On the Index of Fredholm Pairs of Idempotents  

作　　者：Yue Qing WANG Hong Ke DU Yan Ni DOU 

机构地区：[1]Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 400042, P. R. China [2]College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2009年第4期679-686,共8页数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China （10871224）

摘　　要：Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operatorUsing the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operator

关 键 词：FREDHOLM orthogonal projection positive operator 

分 类 号：O175.5[理学—数学] O152.7[理学—基础数学]