k-拟-*-A类压缩算子的性质  

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

作　　者：李晓春[1] 高福根[1] 

机构地区：[1]河南师范大学数学与信息科学学院,河南新乡453007

出　　处：《数学物理学报（A辑）》2014年第4期823-827,共5页Acta Mathematica Scientia

基　　金：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);河南师范大学青年基金资助

摘　　要：设T是一个Hilbert空间算子,若满足T^(*k)(|T^2|-|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)(|T^2|-|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~*.A Hilbert space operator T belongs to k-quasi-＊-class A if T^＊k（｜T2｜-｜T＊｜2）Tk≥0. The famous Fuglede-Putnam＇s theorem is as follows： the operator equation AX =XB ironies 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）Tk is 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-,-elass A operators such that AX = XB, then A＊X = XB^＊.

关 键 词：k-拟-＊-A类算子 压缩算子 Fuglede-Putnam定理 

分 类 号：O177.1[理学—数学]