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Generalized Drazin spectrum of operator matrices

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Abstract

Let A ∈ B(X) and B ∈ B(Y), MC be an operator on Banach space X ⊕ Y given by \(\left( \begin{gathered} AC \hfill \\ 0B \hfill \\ \end{gathered} \right) \). A generalized Drazin spectrum defined by σ gD (T) = {λ ∈ ℂ: T − λI is not generalized Drazin invertible} is considered in this paper. It is shown that

$\sigma _{gD} (A) \cup \sigma _{gD} (B) = \sigma _{gD} (M_C ) \cup W_{gD} (A,B,C)$

, where W gD (A,B,C) is a subset of σ gD (A) ∩ σ gD (B) and a union of certain holes in σ gD (MC). Furthermore, several sufficient conditions for σ gD (A) ∪ σ gD (B) = σ gD (MC) holds for every C ∈ B(Y,X) are given.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China

    Shi-fang Zhang & Huai-jie Zhong

  2. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350002, China

    Li-qiong Lin

Authors
  1. Shi-fang Zhang
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  2. Huai-jie Zhong
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  3. Li-qiong Lin
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Correspondence to Shi-fang Zhang.

Additional information

Supported by the National Natural Science Foundation of China (11301077, 1117131, 11171066 and 11226113), Foundation of the Education Department of Fujian Province (JA12074) and the Natural Science Foundation of Fujian Province (2012J05003).

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Zhang, Sf., Zhong, Hj. & Lin, Lq. Generalized Drazin spectrum of operator matrices. Appl. Math. J. Chin. Univ. 29, 162–170 (2014). https://doi.org/10.1007/s11766-014-3142-1

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  • DOI: https://doi.org/10.1007/s11766-014-3142-1

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