
Affiliation:
Department of Mathematics, University of Toronto, Toronto, CanadaM5S 1A1
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For a (bounded, linear) operator A in a (complex, infinite-dimensional, separable) Hilbert space ℋ, the inner derivation DA as an operator on ℬ(ℋ), is defined by DAX = AX – XA. Johnson and Williams [4] showed that, when A is a normal operator, range inclusion DBℬ(ℋ)⊆DA(ℋ)⊆ is equivalent to the condition that B = f(A), where f is a Lipschitz function on σ(A) such that t(z, w)(f(z)–f(w))/(z–w) is a trace class kernel on L2(μ) whenever t(z, w) is such a kernel. (Here μ is the dominating scalar valued spectral measure of A constructed in multiplicity theory). This result is deep and its proof is difficult. In the present paper, we establish the following analogous result which is easier to prove: for a normal operator A, range inclusion DB℘2(ℋ) holds if and only if B = f(A) for some Lipschitz function f on σ(A). Here ℘(ℋ) stands for the Hilbert-Schmidt class of operators on ℋ. As by-products of our argument, we generalize some results in [4], [8], [9] concerning the non-existence of a one-sided ideal contained in certain derivation ranges; for example, we show that if A is hyponormal and if the point spectrum σP(A*) of A* is empty, then DAℬ(ℋ) does not contain any nonzero right ideal.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 2 , July 1984 , pp. 255 - 262
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

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