×

The quasi-linear operator outer generalized inverse with prescribed range and kernel in Banach spaces. (English) Zbl 1314.47004

Let \(\mathcal X\) and \(\mathcal Y\) be real Banach spaces. A subset \(D \subseteq {\mathcal X}\) is said to be homogeneous if, for any \(x \in D\) and \(\lambda \in \mathbb{R}\), we have \(\lambda x \in D\). Let \(A:{\mathcal X} \rightarrow {\mathcal Y}\) be a mapping whose domain is denoted by \({\mathcal D}(A)\). If, for all \(x \in {\mathcal D}(A)\) and for all \(\lambda \in \mathbb{R}\), it holds that \(A(\lambda x) = \lambda A(x)\), then \(A\) is called a homogeneous operator on \({\mathcal D}(A)\). For a subset \(M \subseteq {\mathcal X}\), \(A:{\mathcal X} \rightarrow {\mathcal Y}\) is called quasi-additive on \(M\) if \(A(x+z)=A(x) + A(z)\) for all \(x \in \mathcal X\) and for all \(z \in M\). For homogeneous subsets \(T \subseteq {\mathcal X}\) and \(S \subseteq {\mathcal Y}\) and for a bounded linear operator \(A\) from \(\mathcal X\) into \(\mathcal Y\), the authors study the problem of finding a homogeneous \(B:{\mathcal Y} \rightarrow {\mathcal X}\) which is also quasi-additive on the range space of \(A\) such that \(ABA=A, BAB=B\), the range space of \(B\) equals \(T\) and the null space of \(B\) equals \(S\). Necessary and sufficient conditions are established and a perturbation question is also considered. No illustrating examples are presented.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] X. Y. Bai, Y. W. Wang, G. Q. Liu, and J. Xia, “Definition and criterion for a homogeneous generalized inverse,” Acta Mathematica Sinica, vol. 52, no. 2, pp. 353-360, 2009. · Zbl 1199.47004
[2] Y. Wang and S. Pan, “An approximation problem of the finite rank operator in Banach spaces,” Science in China A, vol. 46, no. 2, pp. 245-250, 2003. · Zbl 1218.47028 · doi:10.1360/03ys9026
[3] Y. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, China, 2005.
[4] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, New York, NY, USA, 2nd edition, 2003. · Zbl 1026.15004
[5] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific Publishing, 2012. · Zbl 1241.47001
[6] D. S. Djordjević and P. S. Stanimirović, “On the generalized Drazin inverse and generalized resolvent,” Czechoslovak Mathematical Journal, vol. 51, no. 3, pp. 617-634, 2001. · Zbl 1079.47501 · doi:10.1023/A:1013792207970
[7] D. S. Djordjević and Y. Wei, “Outer generalized inverses in rings,” Communications in Algebra, vol. 33, no. 9, pp. 3051-3060, 2005. · Zbl 1111.15007 · doi:10.1081/AGB-200066112
[8] F. Du and Y. Xue, “Perturbation analysis of AT,S(2) on Banach spaces,” Electronic Journal of Linear Algebra, vol. 23, pp. 586-598, 2012. · Zbl 1251.47002
[9] Y. Wei, “A characterization and representation of the generalized inverse AT,S(2) and its applications,” Linear Algebra and Its Applications, vol. 280, no. 2-3, pp. 87-96, 1998. · Zbl 0934.15003 · doi:10.1016/S0024-3795(98)00008-1
[10] Y. Wei and H. Wu, “The representation and approximation for the generalized inverse AT,S(2),” Applied Mathematics and Computation, vol. 135, no. 2-3, pp. 263-276, 2003. · Zbl 1027.65048 · doi:10.1016/S0096-3003(01)00327-7
[11] Y. Wei and H. Wu, “On the perturbation and subproper splittings for the generalized inverse AT,S(2) of rectangular matrix A,” Journal of Computational and Applied Mathematics, vol. 137, no. 2, pp. 317-329, 2001. · Zbl 0993.15003 · doi:10.1016/S0377-0427(00)00710-X
[12] Y. Yu and G. Wang, “The generalized inverse AT,S(2) over commutative rings,” Linear and Multilinear Algebra, vol. 53, no. 4, pp. 293-302, 2005. · Zbl 1086.15004 · doi:10.1080/03081080500079510
[13] J. Cao and Y. Xue, “Perturbation analysis of bounded homogeneousgeneralized inverses on Banach spaces,” arXiv preprint, http://arxiv.org/abs/1302.3965.
[14] J. Cao and Y. Xue, “On the simplest expression of the perturbed Moore-Penrose metric generalized inverse,” Preprint. · Zbl 1324.47002
[15] P. Liu and Y. W. Wang, “The best generalised inverse of the linear operator in normed linear space,” Linear Algebra and Its Applications, vol. 420, no. 1, pp. 9-19, 2007. · Zbl 1114.47001 · doi:10.1016/j.laa.2006.04.024
[16] M. Z. Nashed and G. F. Votruba, “A unified operator theory of generalized inverses,” in Generalized Inverses and Applications, M. Z. Nashed, Ed., pp. 1-109, Academic Press, New York, NY, USA, 1976. · Zbl 0356.47001
[17] R. X. Ni, “Moore-Penrose metric generalized inverses of linear operators in arbitrary Banach spaces,” Acta Mathematica Sinica, vol. 49, no. 6, pp. 1247-1252, 2006. · Zbl 1113.46308
[18] H. Wang and Y. Wang, “Metric generalized inverse of linear operator in Banach space,” Chinese Annals of Mathematics B, vol. 24, no. 4, pp. 509-520, 2003. · Zbl 1048.46022 · doi:10.1142/S0252959903000517
[19] J. Ding, “New perturbation results on pseudo-inverses of linear operators in Banach spaces,” Linear Algebra and Its Applications, vol. 362, pp. 229-235, 2003. · Zbl 1044.47011 · doi:10.1016/S0024-3795(02)00493-7
[20] H. Ma, Construction of some generalized inverses of operators between Banach spaces and their selections, perturbations and applications [Ph.D. thesis], 2012.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.