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Expression for the multiplicative perturbation of the Moore-Penrose inverse. (English) Zbl 1390.15008

Let \(\mathcal{B}(\mathcal{H} ,\mathcal{K})\) be the algebra of bounded linear operators from an infinite-dimensional separable complex Hilbert space into another. If \(T \in \mathcal{B}(\mathcal{H} ,\mathcal{K})\) then it has a Moore-Penrose (MP) inverse \(T^{† }\) if there exists a unique \(X \in \mathcal{B}(\mathcal{K} ,\mathcal{H})\) such that \(TXT =T\), \(XTX =X\), \((TX)^{ \ast } =TX\) and \((XT)^{ \ast } =XT\). It is known that \(T^{† }\) exists if and only if the range \(\mathcal{R}(T)\) is closed and that in this case \(E_{T} : =I -TT^{† }\) and \(F_{T} : =I -T^{† }T\) are orthogonal projectors onto \(\mathcal{R}(T)^{ \bot }\) and \(\mathcal{R}(T^{ \ast })^{ \bot }\), respectively. Now suppose that \(\mathcal{H} =\mathcal{H}_{1} \oplus \mathcal{H}_{2}\) and \(M \in \mathcal{B}(\mathcal{H} ,\mathcal{H})\) is written in the form \[ M =\begin{bmatrix} A & B \\ C & D\end{bmatrix} \] corresponding to the direct sum decomposition. The aim of the present paper is to investigate conditions under which \(M\) (or more generally \(XMY\) with \(X\) and \(Y\) invertible) is MP-invertible. Some remarkable results are:
1) Put \(B_{0} : =E_{A}B\), \(C_{0} : =CF_{A}\) and \(S_{A} : =D -CA^{† }B\) and suppose that \(A\), \(B_{0}\) and \(C_{0}\) have closed ranges; then \(M\) is MP-invertible if and only if \(E_{C_{0}}S_{A}F_{B_{0}}\)is MP-invertible.
2) If \(C =0\) and \(A\) is MP-invertible, then \(M\) is MP-invertible if and only if \(\mathcal{R}(B_{0}^{ \ast }) +\mathcal{R}(D^{ \ast })\) is closed.
3) Suppose that \(A\) and \(S_{A}\) are MP-invertible and put \(Z : =B_{0}S_{A}^{† }C_{0}\); then \(M\) is MP-invertible if and only if \(Z\) is MP-invertible. In each case the MP-inverse of \(M\) is given explicitly.

MSC:

15A09 Theory of matrix inversion and generalized inverses
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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[1] Conway, J., A course in functional analysis (1990), New York (NY): Spring-Verlag, New York (NY) · Zbl 0706.46003
[2] Murphy, Gj, C*-Algebras and operator theory (1990), San Diego: Academic Press, San Diego · Zbl 0714.46041
[3] Ben-Israel, A.; Greville, Tne, Generalized inverses: theory and applications (2003), New York (NY): Springer-Verlag, New York (NY) · Zbl 1026.15004
[4] Campbell, Sl; Meyer, Cd, Generalized inverses of linear transformations (1979), London: Pitman Publishing Limited, London · Zbl 0417.15002
[5] Xu, Q.; Wei, Y.; Gu, Y., Sharp norm-estimations for Moore-Penrose inverses of stable perturbations of Hilbert C*-module operators, SIAM J Numer Anal, 47, 4735-4758 (2010) · Zbl 1226.47004
[6] Lancaster, P.; Tismenetsky, M., The theory of matrices (1985), San Diego: Academic Press, San Diego · Zbl 0516.15018
[7] Bai, Zj; Bai, Zz, On nonsingularity of block two-by-two matrices, Linear Alg Appl, 439, 2388-2404 (2013) · Zbl 1283.15009
[8] Lu, Tt; Shiou, Sh, Inverses of 2 × 2 block matrices, Comput Math Appl, 43, 119-129 (2002) · Zbl 1001.15002
[9] Tian, Y., The Moore-Penrose inverses of m × n block matrices and their applications, Linear Algebra Appl, 283, 35-60 (1998) · Zbl 0932.15004
[10] Wei, Y.; Ding, J., Representations for Moore-Penrose inverses in Hilbert spaces, Appl Math Lett, 14, 599-604 (2001) · Zbl 0982.47003
[11] Graybill, Fa, Matrices with applications in statistics (1983), Belmont (CA): Wadsworth, Belmont (CA) · Zbl 0496.15002
[12] Xu, Q.; Song, C.; Wang, G., Multiplicative perturbations of matrices and thegeneralized triple reverse order law for the Moore-Penrose inverse, Linear Algebra Appl, 530, 366-383 (2017) · Zbl 1368.15006
[13] Zhang, X.; Fang, X.; Song, C.; Xu, Q., Representations and norm estimations for the Moore-Penrose inverse of multiplicative perturbations of matrices, Linear Multilinear Algebra, 65, 555-571 (2017) · Zbl 1362.15002
[14] Deng, C.; Du, H., Representations of the Moore-Penrose inverse of 2 by 2 block operator valued matrices, J Korean Math Soc, 46, 1139-1150 (2009) · Zbl 1189.47002
[15] Fillmore, Pa; Williams, Ip, On operator ranges, Adv Math, 7, 244-281 (1971) · Zbl 0224.47009
[16] Douglas, Rg, On majorization, factorization, and range inclusion of operators on Hilbert spaces, Proc Am Math Soc, 17, 413-416 (1966) · Zbl 0146.12503
[17] Khadivi, M., Range inclusion and operator equations, J Math Anal Appl, 197, 630-633 (1996) · Zbl 0868.47014
[18] Deng, C.; Du, H., Representations of the Moore-Penrose inverse for a class of 2-by-2 block operator valued partial matrices, Linear Multilinear Algebra, 58, 15-26 (2010) · Zbl 1187.47005
[19] Castro-González, N.; Martínez-Serrano, Mf; Robles, J., Expressions for the Moore-Penrose inverse of block matrices involving the Schur complement, Linear Algebra Appl, 471, 353-368 (2015) · Zbl 1310.15007
[20] Hartwig, Re, Block generalized inverses, Arch Ration Mech Anal, 61, 197-251 (1976) · Zbl 0335.15004
[21] Baksalary, Jk; Styan, Gph, Generalized inverses of partitioned matrices in Banachiewicz-Schur form, Linear Algebra Appl, 354, 41-47 (2002) · Zbl 1022.15006
[22] Deng, C., A note on the Drazin inverses with Banachiewicz-Schur forms, Appl Math Comput, 213, 230-234 (2009) · Zbl 1182.47001
[23] Tian, Y.; Takane, Y., More on generalized inverses of partitioned matrices with Banachiewicz-Schur forms, Linear Algebra Appl, 430, 1641-1655 (2009) · Zbl 1162.15002
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