On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

[EN] For two given Hilbert spaces H and K and a given bounded linear operator A is an element of L(H, K) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive g-inverse G is an element of L ( K; H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, l-quasi-normal and r-quasi-normal and l-quasi-EP and r-quasi-EP operators.Research partially supported by Ministerio de Economia y Competitividad of Spain (grant DGI MTM2013-43678-P and Red de Excelencia MTM2015-68805-REDT)Malik, SB.; Thome, N. (2017). On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces. Filomat. 31(7):1927-1931. https://doi.org/10.2298/FIL1707927MS1927193131

The mm-weak core inverse

Since the day the core inverse has been known in a paper of Bakasarly and Trenkler, it has been widely researched. So far, there are four generalizations of this inverse for the case of matrices of an arbitrary index, namely, the BT inverse, the DMP inverse, the core-EP inverse and the WC inverse. In this paper we introduce a new type of generalized inverse for a matrix of arbitrary index to be called $m$-weak core inverse which generalizes the core-EP inverse, the WC inverse, and therefore the core inverse. We study several properties and characterizations of the $m$-weak core inverse by using matrix decompositions

The class of m-EP and m-normal matrices

The well-known classes of EP matrices and normal matrices are de- fined by the matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively. This paper investigates the class of m-EP matrices and m-normal matrices that provide a generalization of EP matrices and normal matrices, respectively, and analyzes both of them for their properties and characterizations.Third author was partially supported by Ministerio de Economia y Competitividad of Spain [grant number DGI MTM2013-43678-P], [Red de Excelencia MTM2015-68805-REDT].Malik, SB.; Rueda, L.; Thome, N. (2016). The class of m-EP and m-normal matrices. Linear and Multilinear Algebra. 64(11):2119-2132. https://doi.org/10.1080/03081087.2016.1139037S21192132641

One sided Star and Core orthogonality of matrices

We investigate two one-sided orthogonalities of matrices, the first of which is left (right) $*$-orthogonality for rectangular matrices and the other is left (right) core-orthogonality of index $1$ matrices. We obtain some basic results for these matrices, their canonical forms, and characterizations. Also, relations between left (right) orthogonal matrices and parallel sums are investigated. Finally under these one-sided orthogonalities we explore the conditions of additivity of the Moore-Penrose inverse and the core inverse

On a new generalized inverse for matrices of an arbitrary index

[EN] The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by Baksalary and Trenkler for matrices of index at most 1 in (Baksalary and Trenkler (2010) [1]) to matrices of an arbitrary index. DMP inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Malik, SB.; 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.2013.10.060S57558022

Further properties on the core partial order and other matrix partial orders

This paper carries further the study of core partial order initiated by Baksalary and Trenkler [Core inverse of matrices, Linear Multilinear Algebra. 2010;58:681-697]. We have extensively studied the core partial order, and some new characterizations are obtained in this paper. In addition, simple expressions for the already known characterizations of the minus, the star (and one-sided star), the sharp (and one-sided sharp) and the diamond partial orders are also obtained by using a Hartwig-Spindelbck decomposition.This author was partially supported by Ministry of Education of Spain [grant number DGI MTM2010-18228] and by Universidad Nacional de La Pampa, Argentina, Facultad de Ingenieria [grant number Resol. No 049/11].Malik, SB.; Rueda, LC.; Thome, N. (2014). Further properties on the core partial order and other matrix partial orders. Linear and Multilinear Algebra. 62(12):1629-1648. https://doi.org/10.1080/03081087.2013.839676S162916486212Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, J. K., & Hauke, J. (1990). A further algebraic version of Cochran’s theorem and matrix partial orderings. Linear Algebra and its Applications, 127, 157-169. doi:10.1016/0024-3795(90)90341-9Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Baksalary, J. K., Baksalary, O. M., & Liu, X. (2003). Further properties of the star, left-star, right-star, and minus partial orderings. Linear Algebra and its Applications, 375, 83-94. doi:10.1016/s0024-3795(03)00609-8Groβ, J., Hauke, J., & Markiewicz, A. (1999). Partial orderings, preorderings, and the polar decomposition of matrices. Linear Algebra and its Applications, 289(1-3), 161-168. doi:10.1016/s0024-3795(98)10108-8Mosić, D., & Djordjević, D. S. (2012). Reverse order law for the group inverse in rings. Applied Mathematics and Computation, 219(5), 2526-2534. doi:10.1016/j.amc.2012.08.088Patrício, P., & Costa, A. (2009). On the Drazin index of regular elements. Open Mathematics, 7(2). doi:10.2478/s11533-009-0015-6Rakić, D. S., & Djordjević, D. S. (2012). Space pre-order and minus partial order for operators on Banach spaces. Aequationes mathematicae, 85(3), 429-448. doi:10.1007/s00010-012-0133-2Tošić, M., & Cvetković-Ilić, D. S. (2012). Invertibility of a linear combination of two matrices and partial orderings. Applied Mathematics and Computation, 218(9), 4651-4657. doi:10.1016/j.amc.2011.10.052Hartwig, R. E., & Spindelböck, K. (1983). Matrices for whichA∗andA†commute. Linear and Multilinear Algebra, 14(3), 241-256. doi:10.1080/03081088308817561Baksalary, O. M., Styan, G. P. H., & Trenkler, G. (2009). On a matrix decomposition of Hartwig and Spindelböck. Linear Algebra and its Applications, 430(10), 2798-2812. doi:10.1016/j.laa.2009.01.015Mielniczuk, J. (2011). Note on the core matrix partial ordering. Discussiones Mathematicae Probability and Statistics, 31(1-2), 71. doi:10.7151/dmps.1134Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.978089871951

TNFα-stimulated gene-6 (TSG6) activates macrophage phenotype transition to prevent inflammatory lung injury

TNFα-stimulated gene-6 (TSG6), a 30-kDa protein generated by activated macrophages, modulates inflammation; however, its mechanism of action and role in the activation of macrophages are not fully understood. Here we observed markedly augmented LPS-induced inflammatory lung injury and mortality in TSG6−/− mice compared with WT (TSG6+/+) mice. Treatment of mice with intratracheal instillation of TSG6 prevented LPS-induced lung injury and neutrophil sequestration, and increased survival in mice. We found that TSG6 inhibited the association of TLR4withMyD88, thereby suppressing NF-κB activation. TSG6 also prevented the expression of proinflammatory proteins (iNOS, IL-6, TNFα, IL-1β, and CXCL1) while increasing the expression of antiinflammatory proteins (CD206, Chi3l3, IL-4, and IL-10) in macrophages. This shift was associated with suppressed activation of proinflammatory transcription factors STAT1 and STAT3. In addition, we observed that LPS itself up-regulated the expression of TSG6 in TSG6+/+ mice, suggesting an autocrine role for TSG6 in transitioning macrophages. Thus, TSG6 functions by converting macrophages from a proinflammatory to an anti-inflammatory phenotype secondary to suppression of TLR4/NF-κB signaling and STAT1 and STAT3 activation

Some new results on the core partial order

The current research investigates the core partial order for its further properties. We introduce a new property for matrices of index at most 1, namely (Formula presented.) and is called as ‘the core-subtractivity’. This property then is used to explore relations between the powers and subtractivity properties of the core partial order. In particular, we prove that (Formula presented.) is equivalent to (Formula presented.) under the core-subtractivity property. We also examine some new conditions under which the core partial order is equivalent to the minus and diamond partial orders.Fil: Ferreyra, David Eduardo. Universidad Nacional de Río Cuarto; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Malik, Saroj B.. Ambedkar University; Indi

On star-dagger matrices and the core-EP decomposition

The concept of star-dagger matrices was introduced in 1984 by Hartwig and Spindelböck. While they completely characterized the star-dagger matrices by using a block decomposition of the form PQ00, they also proposed the following open problem: “Can the triangular form PQ0Rbe used to obtain further results on the star-dagger matrices?” In this paper, we have attempted this open problem by using an upper-triangularization of Schur's type for a square matrix, namely, the core-EP decomposition. Furthermore, similar problems regarding bi-dagger and bi-EP matrices are investigated.Fil: Ferreyra, David Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentina. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; ArgentinaFil: Levis, Fabián Eduardo. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Malik, Saroj B.. Ambedkar University; IndiaFil: Priori, Albina Natalia. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentin