Invariant and hyperinvariant subspaces for amenable operators

TL;DR

This paper investigates the structure of amenable operators on Hilbert spaces, exploring their invariant subspaces and decompositions to address a long-standing conjecture about their similarity to normal operators.

Contribution

It establishes equivalences related to the conjecture and provides two decompositions for amenable operators supporting the hypothesis.

Findings

The conjecture is equivalent to amenable operators having non-trivial hyperinvariant subspaces.

Amenable operators are shown to be similar to reducible operators.

Two decompositions for amenable operators are proposed, supporting the conjecture.

Abstract

There has been a long-standing conjecture in Banach algebra 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 every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to 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, which supporting the conjecture.