Abstract
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 equiv- alent 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 de- compositions for amenable operators, supporting the conjecture.
Highlights
Throughout this paper, H denotes a complex separable infinite-dimension Hilbert space and B(H) denotes the bounded linear operators on H
If T ∈ B(H), we say that a subspace M of H is a hyperinvariant subspace for T if M is invariant under each operator in A′T ; M is a reducible subspace for T if M, M⊥ ∈ LatT
Suppose that T is amenable operator and A′T contains a subalgebra which is similar to an abelian von Neumman algebra with no direct summand of uniform multiplicity infinite, T is similar to a normal operator
Summary
Throughout this paper, H denotes a complex separable infinite-dimension Hilbert space and B(H) denotes the bounded linear operators on H.
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