Spectrum perturbations of compact operators in a Banach space

Abstract

For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property [formula presented] where λk(A) (k = 1, 2, ...) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã A Γp and [formula presented] where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I - Ap be boundedly invertible and [formula presented] where ψp(A) = infk=1,2,... |1 - λkp $ begin{array}{} displaystyle λ kp end{array}$(A)|. Then I - Ãp is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.