Zong Yao WANG,Zhi Gang FENG
Suppose that H is a complex separable infinite dimensional Hilbert space, a bounded operator T is said to be strongly irreducible if the only idempotents that commute with T are be 0 or I. Wang Zongyao, Jiang Chunlan and Ji Youqing have proved that there are a lot of strongly irreducible operators in the nest algebra of any nests and have got the closure of the unitary orbit of them. This paper discusses the existence of strongly irreducible operators in the algebra of the tensor product of finitely many nests. We prove that for any connected perfect set a in the complex plane, there exists a diagonal operator N and its compact perturbation T= N + K, the norm of K can be arbitrarily small, such that T is a strongly irreducible operator. T is in the algebra of the tensor product of finitely many well-ordered nests and σ(T) = σlr(T)=σ(N)=σlre(N) = σ. Furthermore. this paper discusses the operator with singleton spectrum and the algebra of the tensor product of well-ordered nests and the nests whose orthogonal complement is well-ordered, and obtains some results.
