Perturbations of regularized determinants of operators in a banach space

Let X be a separable Banach space with the approximation property. For an integer p ≥ 1, let Γ p be a quasinormed ideal of compact operators in X with a quasinorm N Γ p, such that ∑ k = 1 ∞ k (A) p ≤ a p N Γ p p (A) (A ϵ Γ p), where k (A) are the eigenvalues of A and a p is a constant independent of A. We suggest upper and lower bounds for the regularized determinants of operators from Γ p as well as bounds for the difference between determinants of two operators. Applications to the p -summing operators, Hille-Tamarkin integral operators, Hille-Tamarkin matrices, Schatten-von Neumann operators, and Lorentz operator ideals are discussed.