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EP Operators and Generalized Inverses

Published online by Cambridge University Press:  20 November 2018

Stephen L. Campbell  and
Carl D. Meyer
Stephen L. Campbell
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27607
Carl D. Meyer
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27607
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Abstract

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The relationship between properties of the generalized inverse of A, A†, and of the adjoint of A, A*, are studied. The property that A†A and AA† commute, called (E4), is investigated. (E4) generalizes the property of A being EPr. A canonical form and a formula for A† are given if a matrix A is (E4). Results are in a Hilbert space setting whenever possible. Examples are given.

Keywords

15A0915A2147A65Generalized inverseEP operatorsEPr matricescanonical form
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 3 , August 1975 , pp. 327 - 333
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Baskett, T. S. and Katz, I. J., Theorems on products ofEPr matrices, Linear Algebra and Its Applications, 2 (1969), 87-103.Google Scholar
2. Beutler, F. J., The operator theory of the pseudoinverse I: bounded operators, J. Math. Anal. Appl. 10 (1965), 451-470.Google Scholar
3. Brown, Arlen, On a class of operators, Proc. Amer. Math. Soc. (1953), 723-728.Google Scholar
4. Campbell, Stephen L., Linear operators for which T*T and TT* commute, Proc. Amer. Math. Soc, 34 (1972), 177-180.Google Scholar
5. Desoer, C. A. and Wahlen, B. H., A note on pseudoinverses, SIAM J. Appl. Math., 11 (1963), 442-447.Google Scholar
6. Dunford, Nelson and Schwartz, Jacob T., Linear Operators Part II, Interscience Publishers, New York, 1963.Google Scholar
7. Embry, Mary R., Conditions implying normality in Hilbert space, Pacific J. Math., 18 (1966), 457-460.Google Scholar
8. Halmos, Paul R., A Hilbert Space Problem Book, D. Van Nostrand Co., Princeton, New Jersey, 1967.Google Scholar
9. Katz, I. J. and Pearl, M. H., On EPr and normal EPr matrices, J. of Research of the N.B.S., Series B, 70B (1966), 47-75.Google Scholar
10. Meyer, Carl D., Some remarks on EPr matrices and generalized inverses, Linear Algebra and Its Applications, 3 (1970), 275-278.Google Scholar
11. Pearl, Martin H., On normal and EPr matrices, Mich. Math. J., 6 (1959), 1-5.Google Scholar
12. Pearl, Martin H., On generalized inverses of matrices, Proc. Cambridge Phil. Soc. 62 (1966), 673-677.Google Scholar
13. Petryshyn, W. V., On generalized inverses and on the uniform convergence of(I—BK)n with applications to iterative methods, J. Math. Anal. Appl., 18 (1967), 417-439.Google Scholar
14. Schwerdtfeger, H., Introduction to Linear Algebra and the Theory of Matrices, P. Noordhoff N.V., Groningen-Holland, 1961.Google Scholar
15. Yosida, Kôsaku, Functional Analysis, Springer Verlag, New York, 1968.Google Scholar