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 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 ∑_kx_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.