On Decomposability of Compact Perturbations of Operators

Let A be a Hilbert-space operator satisfying the growth condition ||(z —/4) || < expi&[dist(z, /)]~SS, z //.where / is a C Jordan curve, and K > 0, s £ (0, 1) are two constants. Let T = A + B for some fi e C , I < p < oc. It is shown that T is strongly decomposable if and only if cr(T') does not fill the "interior" of /. 1, H. Radjavi and the author [13] showed that if a Hilbert-space operator T is the sum of a unitary operator A and an operator B of a Schatten class C (1 < p < oc) such that o(T) does not fill the unit disc, then T is strongly decomposable. In the present paper we show that the above statement remains true if (a) the unit disc is replaced by a domain whose boundary is a C Jordan curve /, (b) the unitary operator is replaced by an operator A whose resolvent satisfies the growth condition (1) \\(zA)"1!! 0 are two constants. (By S~ we mean it exists and is bounded.) (Recall that (i) an invariant subspace Y of T is called a maximal spectral subspace of T it M C Y tor all invariant subspaces M of T such that cr(T\M) C o(T\Y), (ii) T is said to be decomposable if for every finite open covering Gv G2> . . . , G of o(T) there is a family Yy Y2, . . . , Yn of maximal spectral subspaces of T such that <7\T|Y.)CG., i = 1, 2,...,«, and iV = Yj + Y2 + • • •+ Y where H is the underlying Hilbert space, (iii) T is said to be strongly decomposable if the restriction of T to an arbitrary maximal spectral subspace of T is again decomposable.) Let us begin with a detailed introduction. Many authors have proved the existence of a nontrivial invariant subspace for the perturbation of a Hermitian operator A by an operator B of a class C (1 < p < oc); a complete list of these authors is given in [6, p. 2122], These results were further generalized by K. Kitano fe] to the case where A is a normal operator Received by the editors September 9, 1974. AMS (MOS) subject classifications (1970). Primary 47A15, 47B05, 47B15.