A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses

Abstract

In this paper, we give a new characterization of generalized Browder's theorem by considering equality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazinmeromorphic spectrum. Also, we generalize Cline's formula to the case of generalized Drazinmeromorphic invertibility under the assumption that A k B k A k = A k+1 for some positive integer k. Mathematics Subject Classification: 47A10, 47A53. Keywords: SVEP, generalized Drazin-meromorphic invertible, meromorphic operators, operator equation.semi-Fredholm, Fredholm, lower semi-Weyl , upper semi-Weyl and Weyl spectra are defined byA bounded linear operator T is said to be bounded below if it is injective and R(T ) is closed. For T ∈ B(X) the ascent denoted by p(T ) is the smallest non negative integer p such that kerT p = kerT p+1 . If no such integer exists we set p(T ) = ∞. For T ∈ B(X) the descent denoted by q(T ) is the smallest non negative integer q such that R(T q ) = R(T q+1 ). If no such integer exists we set q(T ) = ∞. By [1, Theorem 1.20] if both p(T ) and q(T ) are finite then p(T ) = q(T ). An operatorcalled upper semi-Browder if it is an upper semi-Fredholm and p(T ) < ∞. An operator T ∈ B(X) is called lower semi-Browder if it is an lower semi-Fredholm and q(T ) < ∞. We say that an operator T ∈ B(X) is Browder if it is upper semi-Browder and lower semi-Browder. The lower semi-Browder, upper semi-Browder and Browder spectra are defined by σ lb (T ) : = {λ ∈ C : λI − T is not lower semi-Browder}, σ ub (T ) : = {λ ∈ C : λI − T is not upper semi-Browder}, σ b (T ) : = {λ ∈ C : λI − T is not Browder}, respectively. Clearly, every Browder operator is Drazin invertible. An operator T ∈ B(X) is said to possess the single-valued extension property (SVEP) at λ 0 ∈ C if for every neighbourhood V of λ 0 the only analytic function f : V → X which satisfies the equation (λI − T )f (λ) = 0 is the function f = 0. If an operator T has SVEP at every λ ∈ C, then T is said to have SVEP. Morever, the set of all points λ ∈ C such that T does not have SVEP at λ is an open set contained in interior of σ(T ). Therefore, if T has SVEP at each point of an open punctured disc D \ {λ 0 } centered at λ 0 , T also has SVEP at λ 0 . p(λI − T ) < ∞ ⇒ T has SVEP at λ and q(λI − T ) < ∞ ⇒ T * has SVEP at λ. An operator T ∈ B(X) is called Riesz if λI −T is Browder for all λ ∈ C\{0}. An operator T ∈ B(X) is called meromorphic if λI − T is Drazin invertible for all λ ∈ C \ {0}. Clearly, every Riesz operator is meromorphic. A subspace M of X is said to be T -invariant if T (M) ⊂ M. For a T -invariant subspace M of X we define T M : M → M by T M (x) = T (x), x ∈ M. We say T is completely reduced by the pair (M, N) (denoted by (M, N) ∈ Red(T )) if M and N are two closed T -invariant subspaces of X such that X = M ⊕ N. An operator T ∈ B(X) is called semi-regular if R(T ) is closed and ker(T ) ⊂ R(T n ) for every n ∈ N. An operator T ∈ B(X) is called nilpotent if T n = 0 for some n ∈ N and called quasi-nilpotent if ||T n || 1 n → 0, i.e λI − T is invertible for ...

On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

“…Our next theorem gives new characterization of some Browder's theorem type classes in terms of spectra introduced and studied in this paper. This theorem is an improvement of some recent results dressed in [19,22]. The next theorem gives a sufficient condition for an operator T ∈ L(X) to have the W-SVEP.…”

“…The other implications are already done in [1]. The assertions (b) and (c) (in which some implications are already done in [1,6,19,22]) go similarly with (a). on T all other statements and under the assumption int σ gz w (T ) = ∅, all statements imply the statement (i).…”

Generalized Drazin-Meromorphic Invertible Operators and Browder's Type Theorems

“…by [10,Theorem 2.8] we get σ gDM (T ) = σ gDM W (T ) if and only if T satisfies generalized Browder's theorem.…”