The (b,c)-inverse for products and lower triangular matrices

Ke, Yuanyuan; Wang, Zhou; Chen, Jianlong

doi:10.1142/s021949881750222x

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…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: Definition 21 ([8 Definition 13])mentioning

confidence: 99%

“…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%

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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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Section: Definition 21 ([8 Definition 13])mentioning

confidence: 99%

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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“…The set of all (b, c)-invertible elements of R will be denoted by R (b,c) . For more results of the (b, c)-inverse, we refer the reader to see [5,6,16,24]. Drazin [4] introduced an outer generalized inverse relative to a pair of idempotents e, f ∈ R which intermediates between the Bott-Duffin inverse and the (b, c)-inverse.…”

Section: Introductionmentioning

confidence: 99%

The reverse order law of the $(b, c)$-inverse in rings

Ke,

Mosić,

Chen

2016

Preprint

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We present equivalent conditions of reverse order law for the (b, c)-inverse (aw) (b,c) = w (b,s) a (t,c) to hold in a ring. Also, we study various mixed-type reverse order laws for the (b, c)-inverse. As a consequence, we get results related to the reverse order law for the inverse along an element. More general case of reverse order law (a 1 a 2 ) (b3,c3) = a (b2,c2) 2 a (b1,c1) 1 is considered too.

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The (b, c)-Inverse in Rings and in the Banach Context

Boasso¹,

Kantún-Montiel

2017

Mediterr. J. Math.

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In this article the (b, c)-inverse will be studied. Several equivalent conditions for the existence of the (b, c)-inverse in rings will be given. In particular, the conditions ensuring the existence of the (b, c)-inverse, of the annihilator (b, c)-inverse and of the hybrid (b, c)-inverse will be proved to be equivalent, provided b and c are regular elements in a unitary ring R. In addition, the set of all (b, c)-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the (b, c)-inverse and the Bott-Duffin inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally the continuity of the (b, c)-inverse will be characterized.Mathematics Subject Classification. 15A09, 16B99, 16U99, 46H05.

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The (b,c)-inverse for products and lower triangular matrices

Cited by 7 publications

References 14 publications

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

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

The reverse order law of the $(b, c)$-inverse in rings

The (b, c)-Inverse in Rings and in the Banach Context

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