Inequalities of the Carleman type for Schatten-von Neumann operators

Let S _p be the ideal of the Schatten-von Neumann operators with the norm N _p. For an integer p ≥ 3 and an A ∈ S _p, the inequalities of the type ||(I - A) ^-1det _p(I - A)|| ≤ exp [a _p N _p ^p(A) + c _p] are proved, where det _p(I A) is the regularized determinant, I is the unit operator and a _p, and b _p are explicitly pointed constants. Applications of these inequalities to spectrum perturbations of operators, as well as to invertibility and positive invertibility of infinite matrices are also discussed.