k -拟-*-A 类压缩算子的性质

数学物理学报 ›› 2014, Vol. 34 ›› Issue (4): 823-827.

k -拟-*-A 类压缩算子的性质

李晓春|高福根

河南师范大学数学与信息科学学院河南新乡 453007

收稿日期:2013-04-23 修回日期:2014-03-28 出版日期:2014-08-25 发布日期:2014-08-25
基金资助:
国家自然科学基金(11301155, 11271112)、河南省教育厅科学技术研究重点项目(13B110077)、河南师范大学博士科研启动费支持课题(qd12102)和河南师范大学青年基金资助

On Properties for k-Quasi-*-class A Contractions

LI Xiao-Chun, GAO Fu-Gen

College of Mathematics and Information Science, Henan Normal University, Henan Xinxiang |453007

Received:2013-04-23 Revised:2014-03-28 Online:2014-08-25 Published:2014-08-25
Supported by:
国家自然科学基金(11301155, 11271112)、河南省教育厅科学技术研究重点项目(13B110077)、河南师范大学博士科研启动费支持课题(qd12102)和河南师范大学青年基金资助

摘要/Abstract

摘要：

设T是一个Hilbert空间算子, 若满足T^*k(|T²|-|T^*|²)T^k≥0, 则称T为k -拟-*-A 类算子. 著名的Fuglede-Putnam定理: 若AX=XB, 则A^*X=XB^*, 其中A和B是正规算子. 该文中, 首先证明了若T是一个压缩的k -拟-*-A 类算子, 则T有非平凡的不变子空间或者T 是真压缩算子, 且正算子D=T^*k(|T²|-|T*|²)T^k是强稳定压缩算子; 其次证明了k -拟-*-A 类算子不是超循环算子; 最后证明了若X是Hilbert-Schmidt算子, A 和(B*)^-1是k -拟-*-A 类算子, 满足AX=XB, 则A*X=XB^*.

关键词: k -拟-*-A 类算子, 压缩算子, Fuglede-Putnam 定理

Abstract:

A Hilbert space operator T belongs to k-quasi-*-class A if T*^k(|T²|-|T*|²)T^k≥0. The famous Fuglede-Putnam's theorem is as follows: the operator equation AX=XB implies A*X=XB* when A and B are normal operators. In this paper, firstly we prove that if T is a contraction of k-quasi-*-class A
operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D=T*^k(|T²|-|T*|²)T^kis a strongly stable contraction; secondly we prove that k-quasi-*-class A operators are not supercyclic; at last we show that if X is a Hilbert-Schmidt operator, A and (B*)^-1 are k-quasi-*-class A operators such that AX=XB, then A*X=XB*.

Key words: k-quasi-*-class A operators, Contraction operator, The Fuglede-Putnam theorem

中图分类号:

47B20 47A63

李晓春, 高福根. k -拟-*-A 类压缩算子的性质[J]. 数学物理学报, 2014, 34(4): 823-827.

LI Xiao-Chun, GAO Fu-Gen. On Properties for k-Quasi-*-class A Contractions[J]. Acta mathematica scientia,Series A, 2014, 34(4): 823-827.

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