Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators

Abstract

We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kato–Riesz decomposition if there exists a pair of T-invariant closed subspaces (M, N) such that , the reduction is Kato and is Riesz. In this paper, we define and investigate the generalized Kato–Riesz spectrum of an operator. For T is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator S acting on X such that , , is Riesz. We investigate generalized Drazin-Riesz invertible operators and also characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular, we characterize the single-valued extension property at a point in the case that admits a generalized Kato–Riesz decomposition.

Keywords:

• Banach space
• Kato operators
• Riesz operators
• single-valued extension property
• approximate point (surjective) spectrum
• essential spectra

AMS Subject Classifications:

• 47A53
• 47A10

Acknowledgements

The authors wish to thank Professors Bhagwati P. Duggal and Vladimir Pavlović for helpful conversations concerning the paper. The authors would also like to thank the referee for useful suggestions which helped improve the original version of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Ministry of Education, Science and Technological Development, Republic of Serbia [grant number 174007].