Let B(X) the Banach algebra of all bounded operators on a Banach space X and let T ∈ B(X). We denote by R alc (X) = {T ∈ B(X) : C(T ) = {0}} and R ac (X) = {T ∈ B(X) : K(T ) = {0}} where C(T ) and K(T ) are respectively the algebraic core and the analytic core. In this paper we show that R alc (X) and R ac (X) are a regularities in Kordula-Müller's sense.
Mathematics Subject Classification : 47A10
In this paper, we continue the study of the pseudo B-Fredholm operators of Boasso, and the pseudo B-Weyl spectrum of Zariouh and Zguitti; in particular we find that the pseudo B-Weyl spectrum is empty whenever the pseudo B-Fredholm spectrum is, and look at the symmetric differences between the pseudo B-Weyl and other spectra.
In this paper, we investigate the limit points set of surjective and approximate point spectra of upper triangular operator matrices M C = µ A C 0 B ¶. We prove that σ * (M C) ∪ W = σ * (A) ∪ σ * (B) where W is the union of certain holes in σ * (M C), which happen to be subsets of σ lgD (B) ∩ σ rgD (A), σ * ∈ {σ lgD , σ rgD } are the limit points set of surjective and approximate point spectra. Furthermore, several sufficient conditions for σ * (M C) = σ * (A) ∪ σ * (B) holds for every C ∈ B(Y, X) are given.
We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.
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