Lattice properties of the core-partial order

Djikić, Marko S.

doi:10.1215/17358787-0000010x

Cited by 6 publications

(4 citation statements)

References 14 publications

(28 reference statements)

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“…The second reason for such generalization comes from the results of [13,11,14,12] which describe different properties of operators A and B which coincide on R(A * ) ∩ R(B * ) (a generalization of Werener's condition of weak complementarity, see [20]). Accordingly, we will present different properties of CoR operators, regarding range additivity, some additive results for the Moore-Penrose inverse, etc.…”

Section: Motivation and Preliminariesmentioning

confidence: 99%

Operators with compatible ranges

Djikić

2017

Filomat

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A bounded operator T on a finite or infinite-dimensional Hilbert space is called a disjoint range (DR) operator if R(T ) ∩ R(T * ) = {0}, where T * stands for the adjoint of T , while R(·) denotes the range of an operator. Such operators (matrices) were introduced and systematically studied by Baksalary and Trenkler, and later by Deng et al. In this paper we introduce a wider class of operators: we say that T is a compatible range (CoR) operator if T and T * coincide on R(T ) ∩ R(T * ). We extend and improve some results about DR operators and derive some new results regarding the CoR class.

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Section: Motivation and Preliminariesmentioning

confidence: 99%

Operators with compatible ranges

Djikić

2017

Filomat

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“…Generalized inverses of matrices are applied in areas as varied as Markov chains [4], coding theory [29], chemical equations [23], robotics [9], geology [5], etc. Matrix partial orders is another important area in which generalized inverses are an essential tool towards which attention is directed [7,8,20,25,32].…”

Section: Introductionmentioning

confidence: 99%

The weak core inverse

et al. 2020

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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 O.M. Baksalary and G. Trenkler in 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.

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“…Furthermore, generalized inverses play an important role in the study of matrix partial orders, as we can see in [13]. Interesting applications of partial orders were investigated, for example, in [1,6,7,15]. Let C m×n be the set of m × n complex matrices.…”

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confidence: 99%

Solving an Open Problem About the G-Drazin Partial Order

Ferreyra

Lattanzi

Levis

et al. 2020

ELA

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G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: if A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.

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Lattice properties of the core-partial order

Cited by 6 publications

References 14 publications

Operators with compatible ranges

Operators with compatible ranges

The weak core inverse

Solving an Open Problem About the G-Drazin Partial Order

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