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Quasi-Fredholm spectrum for operator matrices

Ouali El, El Ech-Cherif, Mohammed Karmouni

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Abstract

For A ? L(X), B ? L(Y) and C ? L(Y,X) we denote by MC the operator matrix defined on X ? Y by MC = (A C 0 B). In this paper, we prove that ?qF(A) ? ?qF(B) ? [ C?L(Y,X) ?qF(MC) ? ?p(B) ? ?p(A?), where ?qF(.) (resp. ?p(.)) denotes the quasi-Fredholm spectrum (resp. the point spectrum). Furthermore, we consider some sufficient conditions for MC to be quasi-Fredholm and sufficient conditions to have ?qF(A) ? ?qF(B) = ? C?L(Y,X) ?qF(MC).

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For A ? L(X), B ? L(Y) and C ? L(Y,X) we denote by MC the operator matrix defined on X ? Y by MC = (A C 0 B). In this paper, we prove that ?qF(A) ? ?qF(B) ? [ C?L(Y,X) ?qF(MC) ? ?p(B) ? ?p(A?), where ?qF(.) (resp. ?p(.)) denotes the quasi-Fredholm spectrum (resp. the point spectrum). Furthermore, we consider some sufficient conditions for MC to be quasi-Fredholm and sufficient conditions to have ?qF(A) ? ?qF(B) = ? C?L(Y,X) ?qF(MC).

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Available abstract

For A ? L(X), B ? L(Y) and C ? L(Y,X) we denote by MC the operator matrix defined on X ? Y by MC = (A C 0 B). In this paper, we prove that ?qF(A) ? ?qF(B) ? [ C?L(Y,X) ?qF(MC) ? ?p(B) ? ?p(A?), where ?qF(.) (resp. ?p(.)) denotes the quasi-Fredholm spectrum (resp. the point spectrum). Furthermore, we consider some sufficient conditions for MC to be quasi-Fredholm and sufficient conditions to have ?qF(A) ? ?qF(B) = ? C?L(Y,X) ?qF(MC).

Key concepts: Operator matrix, Mathematics, Spectrum (functional analysis), Operator (biology), Fredholm determinant, Fredholm integral equation, Matrix (chemical analysis), Combinatorics

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