Local commutants and ultrainvariant subspaces

Abstract

For an operator A on a complex Banach space X and a closed subspace M⊆X, the local commutant of A at M is the set C(A;M) of all operators T on X such that TAx=ATx for every x∈M. It is clear that C(A;M) is a closed linear space of operators, however it is not an algebra, in general. For a given A, we show that C(A;M) is an algebra if and only if the largest subspace MA of X such that C(A;M)=C(A;MA) is invariant for every operator in C(A;M). We say that a closed subspace M⊆X is ultrainvariant for A∈B(X) if it is invariant for every operator in C(A;M). For several types of operators we prove that they have non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the family of all ultrainvariant subspaces of a nilpotent operator can be a proper subset of the lattice of all hyperinvariant subspaces.