THAT COMMUTE WITH COMPACT OPERATORS

It is shown that if T is a completely reducible operator on a Banach space and TK = KT, where K is an injective compact operator with a dense range, then T is a scalar type spectral operator. Other related results are also obtained. Let A be an algebra of bounded linear operators on a Banach space X. lat A is the lattice of (closed) invariant subspaces of A. We say that A is completely reducible if for every M c lat A there is N c lat A with M -+ N = X (that is, M n N = 0 and the algebraic sum M + N coincides with X). An operator T is completely reducible if the algebra generated by T is. It is unknown whether a weakly closed unital completely reducible algebra must be reflexive; that is, must contain every operator which leaves invariant its invariant subspaces. Some partial solutions of this problem can be found in (1, 6, 7). In this paper we show that every completely reducible operator commuting with an injective compact operator with a dense range is a scalar type spectral operator. In particular, the weakly closed unital algebra generated by such an operator must be reflexive. This result seems to be unknown even for operators on a Hilbert space. Also, we show that every compact completely reducible operator must be a scalar type spectral operator. This answers a question raised by E. Azoff and A. Lubin