INVARIANT AND HYPERINVARIANT SUBSPACES FOR AMENABLE OPERATORS

There has been a long-standing conjecture in Banach algebras that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to the statement that every non-scalar amenable operator has a non-trivial hyperinvariant subspace. It is also equivalent to the statement that every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, supporting the conjecture.