Journal of Operator Theory

Volume 36, Issue 1, Summer 1996 pp. 3-19.

A class of strongly irreducible operators with nice property

Authors: Chunlan Jiang (1) and Zongyao Wang (2)
Author institution:(1) Department of Mathematics, Jilin University, Chang Chun, 130023, P.R. CHINA
(2) Department of Mathematics, East China University, 130 Mei Long Road, Shanghai 200237, P.R. CHINA

Summary: A bounded linear operator T on Hilbert space H is strongly irreducible if T does not commute with any non-trivial idempotent. T is said to have nice property if either $\mathcal A'(T)$ or $\mathcal A'(T*)$ is a commutative strictly cyclic operator algebra. This paper uses the multiplication operators on Sobolev space to construct a class of strongly irreducible operators with nice property and proves that the set of operators similar to orthogonal direct sums of finitely many strongly irreducible operators with nice property is dense in $\mathcal L(H)$. The paper also proves that for each essentially normal operator T with connected $\sigma _{\rm{e}} (T)$ and ${\rm{ind}}(T - \lambda ) = 0(\lambda \in \rho _{{\rm{S}} - {\rm{F}}} (T))$, there exists a compact operator K such that T + K is strongly irreducible.

Keywords: Strongly irreducible operator, Sobolev space, spectrum, strictly cyclic operator algebra.

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