Abstract
In this paper, we investigate the limit points set of essential spectrum of upper triangular operator matrices We prove that accσe(MC) ∪ W = accσe(A) ∪ accσe(B) where W is the union of certain holes in accσe(MC), which happen to be subsets of accσe(B) ∩ accσe(A). Also, several sufficient conditions for accσe(MC) = accσe(A) ∪ accσe(B) holds are given.
Highlights
Introduction and PreliminariesLet X, Y be infinite dimensional complex Banach spaces and B(X, Y ) denote the complex algebra of all bounded linear operators from X to Y
If T ∈ B(X), we denote by T ∗, N (T ), R(T ), σap(T ), σsu(T ), σ(T ), respectively the adjoint, the null space, the range, the approximate point spectrum, the surjectivity spectrum and the spectrum of T
The index of a semi Fredholm operator T is defined by ind(T ) = α(T ) − β(T )
Summary
Let X, Y be infinite dimensional complex Banach spaces and B(X, Y ) denote the complex algebra of all bounded linear operators from X to Y . T is semi-Fredholm if it is a lower or upper semiFredholm operator. The essential spectrum of T is the subset of C defined by: σe(T ) = {λ ∈ C : T − λI is not a Fredholm operator}
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