Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-16T22:34:35.697Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1.Clancy, K. F., On the local spectra of seminormal operators, Proc. Amer. Math. Soc. 72 (1978), 473–479.CrossRefGoogle Scholar
2.Embry, M. R., Factorizations of operators on Banach space, Proc. Amer. Math. Soc. 38 (1973), 587–590.CrossRefGoogle Scholar
3.Johnson, B. E., Continuity of linear operators commuting with continuous linear operators, Trans. Amer. Math. Soc. 128 (1967), 88–102.CrossRefGoogle Scholar
4.Johnson, B. E. and Williams, J. P., The range of a normal derivation, Pacific J. Math. 58 (1975), 105–122.CrossRefGoogle Scholar
5.Putnam, C. R., Ranges of normal and subnormal operators, Michigan Math. J. 18 (1971), 33–36.CrossRefGoogle Scholar
6.Schatten, R., Norm ideals of completely continuous operators (Springer-Verlag, 1960).CrossRefGoogle Scholar
7.Stampfli, J. G., Derivations on ℋ(ℋ): the range, Illinois J. Math. 17 (1973), 518–524.Google Scholar
8.Weber, R. E., The range of a derivation and ideals, Pacific J. Math. 50 (1974), 617–624.CrossRefGoogle Scholar
9.Williams, J. P., On the range of a derivation II, Proc. Roy. Irish Acad. Sect. A 74 (1974), 299–310.Google Scholar