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On the index of Fredholm pairs of idempotents

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Abstract

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(ℛ(P) ∩ ℛ(Q)⊥) − dim(ℛ(P)⊥ ∩ ℛ(Q)), for all m >- n. Moreover, we prove that dim(ℛ(P) ∩ ℛ(Q)⊥) = dim(ℛ(P)⊥ ∩ ℛ(Q)) if and only if there exists a unitary U such that UP = QU and PU = UQ, where ℛ(T) denotes the range of T.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing, 400042, P. R. China

    Yue Qing Wang

  2. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, P. R. China

    Yue Qing Wang, Hong Ke Du & Yan Ni Dou

Authors
  1. Yue Qing Wang
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  2. Hong Ke Du
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  3. Yan Ni Dou
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Correspondence to Hong Ke Du.

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

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Wang, Y.Q., Du, H.K. & Dou, Y.N. On the index of Fredholm pairs of idempotents. Acta. Math. Sin.-English Ser. 25, 679–686 (2009). https://doi.org/10.1007/s10114-009-7067-1

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  • DOI: https://doi.org/10.1007/s10114-009-7067-1

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