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A generalized inverse for arbitrary operators between Hilbert spaces†

Published online by Cambridge University Press:  24 October 2008

I. Erdelyi
I. Erdelyi
Affiliation:
Temple University, Philadelphia
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Abstract

A function analytic approach to the generalized inversion problem, for arbitrary operators between Hilbert spaces, is investigated in the present paper.

The generalized inverse is defined as the inverse of the largest invertible restriction of the given operator and it is extended by zero to the orthogonal complement of the range of the invertible restriction. This domain-dense operator satisfies, on some restricted manifolds, the defining properties of the generalized inverse in some special cases, and it provides the least extremal solution of possibly inconsistent linear equations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 43 - 50
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Arghiriade, E.Sur l'inverse généralisée d'un opérateur linéaire dans les espaces de Hilbert. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. ser. VIII, Vol. 45 (1968), 471–477.Google Scholar
(2)Ben-Israel, A. and Charnes, A.Contributions to the theory of generalized inverses. J. Soc. Indust. Appl. Math. 11 (1963), 667–699.CrossRefGoogle Scholar
(3)Beutler, F. J.The operator theory of the pseudo-inverse. II Unbounded operators with arbitrary range. J. Math. Anal. Appl. 10 (1965), 471–493.CrossRefGoogle Scholar
(4)Erdelyi, I. and Ben-Israel, A. Least external solutions of linear equations and generalized inversion between Hilbert spaces. J. Math. Anal. Appl. (In Press).Google Scholar
(5)Hestenes, M. R.Relative self-adjoint operators in Hilbert space. Pacific J. Math. 11, (1961), 1315–1357.CrossRefGoogle Scholar
(6)Moore, E. H.General Analysis I. Mem. Amer. Phil. Soc. I., 1935.Google Scholar
(7)Tseng, Y. Y.Generalized inverses of unbounded operators between two unitary spaces (in Russian). Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 431–434(reviewed in Math. Rev. 11 (1950), 115).Google Scholar
(8)von Neumann, J., Functional Operators, Vol. II: The Geometry of Orthogonal Spaces. Annals of Mathematics Studies No. 22, Princeton University Press (Princeton, n Jersey, 1950).Google Scholar