## Abstract

This chapter is a survey of the recent results of the author on the spectrum perturbations of linear operators in a Banach space. It consists of three parts. In the first part, for an integer p ≥ 1, we introduce the approximative quasi-normed ideal Γ_{p} of compact operators A with a quasi-norm NΓp(.) and the property ∑k|λk(A)|p≤apNΓpp(A), where λ_{k}(A) (k = 1, 2, …) are the eigenvalues of A and a_{p} is a constant independent of A. Let I be the unit operator. Assuming that A ∈ Γ_{p} and I − A^{p} is boundedly invertible, we obtain invertibility conditions for perturbed operators. Applications of these conditions to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples, in the first part of the chapter, we consider the Hille–Tamarkin integral operators and Hille–Tamarkin infinite matrices. The second part of the chapter deals with the ideal of nuclear operators A in a Banach space satisfying the condition ∑_{k}x_{k}(A) < ∞, where x_{k}(A) (k = 1, 2, …) are the Weyl numbers of A. The inequality between the resolvent and determinant of A is derived. That inequality gives us new perturbation results. The third part of the chapter is devoted to non-compact operators in a Banach space having maximal chains of invariant subspaces and admitting the so-called triangular representation. The representation for the resolvents of such operators via multiplicative operator integrals is established. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. In addition, a norm estimate for the resolvent of operators admitting triangular representation is derived. It enables us to obtain a perturbation bound for the spectral variations and to show that the considered operators are Kreiss-bounded. Applications to operators in L^{p} are also discussed. In particular, a new bound for the spectral radius of an integral operator is obtained. Some of the results presented in this chapter are new.

Original language | English |
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Title of host publication | Springer Optimization and Its Applications |

Publisher | Springer |

Pages | 297-333 |

Number of pages | 37 |

DOIs | |

State | Published - 1 Jan 2021 |

### Publication series

Name | Springer Optimization and Its Applications |
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Volume | 179 |

ISSN (Print) | 1931-6828 |

ISSN (Electronic) | 1931-6836 |

## Keywords

- Absolutely (p, 2)-summing operators
- Absolutely p-summing operators
- Banach space
- Compact operators
- Determinant
- Infinite matrices
- Integral operators
- Invariant projections
- Linear operators
- Perturbations
- Resolvent

## ASJC Scopus subject areas

- Control and Optimization