In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp
estimates for f(A)−f(B) were obtained for self-adjoint operators A and B
and for various classes of functions f on the real line R. In this paper
we extend those results to the case of functions of normal operators. We show
that if a function f belongs to the H\"older class \L_\a(\R^2), 0<\a<1,
of functions of two variables, and N1 and N2 are normal operators, then
\|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a. We obtain a more
general result for functions in the space
\L_\o(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\} for an
arbitrary modulus of continuity \o. We prove that if f belongs to the Besov
class B_{\be1}^1(\R^2), then it is operator Lipschitz, i.e.,
\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|. We also study
properties of f(N1)−f(N2) in the case when f\in\L_\a(\R^2) and N1−N2
belongs to the Schatten-von Neuman class \bS_p.Comment: 32 page