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Journal of Operator Theory

Volume 69, Issue 1, Winter 2013  pp. 87-100.

Invariant and hyperinvariant subspaces for amenable operators

Authors Luo Yi Shi (1) and Yu Jing Wu (2)
Author institution: (1) Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, P.R. China
(2) Tianjin Vocational Institute, Tianjin 300410, P.R. China

Summary:  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.

Keywords:  amenable, invariant subspaces, hyperinvariant subspaces, reduction property

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