The Bott–Duffin (e,f)-inverses and their applications

Ke, Yuanyuan; Chen, Jianlong

doi:10.1016/j.laa.2015.10.009

Cited by 11 publications

(5 citation statements)

References 14 publications

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“…After their introduction, an impressive number of papers dealt with (b, c)-inverses and inverses along an element. They have been most studied in the contexts of rings [4,5,16,36,[38][39][40]53,67,68,75,[80][81][82][83][84] and semigroups [2,14,16,[24][25][26][27][44][45][46]49], and they have been also studied in the context of Banach algebras [8,9,45], residuated semigroups and quantales [33], matrices over a field [6,15,61,70], matrices over a ring [37], tensors [59], and fuzzy matrices with entries in a complete residuated lattice [18]. Core and dual core inverses were studied in [1,43,57,69,76], in the contexts of matrices and rings, and inner and outer inverses with prescribed idempotents and ideals have been considered in [13,22,…”

Section: Introductionmentioning

confidence: 99%

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Ćirić,

Ignjatović,

Stanimirović

2023

Semigroup Forum

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We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green's R-, L -and H -classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green's R-and L -classes also represent extensions of Drazin's (b, c)-inverses and Mary's inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper.

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Section: Introductionmentioning

confidence: 99%

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Ćirić,

Ignjatović,

Stanimirović

2023

Semigroup Forum

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“…It is worth noticing one of the main properties of these inverses, namely, they encompass several well known outer inverses such as the Drazin inverse, the group inverse and the Moore-Penrose inverse. Furthermore, several authors have studied these notions, see for the (b, c)-inverse [4,6,8,9,14,15,25,30] and for the invese along an element [1, 2, 19-22, 32, 33]. In particular, in [4] and [2] several properties of the (b, c)-inverse and the inverse along an element were studied in the Banach context, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…According to what has been said, if a ∈ A is Bott-Duffin (p, q)-invertible, then the element y in Definition 2.2 will be denoted by a −(p, q) . To learn more on the outer inverses recalled in Definition 2.1 and Definition 2.2, see [4,6,8,9,14,15,25,30].…”

Section: Introductionmentioning

confidence: 99%

Further results on the (b, c)-inverse, the outer inverse AT,S(2) and the Moore–Penrose inverse in the Banach context

Boasso¹

2018

Linear and Multilinear Algebra

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In this article properties of the (b, c)-inverse, the inverse along an element, the outer inverse with prescribed range and null space A(2) T,S and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and C * -algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the (b, c)-inverse and the outer inverse A(2)T,S will be also characterized. Keywords: (b, c)-inverse; outer inverse A (2) T,S ; Moore-Penrose inverse; Banach algebra; C * -algebra; Banach space operator AMS Subjects Classifications: 46H05; 46L05; 47A05; 15A09proved to be open. The continuity of the (b, c)-inverse of Banach space operators and of Banach algebra and C * -algebra elements will be characterized in section 5; two main notions will be used to accomplish this aim: the gap between two subspaces and the Moore-Penrose inverse. The diffrentiability of the (b, c)-inverse in Banach algebras and C * -algebras will be studied in section 6; the Moore-Penrose inverse will be also applied in this section. Finally, the continuity and differentiability of the outer inverse AT,S will be characterized in section 7 using again the gap between subspaces and the Moore-Penrose inverse. In addition, in section 5 and 6, as an application of the main results of these sections, the continuity and the differentiability of the Moore-Penrose inverse for Banach algebra elements and Banach space operators will be studied, respectively. Preliminary definitionsFrom now on, A will denote a unitary Banach algebra with unit ½ while A −1 and A • will stand for the set of invertible elements and the set of idempotents of A, respectively. A particular case is L(X), the Banach algebra of all linear and bounded maps defined on and with values in the Banach space X. However, in the present work it will be necessary to consider the Banach space of all operators defined on the Banach space X with values in the Banach space Y, which will be denoted by L(X, Y). Note that if T ∈ L(X, Y), then N(T ) ⊆ X and R(T ) ⊆ Y will stand for the null space and the range of the operator T , respectively. For example, when A is a unitary Banach algebra and x ∈ A, the operators L x : A → A and R x : A → A are the maps defined as follows: given z ∈ A, L x (z) = xz and R x (z) = zx. Observe that since A is unitary, then L a = a = R a . Moreover, the following notation will be used:x −1 (0) = N(L x ), xA = R(L x ), x −1 (0) = N(R x ), Ax = R(R x ).

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“…As it was pointed out in [9], this inverse generalizes among others the standard inverse, the Drazin inverse, and the Moore-Penrose inverse. To learn more on this inverse, see [5,6,9,10,12].…”

Section: Introduction and Notationmentioning

confidence: 99%

On One-Sided (B;c)-Inverses of Arbitrary Matrices

Benítez¹,

Boasso²,

Jin³

2017

ELA

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In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also researched in the context of rectangular matrices.Keywords: One-sided (b, c)-inverse; One-sided inverse along an element; (b, c)-inverse; Inverse along an element; Matrices AMS classification: 15A09, 15A23, 15A60, 65F99 Definition 2.7. Let A ∈ n,m and D, E ∈ m,n .(i) The matrix A is said to be left (D, E)-invertible, if there exists C ∈ m,n such that CAD = D and N(E) ⊆ N(C). Any matrix C satisfying these conditions is said to be a left (D, E)-inverse of A.The proofs of the following results are straightforward and they are left to the reader.Remark 2.8. Let A ∈ n,m and D, E ∈ m,n . The following statements holdn such that R(D ′ ) = R(D) and N(E ′ ) = N(E). The following statement holds. (iii) The matrix A is left (D, E)-invertible if and only if it is left (D ′ , E ′ )-invertible. In addition, in this case C ∈ m,n is a left (D, E)-inverse of A if and only if it is a left (D ′ , E ′ )-inverse of A. (iv) The matrix A is right (D, E)-invertible if and only if it is right (D ′ , E ′ )-invertible. Moreover, in this case B ∈ m,n is a right (D, E)-inverse of A if and only if it is a right (D ′ , E ′ )-inverse of A.When the matrices D, E ∈ m,n in Definition 2.7 coincide, the notions of left and right inverse along a matrix can be introduced.Definition 2.9. Let A ∈ n,m and D ∈ m,n .(i) The matrix A is said to be left invertible along D, if there exists C ∈ m,n such that CAD = D and N(D) ⊆ N(C). Any matrix C satisfying these conditions is said to be a left inverse of A along D.(ii) The matrix A is said to be right invertible along D, if there exists B ∈ m,n such that DAB = D and R(B) ⊆ R(D). Any matrix B satisfying these conditions is said to be a right inverse of A along D.Note that similar results to the ones in Remark 2.8 for the case D = E ∈ m,n hold for left and right invertible matrices along a matrix. The details are left to the reader.Recall that given a ring R and a, b, c ∈ R, in [13, Definition 2.3] the left and right annihilator (b, c)-inverses of the element a were introduced. However, in the case of matrices, as under the conditions of [13, Proposition 2.5], these notions coincide with the ones in Definition 2.7.

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The Bott–Duffin (e,f)-inverses and their applications

Cited by 11 publications

References 14 publications

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Further results on the (b, c)-inverse, the outer inverse AT,S(2) and the Moore–Penrose inverse in the Banach context

On One-Sided (B;c)-Inverses of Arbitrary Matrices

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