Multilinear Schur multipliers and Schatten properties of operator Taylor remainders

作者：Potapov Denis^*; Skripka Anna; Sukochev Fedor; Tomskova Anna

来源：Advances in Mathematics, 2017, 320: 1063-1098.

DOI：10.1016/j.aim.2017.09.012

• Multilinear Schur multipliers
• Taylor approximations of operator functions

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 C-n and a perturbation is in the pth Schatten class S-p, p > n, then the respective nth order operator Taylor remainder is an element of S-p/n and has an estimate like the one in [16]. We construct examples of f is an element of C-n and perturbations in S-n such that the nth order Taylor remainder of the respective operator function is not in S-1. 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.

• 出版日期2017-11-7