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Representation of Hilbert space operators by (nJ)-matrices

Published online by Cambridge University Press:  24 October 2008

D. R. Smart
D. R. Smart
Affiliation:
Christ's College Cambridge
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Extract

Introduction. Let be the complex separable Hilbert space. We say that the closed linear operator T, with domain dense in. , is represented by the infinite matrix H if T is the operator T˜1(H) defined† by H (with respect to some complete orthonormal set). We define an (nJ)-matrix as a Hermitian matrix H = [hij]i, j ≥ 1 for which hij = 0 when i − j > n and hij ╪ 0 when i − j = n. (Thus a Jacobi matrix is a (1J)-matrix.) If, in addition, hij = 0 when 0 < i − j < n, we call H an (nJ ┴)-matrix.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 304 - 311
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1) Stone, M. H. Linear transformations in Hilbert space and their application to analysis (New York, 1932).Google Scholar
(2) Hamburger, H. L. Hermitian transformations of deficiency index (1, 1), Jacobi matrices and undetermined moment problems. Amer. J. Math. 66 (1944), 489–522.Google Scholar