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A proof of Hartman's theorem on compact Hankel operators

Published online by Cambridge University Press:  24 October 2008

F. F. Bonsall  and
S. C. Power
F. F. Bonsall
Affiliation:
University of Edinburgh
S. C. Power
Affiliation:
University of Edinburgh
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Extract

Let U be a unilateral shift of arbitrary (perhaps uncountable) multiplicity on a Hilbert space. Following Rosenblum (5), an operator A is said to be a Hankel operator relative to U if

Hartman (2) has characterized the compact Hankel operators relative to the unilateral shift of multiplicity one as the Hankel operators with symbol in H∞ + C(T). Using the usual function space model for representing the unilateral shift, Page ((4), theorem 10) has extended Hartman's theorem to unilateral shifts of countable multiplicity. We give a model-free proof of Hartman's theorem which applies to shifts of arbitrary multiplicity. The proof turns on the observation that a compact operator acts compactly on a certain algebra of operators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 3 , November 1975 , pp. 447 - 450
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

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