On Isolated Points of the Spectrum of a Bounded Linear Operator

For a bounded linear operator A on a Banach space we characterize the isolated points in the spectrum of A , the Riesz points of A , and the poles of the resolvent of A . 1. Terminology and introduction Throughout this paper E will be an infinite-dimensional complex Banach space and A will be a bounded linear operator on E. We denote by N(A) the kernel and by A(E) the range of A. The spectrum of A will be denoted by o(A). The resolvent set g(A) of A is the complement of a (A) in the complex plane C. For any X in g(A) the resolvent operator (XI A)~x is denoted by Rx(A). Let Xo be an isolated point in a (A). The spectral projection corresponding to X0 will be denoted by PXo. We have E = PXo(E) © N(PXo). In [3] Mbekhta introduced two important subspaces of E: K(A) = {x e E : there exist c > 0 and a sequence (x„)„>x C E such that Ax\ = x, Axn+X = x„ for all 77 £ N, and ||x„|| < c"||x|| for all 77 £ N}, H0(A) = {x£E: lim H/Txll1/" = 0) I n—>oo J and proved the following Theorem 1. A point Xq £ a (A) is isolated in a (A) if and only if there is a bounded projection P on E such that P(E) = H0(X0I A) and N(P) = K(X0I A). In the present paper we shall prove that Ao £ o(A) is an isolated point of a (A) if and only if K(X0I A) is closed and E = K(X0I -A)® H0(X0I A) (where © denotes the algebraically direct sum). This characterization leads to Received by the editors March 4, 1991 and, in revised form, June 24, 1991. 1991 Mathematics Subject Classification. Primary 47A10; Secondary 47B06.