# American Institute of Mathematical Sciences

September  2022, 12(3): 469-480. doi: 10.3934/naco.2021015

## 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

Received  May 2020 Revised  April 2021 Published  September 2022 Early access  May 2021

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.

Citation: Abdessalam Kara, Said Guedjiba. Some representations of moore-penrose inverse for the sum of two operators and the extension of the fill-fishkind formula. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 469-480. doi: 10.3934/naco.2021015
##### 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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##### 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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