Completely reducible operator algebras and spectral synthesis

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 (see the last page of [1]) and, independently, by V. Lomonosov. Finally, our result generalizes the results of Loginov and Sul'man [2] and Rosenthal [5] on reductive Hilbert space operators that commute with compact operators. The following theorem is the central result of the author's paper [4], where it was stated in a slightly different form: THEOREM 1. Let A be a commutative operator algebra on a Banach space X. If the commutant of A is completely reducible and the ranges of compact operators in A span X, then every operator in A is a scalar type spectral operator. If, in addition, A is a weakly closed unital completely reducible algebra, then A is generated, as a uniformly closed algebra, by a complete totally atomic Boolean algebra of projections. Moreover, A is reflexive and admits spectral synthesis (i.e., every invariant subspace of A is spanned by its one-dimensional invariant subspaces). Thus, in order to prove the result described above, it suffices to show that if A is a commutative completely reducible algebra which has enough hyperinvariant subspaces, then the commutant of A is also completely reducible. This will be done in Theorem 8 below. The above result then follows easily, a sufficient supply of Received by the editors September 14, 1981. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B40. 01987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

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Completely reducible operators that commute with compact operators

The survey is devoted to a circle of problems, grouped around one of the oldest problems of functional analysis, namely the invariant subspace problem. It is shown that the investigation of the algebraic and analytic properties of families of operators touches upon the question of the structure of their invariant subspaces.

Invariant subspaces of operator algebras

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

That commute with compact operators

We prove a theorem on the structure of weakly closed reductive operator algebras. The proof essentially relies on a known result of V. I. Lomonosov on transitive operator algebras containing a nonzero compact operator. We deduce a number of corollaries which apply to the reductivity problem.Bibliography: 20 titles.

V S Šul'man

Papers

On Reductive Operators and Operator Algebras