良序套张量积的代数中的强不可约算子

Strongly Irreducible Operators in the Algebra of the Tensor Product of Well-Ordered Nests

设Ｈ为复的可分无限维Ｈｉｌｂｅｒｔ空间，称有界线性算子Ｔ为强不可约的，如果与Ｔ可交换的幂等算子只有０和Ｉ．王宗尧、蒋春澜、纪有清等人证明了在任何一个套的套代数中都存在大量的强不可约算子，并且找到了它们的酉轨道闭包．本文考虑有限个套的张量积的代数中强不可约算子的存在性问题。证明了：对复平面上任何一个连通完备集σ、总存在一个对角算子Ｎ和它的一个范数可以任意小的紧摄动Ｔ＝Ｘ＋Ｋ，使得Ｔ是一个强不可约算子、Ｔ在有限个良序套的张量积的代数中，并且σ（Ｔ）＝σｌｒｅ（Ｔ）＝σ（Ｎ）＝σｌｒｅ（Ｎ）＝σ进一步，文章还对具有单点谱的算子和良序套与正交补为良序套的张量积的代数进行了讨论，得到了一些结果．

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.