Spectrum perturbations of compact operators in a Banach space

For an integer p ≥ 1, let Γ_p be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓ_p(.) and the property [formula presented] where λ_k(A) (k = 1, 2, ...) are the eigenvalues of A and a_p is a constant independent of A. Let A, Ã A Γ_p and [formula presented] where b_p is the quasi-triangle constant in Γ_p. It is proved the following result: let I be the unit operator, I - A^p be boundedly invertible and [formula presented] where ψ_p(A) = inf_k=1,2,... |1 - λkp $ begin{array}{} displaystyle λ k^p end{array}$(A)|. Then I - Ã^p 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.