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On the resolvent and the principal vectors of a compact linear operator

Published online by Cambridge University Press:  24 October 2008

J. R. Ringrose
J. R. Ringrose
Affiliation:
St John's College, Cambridge
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Extract

Let T be a compact linear operator acting in a complex Hilbert space H. Then there is a simple resolution of the identity {Eλ} in H which reduces T (for a proof of this statement, and for definitions of the terms used, see (l), section 5, especially Theorem 6). In the present paper we give constructions for the resolvent of T, and for a complete set of principal (that is, eigen and adjoined) vectors, in terms of T, {Eλ}, and the diagonal coefficients {αλ} of T. These constructions require no technique more involved than the expansion of a resolvent operator in a Neumann series.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 525 - 531
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1) Ringrose, J. R. Super-diagonal forms for compact linear operators. Proc. London Math. Soc. (3), 12 (1962), 367–384.CrossRefGoogle Scholar
(2) Ringrose, J. R. On the triangular representation of integral operators. Proc. London Math. Soc. (3), 12 (1962), 385–399.CrossRefGoogle Scholar