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Common properties of the operator products in local spectral theory

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Abstract

Let X, Y be Banach spaces, A,D: X → Y and B,C: Y → X be the bounded linear operators satisfying operator equation set

$\left\{ \begin{gathered} ACD = DBD, \hfill \\ DBA = ACA. \hfill \\ \end{gathered} \right. $

. In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property (β) and Dunford property (C).

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China

    Kai Yan & Xiao Chun Fang

Authors
  1. Kai Yan
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  2. Xiao Chun Fang
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Correspondence to Kai Yan.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11371279)

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Yan, K., Fang, X.C. Common properties of the operator products in local spectral theory. Acta. Math. Sin.-English Ser. 31, 1715–1724 (2015). https://doi.org/10.1007/s10114-015-5116-5

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  • DOI: https://doi.org/10.1007/s10114-015-5116-5

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