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In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitly by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevic generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b R. Then a R# with a# = b if and only if (ab)* = ab, ba 2 = a, and ab 2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

In this paper, we introduce two new generalized core inverses, namely, the (p,q,m)-core inverse and the ⟨p,q,n⟩-core inverse; both extend the inverses of the ⟨i,m⟩-core inverse, the (j,m)-core inverse, the core inverse, the core-EP inverse and the DMP-inverse.

This paper not only explores in detail the basic properties of the Kronecker product, such as the application of the union law, the distributive law, the transpose and the conjugate transpose, as well as the trace and the rank of the Kronecker product, but also rigorously proves these properties by means of mathematical induction or direct proofs. In addition, this paper extends these properties to Kronecker n-matrices and explores the applicability of inverse matrices and determinants to Kronecker products. More importantly, we introduce a conjecture related to the long-standing problem of distillability, which is partially solved when the number of nonzero elements is restricted. By showing the framework of the proof of the conjecture and a special case, we provide new perspectives for further research in this area.

We provide existence criteria and characterizations for outer inverses in an involutive semigroup belonging to the prescribed Green’s R -, L - and H -classes. We show that the existence criteria for such outer inverses are simultaneously the existence criteria for various types of generalized inverses, such as { 1 , 4 } -inverses, { 1 , 3 } -inverses, Moore–Penrose inverses, core inverses and dual core inverses. We also provide several new characterizations of the mentioned types of generalized inverses, as well as characterizations of { 2 , 4 } -inverses and { 2 , 3 } -inverses.

Let S be a ∗-semigroup and let a, w ∈ S. The initial goal of this work is to consider commuting properties of a new class of generalized inverses, called the complete w-core inverse of a, extending the EP invertibility of generalized inverses. It is shown that a is completely w-core invertible if and only if a is w-core invertible and a ω # commutes with aw if and only if a is w-EP invertible. Moreover, the representations of complete w-core inverses and w-EP inverses are both presented in rings. Also, the connections between the complete w-core inverse and other generalized inverses are given. More criterions for the complete w-core inverse is derived by units and one-sided ideals in rings.

In this paper we introduce left and right annihilator (b,c)-inverses and we investigate some of theirs properties. Furthermore, here we study some properties of left and right (b,c)-inverses.

Let R be a unital *-ring. For any a, s, t, v, w ∈ R we define the weighted w -core inverse and the weighted dual s -core inverse, extending the w -core inverse and the dual s -core inverse, respectively. An element a ∈ R has a weighted w -core inverse with the weight v if there exists some x ∈ R such that awxvx = x, xvawa = a and ( awx )* = awx . Dually, an element a ∈ R has a weighted dual s -core inverse with the weight t if there exists some y ∈ R such that ytysa = y, asaty = a and ( ysa )* = ysa . Several characterizations of weighted w -core invertible and weighted dual s -core invertible elements are given when weights v and t are invertible Hermitian elements. Also, the relations among the weighted w -core inverse, the weighted dual s -core inverse, the e -core inverse, the dual f -core inverse, the weighted Moore-Penrose inverse and the ( v, w )-( b, c )-inverse are considered.

Inspired by the concepts of Φ-GCEP and Φ-*GCEP inverses as generalizations of the gMP, *gMP and MP inverse, the aim of this paper is to extend notions of inner-gMP, gMP-inner, inner-*gMP, *gMP-inner, 1MP and MP1 inverses using Φ-GCEP and Φ-*GCEP inverses. Thus, four new types of generalized inverses are introduced and called inner-Φ-GCEP, Φ-GCEP-inner, inner-Φ-*GCEP and Φ-*GCEP-inner inverses. As particular kinds of these new inverses, inner-GCEP, GCEP-inner, inner-*GCEP and *GCEP-inner inverses are established. Numerous properties, representations and characterizations of new inverses are presented as well. Applying our new inverses, we solve some linear equations and express their solutions.