American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions
  • NACO Home
  • This Issue
  • Next Article
    Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications
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

Abdessalam Kara and  Said Guedjiba

University of Batna 2, Faculty of Mathematics and Computer Sciences, Department of Mathematics, Algeria

Received  May 2020 Revised  April 2021 Published  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.

Keywords: Moore-Penrose inverse; Closed range operator. Sum of operators, Disjoint ranges.
Mathematics Subject Classification: Primary: 15A09, Secondary: 47A05.
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 & Optimization, 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.  Google Scholar

[2]

A. Ben-Israel and T. N. E Greville, Generalized Inverses, Theory and Applications, 2nd ed Berlin Springer, New York, 2003.  Google Scholar

[3]

S. L. Campbell and C. D Meyer, Generalized Inverses of Linear Transformations, Dover Publ., New York, 1979. doi: 10.1137/1.9780898719048.  Google Scholar

[4]

S. R. Caradus, Generalized Inverses and Operator Theory, Queen's paper in pure and applied mathematics, Queen's University, Kingston, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. S. Dordević and P. S. Stanimirović, General representations of pseudoinverses, Matematicki vesnik, 51 (1999), 69-76.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. R. Hestenes, Relative hermitian matrices, Pacific Journal Math., 11 (1961), 225-245.  doi: 10.2140/pjm.1961.11.225.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Ben-Israel and T. N. E Greville, Generalized Inverses, Theory and Applications, 2nd ed Berlin Springer, New York, 2003.  Google Scholar

[3]

S. L. Campbell and C. D Meyer, Generalized Inverses of Linear Transformations, Dover Publ., New York, 1979. doi: 10.1137/1.9780898719048.  Google Scholar

[4]

S. R. Caradus, Generalized Inverses and Operator Theory, Queen's paper in pure and applied mathematics, Queen's University, Kingston, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. S. Dordević and P. S. Stanimirović, General representations of pseudoinverses, Matematicki vesnik, 51 (1999), 69-76.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. R. Hestenes, Relative hermitian matrices, Pacific Journal Math., 11 (1961), 225-245.  doi: 10.2140/pjm.1961.11.225.  Google Scholar

[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.  Google Scholar

[1]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[2]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[3]

V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481
[4]

Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems & Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064
[5]

Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems & Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217
[6]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, 2021, 15 (2) : 367-386. doi: 10.3934/ipi.2020072
[7]

Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics & Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003
[8]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457
[9]

Sigve Hovda. Closed-form expression for the inverse of a class of tridiagonal matrices. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 437-445. doi: 10.3934/naco.2016019
[10]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[11]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[12]

Pavel Krejčí, Giselle A. Monteiro. Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3051-3066. doi: 10.3934/dcdsb.2018299
[13]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026
[14]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115
[15]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[16]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[17]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[18]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[19]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[20]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139

 Impact Factor: 

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences