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The Decomposability for Operator Matrices and Perturbations

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Abstract

Let X and Y be Banach spaces. For A ∈ L(X), B ∈ L(Y), C ∈ L(Y, X), let MC be the operator matrix defined on X ⊕ Y by \({M_C} = \left( {\begin{array}{*{20}{c}} A&C \\ 0&B \end{array}} \right) \in L(X \oplus Y)\). In this paper we investigate the decomposability for MC. We consider Bishop’s property (β), decomposition property (δ) and Dunford’s property (C) and obtain the relationship of these properties between MC and its entries. We explore how σ*(MC) shrinks from σ*(A) ∪ σ*(B), where σ* denotes σβ,σδ,σC, σdec. In particular, we develop some sufficient conditions for equality σ*(MC) = σ*(A) ∪σ*(B). Besides, we consider the perturbation of these properties for MC and show that in perturbing with certain operators C the properties for MC keeps with A, B. Some examples are given to illustrate our results. Furthermore, we study the decomposability for \(\left( {\begin{array}{*{20}{c}} 0&A \\ B&0 \end{array}} \right)\). Finally, we give applications of decomposability for operator matrices.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, P. R. China

    Xiao Li Wang

  2. Statistics and Mathematics College, Inner Mongolia University of Finance and Economics, Hohhot, 010070, P. R. China

    Xiao Li Wang

  3. Mathematics Science College, Inner Mongolia Normal University, Hohhot, 010022, P. R. China

    Alatancang

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  1. Xiao Li Wang
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  2. Alatancang
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Correspondence to Xiao Li Wang.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11761029), Inner Mongolia Higher Education Science and Technology Research Project (Grant Nos. NJZY22323 and NJZY22324), Inner Mongolia Natural Science Foundation (Grant No. 2018MS07020)

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Wang, X.L., Alatancang The Decomposability for Operator Matrices and Perturbations. Acta. Math. Sin.-English Ser. 39, 497–512 (2023). https://doi.org/10.1007/s10114-023-1265-0

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  • DOI: https://doi.org/10.1007/s10114-023-1265-0

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