Similarity invariants of operators on a class of Gowers-Maurey spaces  

作　　者：LIN LiQiong ZHONG HuaiJie ZHANG YunNan 

机构地区：[1]College of Mathematics and Computer Science,Fuzhou University [2]School of Mathematics and Computer Science,Fujian Normal University [3]Mathematics and Information Science College,Hebei Normal University

出　　处：《Science China Mathematics》2013年第4期803-810,共8页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11171066);Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 2010350311001);Fujian Natural Science Foundation (Grant No. 2009J05002);Scientific Research Foundation of Fuzhou University (Grant No. 022459)

摘　　要：This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, ∑dc spaces, where an infinite dimensional Banach space X is called a ∑dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T1 and T2 on a ∑dc space are similar if and only if theK0-group of the commutant algebra of the direct sum T1 GT2 is isomorphic to the group of integers Z. On a ∑dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in （∑SI）（X）, it further gives a necessary and sufficient condition that two operators in （∑SI）（X） are similar by using the ordered K0-groups. It also proves that every operator in （∑SI）（X） has a unique （SI） decomposition up to similarity on a ∑dc space X, where （∑SI）（X） denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces,dc spaces,where an infinite dimensional Banach space X is called a dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton.It shows that two strongly irreducible operators T1 and T2 on a dc space are similar if and only if the K 0-group of the commutant algebra of the direct sum T1⊕T2 is isomorphic to the group of integers Z.On a dc space X,it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in(ΣSI)(X),it further gives a necessary and sufficient condition that two operators in(ΣSI)(X) are similar by using the ordered K 0-groups.It also proves that every operator in(ΣSI)(X) has a unique(SI) decomposition up to similarity on a dc space X,where(ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.

关 键 词：Gowers-Maurey spaces strongly irreducible operators K0-groups commutant algebras 

分 类 号：O177[理学—数学]