The Decomposability for Operator Matrices and Perturbations.

Let X and Y be Banach spaces. For A ∈ L(X), B ∈ L(Y), C ∈ L(Y, X), let M_C be the operator matrix defined on X ⊕ Y by M C = ( A C 0 B ) ∈ L (X ⊕ Y) . In this paper we investigate the decomposability for M_C. We consider Bishop's property (β), decomposition property (δ) and Dunford's property (C) and obtain the relationship of these properties between M_C and its entries. We explore how σ_*(M_C) shrinks from σ_*(A) ∪ σ_*(B), where σ_* denotes σ_β,σ_δ,σ_C, σ_dec. In particular, we develop some sufficient conditions for equality σ_*(M_C) = σ_*(A) ∪σ_*(B). Besides, we consider the perturbation of these properties for M_C and show that in perturbing with certain operators C the properties for M_C keeps with A, B. Some examples are given to illustrate our results. Furthermore, we study the decomposability for ( 0 A B 0 ) . Finally, we give applications of decomposability for operator matrices. [ABSTRACT FROM AUTHOR]