New results on reverse order law for $\{1,2,3\}$- and $\{1,2,4\}$-inverses of bounded operators

Liu, Xiaoji; Wu, Shuxia; Cvetković-Ilić, Dragana S.

doi:10.1090/s0025-5718-2013-02660-9

Cited by 16 publications

(6 citation statements)

References 17 publications

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“…In 26 and 27 , the authors presented necessary and sufficient conditions for { B {1,2,3} A {1,2,3} } ⊆ {( AB ) {1,2,3} } in the case of matrices and bounded linear operators on Hilbert spaces, respectively. An equivalent condition for the inclusion { B {1,2} A {1,2} } ⊆ {( AB ) {1,2} } is given in 28 for bounded linear operators on complex Hilbert spaces.…”

Section: Reverse Order Law For Outer Inversesmentioning

confidence: 99%

Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

Kelathaya

Karantha

2022

F1000Res

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The reverse order law for outer inverses and the Moore-Penrose inverse is discussed in the context of associative rings. A class of pairs of outer inverses that satisfy reverse order law is determined. The notions of left-star and right-star orders have been extended to the case of arbitrary associative rings with involution and many of their interesting properties are explored. The distinct behavior of projectors in association with the star, right-star, and left-star partial orders led to several equivalent conditions for the reverse order law for the Moore-Penrose inverse.

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Section: Reverse Order Law For Outer Inversesmentioning

confidence: 99%

Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

Kelathaya

Karantha

2022

F1000Res

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“…We first mention that several cases of the set inclusions in (1.10) were considered in the literature, such as,

(cf. [20] , [21] , [23] , [28] , [50] , [52] , [54] , [64] , [77] , [81] , [92] , [114] , [116] , [117] , [119] , [123] , [126] ). Despite the fact that ROLs have been around for a long time, there are still many open problems regarding ROLs under various assumptions.…”

Section: Set Inclusions For {1}- 1 2 mentioning

confidence: 99%

“…In fact, people have put a great deal of time and effort into the investigations of various ROLs for generalized inverses of matrix products, in particular, researches of (1.9) – (1.12) have already been flourished and diversified in contents since that beginning (cf. [1] , [13] , [14] , [21] , [24] , [30] , [33] , [34] , [44] , [51] , [52] , [73] , [74] , [75] , [113] , [114] , [115] , [121] , [122] ). In spite of much effort devoted to the researches to ROLs problems in the past several decades, most parts of (1.9) – (1.12) remain unresolved.…”

Section: Introductionmentioning

confidence: 99%

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Tian

2020

Heliyon

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Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is , where denotes a -generalized inverse of matrix. Because -generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for -generalized inverses of the product AB , from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB . We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of , as well as necessary and sufficient conditions for to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A , B , and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).

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“…For more information on this subject please see [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].…”

Section: Improvements Of Hartwig's Triple Reverse Order Lawmentioning

confidence: 99%

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

Cvetković-Ilić¹,

Hofstadler²,

Poor³

et al. 2020

Preprint

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When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials.

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New results on reverse order law for $\{1,2,3\}$- and $\{1,2,4\}$-inverses of bounded operators

Cited by 16 publications

References 17 publications

Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

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