Algebra matrix and similarity classification of operators (Q862686)

A bounded linear operator \(A\) on a complex separable Hilbert space is said to be a Cowen--Douglas operator of index \(n\) (\(n\) is a given positive integer) if the spectrum of \(A\) contains a complex domain \(\Omega\) such that the operators \(A-z,\;z\in\Omega\), are onto and their kernels have dimension \(n\) and generate a dense subspace. \(A\) is called strongly irreducible if its commutant algebra \({\mathcal A}'(A)\) has no nontrivial idempotents. By Gelfand representation theory and the Silov idempotents theorem, the authors obtain a central idempotents theorem for unital semisimple \(n\)-homogeneous Banach algebras. In view of [\textit{C.\,L.\thinspace Jiang}, Can.\ J.\ Math.\ 56, No.\,4, 742--775 (2004; Zbl 1068.47044)] and as an application of this central idempotent theorem, it is finally shown that two strongly irreducible Cowen--Douglas operators \(A\) and \(B\) are similar if and only if \({\mathcal A}'(A\oplus B)\) has no nontrivial central idempotent elements.