Multilinear Schur multipliers and Schatten properties of operator Taylor remainders

Abstract

We establish sharp conditions on scalar functions and perturbations that guarantee Schatten summability of nth order operator Taylor remainders. In the special case of dimension one, our estimates of these remainders deliver well known classical estimates of scalar Taylor remainders. We prove that if a scalar function f is in the set Cn and a perturbation is in the pth Schatten class Sp, p>n, then the respective nth order operator Taylor remainder is an element of Sp/n and has an estimate like the one in [16]. We construct examples of f∈Cn and perturbations in Sn such that the nth order Taylor remainder of the respective operator function is not in S1. Our construction relies, in particular, on novel dimension dependent estimates for Schatten norms of multilinear Schur multipliers from below that are of interest in their own right. Our results apply to both self-adjoint and unitary operators.