Related papers: Generalized Drazin-Riesz Invertibility for Operato…

In this paper, we give a generalized Cline's formula for the generalized Drazin inverse. Let $R$ be a ring, and let $a,b,c,d\in R$ satisfying $$\begin{array}{c} (ac)^2 = (db)(ac), (db)^2 = (ac)(db);\\ b(ac)a = b(db)a, c(ac)d =…

A Hilbert space operator $T\in B$ is $(m,P)$-expansive, for some positive integer $m$ and operator $P\in B$, if $\sum_{j=0}^m{(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)T^{*j}PT^j}\leq 0$. No Drazin invertible operator $T$ can be…

Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with $\sigma\geq n/2$. The main aim of this paper is to provide necessary and sufficient…

Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $\lambda \in \cit$ such that $T$ does not have the single-valued extension property at $\lambda$. In this note we prove equality up to…

In this paper, we define and study the pseudo upper and lower semi B-Fredholm of bounded operators in a Banach space. In particular, we prove equality up to $S(T)$ between the left generalized Drazin spectrum and the pseudo upper semi…

Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin…

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…

In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an…

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most…

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is…

A Hilbert space operator $A\in\B$ is left $(X,m)$-invertible by $B\in\B$ (resp., $B\in\B$ is an $(X,m)$-adjoint of $A\in\B$) for some operator $X\in\B$ if…

This paper provides new necessary and sufficient conditions for the solvability to the operator equations $ AX-XB=C$ and $AX-YB=C,$ where $A $ and $B $ are group invertible operators defined on an infinite dimensional Hilbert space. In…

For applications like the numerical solution of physical equations a discretization scheme for operators is necessary. Recently frames have been used for such an operator representation. In this paper, we apply fusion frames for this task.…

In this paper, we establish the invertibility of the Berezin transform of the symbol as a necessary and sufficient condition for the invertibility of the Toeplitz operator on the Bergman space $L^2_a(\mathbb{D})$. More precisely, if ${\phi}…

A Drazin invertible Hilbert space operator $T\in \B$, with Drazin inverse $T_d$, is $(n,m)$-power D-normal, $T\in [(n,m) DN]$, if $[T_d^n,T^{*m}]=T^n_dT^{*m}-T^{*m}T_d^n=0$; $T$ is $(n,m)$-power D-quasinormal, $T\in [(n,m) DQN]$, if…

In this paper, we obtain several extensions of semi-Fredholm theory on Hilbert modules by generalizing in this setting their classical counterparts regarding Weyl operators and Drazin invertible operators.

A recent paper of Shemesh shows triangularizability of a pair $\{A, B\}$ of complex matrices satisfying the condition $A [A,B]=[A,B] B=0$, or equivalently, the matrices $A$ and $B$ commute with their product $A B$. In this paper we extend…

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is…

There has recently been renewed recognition of the need to understand the consistency properties that must be preserved when a generalized matrix inverse is required. The most widely known generalized inverse, the Moore-Penrose…

Let $T$ be a bounded linear operator on a Banach space $X$. We give new necessary and sufficient conditions for $T$ to be Drazin or Koliha-Drazin invertible. All those conditions have the following form: $T$ possesses certain decomposition…