Spectrum Perturbations of Linear Operators in a Banach Space

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 N Γ p ( . ) $$N_{\varGamma _p}(.)$$ and the property ∑ k | λ k ( A ) | p ≤ a p N Γ p p ( A ) $$\sum _k |\lambda _k(A)|{ }^p\le a_p N_{\varGamma _p}^p(A)$$ , 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 − Ap 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)