Norm estimates for resolvents of linear operators in a Banach space and spectral variations

Let P_t (a ≤ t ≤ b) be a function whose values are projections in a Banach space. The paper is devoted to bounded linear operators A admitting the representation A = ∫ _a ^b ϕ(t)dP_t + V, where ϕ(t) is a scalar function and V is a compact quasi-nilpotent operator such that P_tV P_t = V P_t (a ≤ t ≤ b). We obtain norm estimates for the resolvent of A and a bound for the spectral variation of A. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in L^p are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.