The closure of unitary orbit for strongly irreducible operators in a class of nest algebras

Abstract

A linear operatorT ∈L(H) is called a strongly irreducible, if there is no non-trivial idempotent linear operator commuting withT. In this paper, denote the set of all strongly irreducible operators by (SI). Let\(\mathfrak{N}\) be a nest with infinite dimensional atoms,\(\Gamma (\mathfrak{N})\) be the nest algebra associated with\(\mathfrak{N}\) and\(U(\mathfrak{N} \cap (SI))\) be the closure of\(\left\{ {UTU^* \left| {T \in } \right.(SI) \cap \Gamma \left( \mathfrak{N} \right)} \right\}\), then the following result is proved $$U\left( {\mathfrak{N} \cap \left( {SI} \right)} \right) = \left\{ {T\left| {T \in } \right.L\left( H \right),} \right.\sigma \left( T \right)is\left. {connected} \right\}.$$ .