K 0-group and approximate similarity invariants of strongly irreducible operators.

Let $$ \mathcal{H} $$ olenote a complex separable Hilbert space and $$ \mathcal{L}(\mathcal{H}) $$ denote the collection of bounded linear operators on $$ \mathcal{H} $$ . An operator T ∈ $$ \mathcal{L}(\mathcal{H}) $$ is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an “approximate inverse” of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ $$ \mathcal{L}(\mathcal{H}) $$ with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that $$ \parallel A - T\parallel < \varepsilon , V(\mathcal{A}'(A)) \cong \mathbb{N}, K_0 (\mathcal{A}'(A)) \cong \mathbb{Z} $$ , and $$ \mathcal{A}'(A) $$ /rad $$ \mathcal{A}'(A) $$ is commutative, where $$ \mathcal{A}'(A) $$ denotes the commutant of A and rad $$ \mathcal{A}'(A) $$ denotes the Jacobson radical of $$ \mathcal{A}'(A) $$ . The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan. [ABSTRACT FROM AUTHOR]