On the continuity and differentiability of the (dual) core inverse in C*-algebras

[EN] The continuity of the core inverse and the dual core inverse is studied in the setting of -algebras. Later, this study is specialized to the case of bounded Hilbert space operators and to complex matrices. In addition, the differentiability of these generalized inverses is studied in the context of -algebras.The third author is supported by the Scientific Research Foundation for doctorate programme at Huaiyin Institute of Technology [grant number Z301B18534]. The third author is grateful to China Scholarship Council for helping him pursue his further study at Universitat Politècnica de València, SpainBenítez López, J.; Boasso, E.; Xu, S. (2020). On the continuity and differentiability of the (dual) core inverse in C*-algebras. Linear and Multilinear Algebra. 68(4):686-709. https://doi.org/10.1080/03081087.2018.1516187S686709684Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Rakić, D. S., Dinčić, N. Č., & Djordjević, D. S. (2014). Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra and its Applications, 463, 115-133. doi:10.1016/j.laa.2014.09.003Rakić, D. S., Dinčić, N. Č., & Djordjević, D. S. (2014). Core inverse and core partial order of Hilbert space operators. Applied Mathematics and Computation, 244, 283-302. doi:10.1016/j.amc.2014.06.112Xu, S., Chen, J., & Zhang, X. (2016). New characterizations for core inverses in rings with involution. Frontiers of Mathematics in China, 12(1), 231-246. doi:10.1007/s11464-016-0591-2Drazin, M. P. (2012). A class of outer generalized inverses. Linear Algebra and its Applications, 436(7), 1909-1923. doi:10.1016/j.laa.2011.09.004Boasso, E., & Kantún-Montiel, G. (2017). The (b, c)-Inverse in Rings and in the Banach Context. Mediterranean Journal of Mathematics, 14(3). doi:10.1007/s00009-017-0910-1Harte, R. (1992). On generalized inverses in C*-algebras. Studia Mathematica, 103(1), 71-77. doi:10.4064/sm-103-1-71-77Harte, R. (1993). Generalized inverses in C*-algebras II. Studia Mathematica, 106(2), 129-138. doi:10.4064/sm-106-2-129-138Penrose, R. (1955). A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 51(3), 406-413. doi:10.1017/s0305004100030401Mbekhta, M. (1992). Conorme et Inverse Généralisé Dans Les C*-Algèbres. Canadian Mathematical Bulletin, 35(4), 515-522. doi:10.4153/cmb-1992-068-8Du, F., & Xue, Y. (2012). Perturbation analysis of A_{T,S}^(2) on Banach Spaces. The Electronic Journal of Linear Algebra, 23. doi:10.13001/1081-3810.1543Benítez, J., Cvetković-Ilić, D., & Liu, X. (2014). On the continuity of the group inverse in $C^*$-algebras. Banach Journal of Mathematical Analysis, 8(2), 204-213. doi:10.15352/bjma/1396640064Benítez, J., & Cvetković-Ilić, D. (2013). On the elements aa† and a†a in a ring. Applied Mathematics and Computation, 222, 478-489. doi:10.1016/j.amc.2013.07.015Koliha, J. J. (2001). Continuity and differentiability of the Moore-Penrose inverse in $C^*$-algebras. MATHEMATICA SCANDINAVICA, 88(1), 154. doi:10.7146/math.scand.a-14320Koliha, J. J., & Rakočević, V. (2004). On the Norm of Idempotents in $C^*$ -Algebras. Rocky Mountain Journal of Mathematics, 34(2). doi:10.1216/rmjm/1181069874Benítez, J., & Liu, X. (2012). On the continuity of the group inverse. Operators and Matrices, (4), 859-868. doi:10.7153/oam-06-55Douglas, R. G. (1966). On majorization, factorization, and range inclusion of operators on Hilbert space. Proceedings of the American Mathematical Society, 17(2), 413-413. doi:10.1090/s0002-9939-1966-0203464-1Boasso, E. (2009). Drazin spectra of Banach space operators and Banach algebra elements. Journal of Mathematical Analysis and Applications, 359(1), 48-55. doi:10.1016/j.jmaa.2009.05.036Rakić, D. S. (2017). A note on Rao and Mitra’s constrained inverse and Drazin’s (b,c) inverse. Linear Algebra and its Applications, 523, 102-108. doi:10.1016/j.laa.2017.02.025Stewart, G. W. (1969). On the Continuity of the Generalized Inverse. SIAM Journal on Applied Mathematics, 17(1), 33-45. doi:10.1137/011700

The weak core inverse

[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu.In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paperFerreyra, DE.; Levis, FE.; Priori, AN.; Thome, N. (2021). The weak core inverse. Aequationes Mathematicae. 95(2):351-373. https://doi.org/10.1007/s00010-020-00752-zS351373952Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. Comput. 236(1), 450–457 (2014)Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)Ceryan, N.: Handbook of Research on Trends and Digital Advances in Engineering Geology, Advances in Civil and Industrial Engineering. IGI Global, Hershey (2018)Chen, J.L., Mosić, D., Xu, S.Z.: On a new generalized inverse for Hilbert sapce operators. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1619104Cvetković-Ilić, D.S., Mosić, D., Wei, Y.: Partial orders on $$B(H)$$. Linear Algebra Appl. 481, 115–130 (2015)Djikić, M.S.: Lattice properties of the core-partial order. Banach J. Math. Anal. 11(2), 398–415 (2017)Doty, K.L., Melchiorri, C., Bonivento, C.: A theory of generalized inverses applied to robotics. Int. J. Robot. Res. 12(1), 1–19 (1993)Drazin, M.P.: Pseudo inverses in associative rings and semigroups. Am. Math. Mon. 65(7), 506–514 (1958)Ferreyra, D.E., Levis, F.E., Thome, N.: Revisiting of the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41(2), 265–281 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Maximal classes of matrices determining generalized inverses. Appl. Math. Comput. 333, 42–52 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Characterizations of $$k$$-commutative equalities for some outer generalized inverses. Linear Multilinear Algebra 68(1), 177–192 (2020)Hartwig, R.E., Spindelböck, K.: Matrices for which $$A^*$$ and $$A^\dagger$$ conmmute. Linear Multilinear Algebra 14(3), 241–256 (1984)Liu, X., Cai, N.: High-order iterative methods for the DMP inverse. J. Math. Article ID 8175935, 6 p (2018)Malik, S., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226(1), 575–580 (2014)Malik, S., Rueda, L., Thome, N.: The class of $$m$$-EP and $$m$$-normal matrices. Linear Multilinear Algebra 64(11), 2119–2132 (2016)Manjunatha Prasad, K., Mohana, K.S.: Core EP inverse. Linear Multilinear Algebra 62(6), 792–802 (2014)Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66(5), 1046–1053 (2018)Mitra, S.K., Bhimasankaram, P., Malik, S.: Matrix Partial Orders, Shorted Operators and Applications, Series in Algebra, vol. 10. World Scientific Publishing Co. Pte. Ltd., Singapore (2010)Mosić, D., Stanimirović, P.S.: Composite outer inverses for rectangular matrices. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1671526Penrose, R.: A generalized inverse for matrices. Math. Proc. Cambr. Philos. Soc. 51(3), 406–413 (1955)Rakić, D.S., Dincić, N.C., Djordjević, D.S.: Core inverse and core partial order of Hilbert space operators. Appl. Math. 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Coherent and precoherent operators

In this thesis we introduce and investigate new classes of operators which we call coherent and precoherent operators. These operators appear as solutions of some problems in the literature, but they also represent a generalization of some frequently studied classes of operators. After we study different properties of these new classes, we continue by considering a few interesting problems in operator theory. We consider problems about the Moore-Penrose inverse and arbitrary reflexive inverse of the sum of operators, range additivity of operators, lattice properties of the star and core partial orders on Hilbert space operators, the connection about the parallel sum of operators and their infimum in different partial orders, and one special type of operators, inspired by recently introduced disjoint range operators. Accordingly, we generalize and improve a number of results from the existing literature. One part of the thesis is dedicated to Rickart *-rings and generalizations of some presented results in the algebraic setting. We included a number of examples in order to demonstrate our statements and their possible extent: reduction of conditions, proving opposite directions, etc. In the end, we propose few problems for further research on these topics

Characterizations of k-commutative equalities for some outer generalized inverses

[EN] In this paper, we present necessary and sufficient conditions for the k-commutative equality , where X is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations.D. E. Ferreyra F. E. Levis Partially supported by a Consejo Nacional de Investigaciones Científicas y Técnicas CONICET s Posdoctoral Research Fellowship, UNRC [grant number PPI 18/C472] and CONICET [grant number PIP 112-201501-00433CO], respectively. N. Thome Partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación Ministerio de Economía, Industria y Competitividad of Spain [grant number DGI MTM2013-43678-P and Grant Red de Excelen- cia PMTM2017-90682-REDT]. D. E. Ferreyra and N. Thome Partially supported Universidad Nacional de La Pampa (UNLPam), Facultad de Ingeniería [grant Resol. No 155/14].Ferreyra, DE.; Levis, F.; Thome, N. (2018). Characterizations of k-commutative equalities for some outer generalized inverses. Linear and Multilinear Algebra. 1-16. https://doi.org/10.1080/03081087.2018.1500994S116Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Manjunatha Prasad, K., & Mohana, K. S. (2013). Core–EP inverse. Linear and Multilinear Algebra, 62(6), 792-802. doi:10.1080/03081087.2013.791690Malik, S. B., & Thome, N. (2014). On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation, 226, 575-580. doi:10.1016/j.amc.2013.10.060Mehdipour, M., & Salemi, A. (2017). On a new generalized inverse of matrices. Linear and Multilinear Algebra, 66(5), 1046-1053. doi:10.1080/03081087.2017.1336200Malik, S. B., Rueda, L., & Thome, N. (2016). The class ofm-EPandm-normal matrices. Linear and Multilinear Algebra, 64(11), 2119-2132. doi:10.1080/03081087.2016.1139037Wang, H. (2016). Core-EP decomposition and its applications. Linear Algebra and its Applications, 508, 289-300. doi:10.1016/j.laa.2016.08.008Wang H, Chen J. Weak group inverse. Available from: http://arxiv.org/abs/1704.08403v1Wei, Y. (1998). A characterization and representation of the generalized inverse A(2)T,S and its applications. Linear Algebra and its Applications, 280(2-3), 87-96. doi:10.1016/s0024-3795(98)00008-1Rakić, D. S., Dinčić, N. Č., & Djordjević, D. S. (2014). Core inverse and core partial order of Hilbert space operators. Applied Mathematics and Computation, 244, 283-302. doi:10.1016/j.amc.2014.06.112Stanimirović, P. S., Katsikis, V. N., & Ma, H. (2016). Representations and properties of theW-Weighted Drazin inverse. Linear and Multilinear Algebra, 65(6), 1080-1096. doi:10.1080/03081087.2016.1228810Ferreyra, D. E., Levis, F. E., & Thome, N. (2017). Revisiting the core EP inverse and its extension to rectangular matrices. Quaestiones Mathematicae, 41(2), 265-281. doi:10.2989/16073606.2017.1377779Deng, C. Y., & Du, H. K. (2009). REPRESENTATIONS OF THE MOORE-PENROSE INVERSE OF 2×2 BLOCK OPERATOR VALUED MATRICES. Journal of the Korean Mathematical Society, 46(6), 1139-1150. doi:10.4134/jkms.2009.46.6.1139Wang, H., & Liu, X. (2014). Characterizations of the core inverse and the core partial ordering. Linear and Multilinear Algebra, 63(9), 1829-1836. doi:10.1080/03081087.2014.97570

Partial inner product spaces, metric operators and generalized hermiticity

Motivated by the recent developments of pseudo-hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.Comment: 23 pages, 4 figures (in LaTeX

Unbounded normal operators in octonion Hilbert spaces and their spectra

Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.Comment: 50 page

Metric operators, generalized hermiticity and lattices of Hilbert lpaces

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.Comment: 51pages; will appear as a chapter in \textit{Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects}; F. Bagarello, J-P. Gazeau, F. H. Szafraniec and M. Znojil, eds., J. Wiley, 201

Fully representable and *-semisimple topological partial *-algebras

We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals and focus on the case where the two notions are completely interchangeable (fully representable partial *-algebras) with the scope of characterizing a *-semisimple partial *-algebra. Finally we describe various notions of bounded elements in such a partial *-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome is that, for an appropriate order relation, one recovers the \M-bounded elements introduced in previous works.Comment: 26 pages, Studia Mathematica (2012) to appea