Study of the property (bz) using local spectral theory methods

Abstract

For a bounded linear operator, by local spectral theory methods, we study the property $(bz),$ which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of $\mathbb{X},$ also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed.

Keywords:

• Property (bz)
• semi-Fredholm operator
• tensor product
• $\mathit{SVEP}$

2020 Mathematics Subject Classification:

• Primary 47A10
• 47A11
• Secondary 47A53
• 47A55

Acknowledgements

We would like to thank the reviewers for their comments and corrections, which have improved the presentation of the article. We thank ESPOL professors Francisca Flores Nicolalde and Marcos Mendoza for their encouragement and support of mathematical research.

Disclosure statement

No potential conflict of interest was reported by the author(s).