Representations of AlgLat(T)

Abstract

For a hyponormal operator T with the property that the boundary of the essential spectrum is of planar Lebesgue measure zero, it is proved that the operator algebra AlgLat(T') generated by the invariant subspace lattice of T is commutative. If in addition T is a pure hyponormal operator, then AlgLat(7') is shown to be contained in the bicommutant of T. These are particular cases of more general results obtained for restrictions and quotients of operators decomposable in the sense of Foia§. An operator T e L(H) on a complex Hilbert space is called reflexive if the operator algebra AlgLat(F) generated by the invariant subspace lattice of T is as small as it can be, namely, coincides with the closure of the algebra of all polynomials in T with respect to the weak operator topology. In (16) Sarason proved that normal operators and analytic Toeplitz operators are reflexive. In (7) Deddens was able to show that all isometries are reflexive. Using the Scott Brown technique Olin and Thomson (15) proved that, more general, all sub- normal operators are reflexive. In 1987 Scott Brown (3) applied his methods to prove invariant subspace results for hyponormal operators. In (4) Chevreau, Exner, and Pearcy formulated the conjecture that all hyponormal operators are reflexive. As a modest step in this direction we shall show that for each hyponormal op- erator T e L(H), for which the boundary of the essential spectrum has planar Lebesgue measure zero, the algebra AlgLat(F) is commutative. If, in addition, T is pure, i.e., has no nontrivial normal reducing parts, then AlgLat(F) is shown to be contained in the bicommutant of T. 1. Fredholm theory Let us denote by Y and Z complex Banach spaces that are dual to each other in the sense that either Z = Y' or Y = Z'. We fix continuous linear operators A e L(Y), Be L(Z) with