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Some representations of moore-penrose inverse for the sum of two operators and the extension of the fill-fishkind formula
University of Batna 2, Faculty of Mathematics and Computer Sciences, Department of Mathematics, Algeria |
In the setting of arbitrary Hilbert spaces, we give a representation of M-P inverse of the sum of linear operators $ A+B $ under suitable conditions. Based on the full-rank decomposition of an operator, we prove that the extension of the Fill-Fishkind formula for $ A $ and $ B $ with closed ranges, remains valid, keeping the same conditions of Fill-Fishkind formula for two matrices, also we obtain an analogous formula under the Fill-Fishkind conditions, beyond we derive some representations of M-P inverse of a 2-by-2 block operator with disjoint ranges.
References:
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M. L. Arias, G. Corach and A. Maestripieri,
Range additivity, shorted operator and the Sherman- Morrison -Woodbury formula, Linear Algebra Appl., 467 (2015), 86-99.
doi: 10.1016/j.laa.2014.11.001. |
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A. Ben-Israel and T. N. E Greville, Generalized Inverses, Theory and Applications, 2nd ed Berlin Springer, New York, 2003. |
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S. L. Campbell and C. D Meyer, Generalized Inverses of Linear Transformations, Dover Publ., New York, 1979.
doi: 10.1137/1.9780898719048. |
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S. R. Caradus, Generalized Inverses and Operator Theory, Queen's paper in pure and applied mathematics, Queen's University, Kingston, 1978. |
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R. E. Cline,
Representations for the generalized inverse of sum of matrices, SIAM J. Numer. Anal., 2 (1965), 99-114.
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M. S. Djikić,
Extensions of the Fill–Fishkind formula and the infimum – parallel sum relation, Linear and Multilinear Algebra, 64 (2016), 2335-2349.
doi: 10.1080/03081087.2016.1155532. |
[8] |
D. S. Djordjević and N. Č. Dinčić,
Reverse order law for Moore-Penrose inverse, Journal Math. Anal. Appl., 361 (2010), 252-261.
doi: 10.1016/j.jmaa.2009.08.056. |
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D. S. Dordević and P. S. Stanimirović,
General representations of pseudoinverses, Matematicki vesnik, 51 (1999), 69-76.
|
[10] |
J. A. Fill and D. E. Fishkind,
The Moore–Penrose generalized inverse for sums of matrices, SIAM J. Matrix Anal. Appl., 21 (1999), 629-635.
doi: 10.1137/S0895479897329692. |
[11] |
J. Groß,
On oblique projection, rank additivity and the Moore-Penrose inverse of the sum of two matrices, Linear and Multilinear Algebra, 46 (1999), 265-275.
doi: 10.1080/03081089908818620. |
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M. R. Hestenes,
Relative hermitian matrices, Pacific Journal Math., 11 (1961), 225-245.
doi: 10.2140/pjm.1961.11.225. |
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S. Izumino, Product of operators with closed range and an extension of the revers order law, Tôhoku. Math. J., 34 (1982), 43–52.
doi: 10.2748/tmj/1178229307. |
show all references
References:
[1] |
M. L. Arias, G. Corach and A. Maestripieri,
Range additivity, shorted operator and the Sherman- Morrison -Woodbury formula, Linear Algebra Appl., 467 (2015), 86-99.
doi: 10.1016/j.laa.2014.11.001. |
[2] |
A. Ben-Israel and T. N. E Greville, Generalized Inverses, Theory and Applications, 2nd ed Berlin Springer, New York, 2003. |
[3] |
S. L. Campbell and C. D Meyer, Generalized Inverses of Linear Transformations, Dover Publ., New York, 1979.
doi: 10.1137/1.9780898719048. |
[4] |
S. R. Caradus, Generalized Inverses and Operator Theory, Queen's paper in pure and applied mathematics, Queen's University, Kingston, 1978. |
[5] |
R. E. Cline,
Representations for the generalized inverse of sum of matrices, SIAM J. Numer. Anal., 2 (1965), 99-114.
doi: 10.1137/0702008. |
[6] |
F. Deutsch, The angle between subspaces of a Hilbert space, in Approximation Theory, Wavelets and Applications(ed. S. P. Singh), Kluwer Academic Publ., (1995), 107–130.
doi: 10.1007/978-94-015-8577-4_7. |
[7] |
M. S. Djikić,
Extensions of the Fill–Fishkind formula and the infimum – parallel sum relation, Linear and Multilinear Algebra, 64 (2016), 2335-2349.
doi: 10.1080/03081087.2016.1155532. |
[8] |
D. S. Djordjević and N. Č. Dinčić,
Reverse order law for Moore-Penrose inverse, Journal Math. Anal. Appl., 361 (2010), 252-261.
doi: 10.1016/j.jmaa.2009.08.056. |
[9] |
D. S. Dordević and P. S. Stanimirović,
General representations of pseudoinverses, Matematicki vesnik, 51 (1999), 69-76.
|
[10] |
J. A. Fill and D. E. Fishkind,
The Moore–Penrose generalized inverse for sums of matrices, SIAM J. Matrix Anal. Appl., 21 (1999), 629-635.
doi: 10.1137/S0895479897329692. |
[11] |
J. Groß,
On oblique projection, rank additivity and the Moore-Penrose inverse of the sum of two matrices, Linear and Multilinear Algebra, 46 (1999), 265-275.
doi: 10.1080/03081089908818620. |
[12] |
M. R. Hestenes,
Relative hermitian matrices, Pacific Journal Math., 11 (1961), 225-245.
doi: 10.2140/pjm.1961.11.225. |
[13] |
S. Izumino, Product of operators with closed range and an extension of the revers order law, Tôhoku. Math. J., 34 (1982), 43–52.
doi: 10.2748/tmj/1178229307. |
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