Weyl spectra of operator matrices

Abstract

In this paper it is shown that if MC = ( iS a 2 x 2 upper triangular operator matrix acting on the Hilbert space 7H E IC and if W(.) denotes the spectrum, then the passage from w(A) U w(B) to w(MC) is accomplished by removing certain open subsets of w(A) n w(B) from the former, that is, there is equality w(A) U w(B) = w(MC) U G, where G is the union of certain of the holes in w(Mc) which happen to be subsets of w (A) n w (B). Let At and IC be Hilbert spaces, let 1C(7-, 1C) denote the set of bounded linear operators from 7t to IC, and abbreviate LC(QH,7) to C(H). When A E LC(t) and B E L(/C) are given we denote by Mc an operator acting on 1t E 1C of the form M A C) Mc:=( 0 ), where C E iC(!, 7-). The invertibility and spectra of Mc were considered by Du and Jin [5]. In this paper we give some conditions for operators A and B to exist an operator C such that Mc is Weyl, and describe the Weyl spectra of Mc. Recall ([7], [8]) that an operator A E 12(X, Y) for Banach spaces X and Y is called regular if there is an operator A' E L(Y, X) for which A = AA'A; then A' is called a generalized inverse for A. In this case, X and Y can be decomposed as follows (cf. [8, Theorem 3.8.2]): A-1(0) e A'A(X) = X and A(X) e (AA')-'(0) = Y. It is familiar ([6], [8]) that A E LC((, IC) is regular if and only if A has closed range. An operator A E LC(7-, IC) is called relatively Weyl if there is an invertible operator A' E 12(IC,7t) for which A = AA'A. It is known ([8, Theorem 3.8.6]) that A is relatively Weyl if and only if A is regular and A-1(0) A(H)', where means a topological isomorphism between spaces. An operator A E LQ(H, IC) is called left-Fredholm if it is regular with finite dimensional null space and rightFredholm if it is regular with its range of finite co-dimension. If A is both leftand right-Fredholm, we call it Fredholm. The index, ind A, of a leftor right-Fredholm Received by the editors November 21, 1997 and, in revised form, May 1, 1998 and March 10, 1999. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.