Functions of perturbed tuples of self-adjoint operators

We generalize earlier results of Aleksandrov and Pellet (2010) [2,3], Aleksandrov et al. (2011) [6], Pellet (1985) [13], Pellet (1990) [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B-infinity,1(1) (R-n), then f is operator Lipschitz and we show that if f satisfies a Holder condition of order alpha, then parallel to f(A(1), ... , A(n)) - f(B-1, ... , B-n)parallel to <= const max(1 <= j <= n) parallel to A(j) - B-j parallel to(alpha) for all n-tuples of commuting self-adjoint operators (A(1), ... , A(n)) and (B-1, ... , B-n). We also consider the case of arbitrary moduli of continuity and the case when the operators A(j) - B-j belong to the Schatten-von Neumann class S-p. (C) 2012 Published by Elsevier Masson SAS on behalf of Academie des sciences.