The Decomposability for Operator Matrices and Perturbations

Let X and Y be Banach spaces. For A is an element of L(X), B is an element of L(Y), C is an element of L(Y, X), let M-C be the operator matrix defined on X circle plus Y by M-C=([(A)(0) (C)(B)) is an element of L(X circle plus Y). In this paper we investigate the decomposability for M-C. We consider Bishop's property (beta), decomposition property (delta) and Dunford's property (C) and obtain the relationship of these properties between M-C and its entries. We explore how sigma(*)(M-C) shrinks from sigma(*)(A) boolean OR sigma(*)(B), where sigma(*) denotes sigma beta, sigma(delta), sigma(C), sigma(dec). In particular, we develop some sufficient conditions for equality sigma(*)(M-C) = sigma(*)(A) boolean OR sigma(*)(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)(B) (A)(0)). Finally, we give applications of decomposability for operator matrices.