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

Abstract

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 ).

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

“…If these statements are true, then a{2} R(b)∩L(c) = {b(cab) (2) c | (cab) (2) ∈ cab{2}}. (8) Relaxing the conditions of the above theorem, we provide an existence condition and a characterization of outer inverses that belong to the principal right ideal R(b) generated by a given element b ∈ S. (ii) ab ∈ S (2) . If these statements are true, then a{2} R(b) = {b(ab) (2) | (ab) (2) ∈ ab{2}}.…”

“…Let us note that S is a regular subsemigroup of M ∅ (R), and according to the remark given at the very end of Section 2, Green's relations R, L and H on S can be represented by means of ranges and null spaces, as in Example 2. 8.…”

A New Non-Full Rank Algorithm for the IMC-Derived d-Step MIMO Structures in the Pole-Free State Space

“…In the literature, we can find a plethora of unique/nonunique generalized inverse formulas precisely dedicated to the parameter/polynomial square/non-square full (normal) rank matrices [8], [31]. At this point, the most important ones should be recalled in forms of the S-, σ-, τ -and H-inverse, in order to involve their in the perfect control design process [8], [29], [32].…”

“…Now, after engaging the above equation (31) to the control design process, together with the consideration of the new defined Theorem 4, we obtain the non-full rank pole-free maximum-speed/maximum-accuracy IMC-oriented closedloop perfect control system, which variables are presented in Fig. 2.…”

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

“…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,…”