Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6426070 | Advances in Mathematics | 2011 | 36 Pages |
In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)âf(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 Hölder class Îα(R2), 0<α<1, of functions of two variables, and N1 and N2 are normal operators, then âf(N1)âf(N2)â⩽constâfâÎαâN1âN2âα. We obtain a more general result for functions in the space ÎÏ(R2)={f:|f(ζ1)âf(ζ2)|⩽constÏ(|ζ1âζ2|)} for an arbitrary modulus of continuity Ï. We prove that if f belongs to the Besov class Bâ11(R2), then it is operator Lipschitz, i.e., âf(N1)âf(N2)â⩽constâfâBâ11âN1âN2â. We also study properties of f(N1)âf(N2) in the case when fâÎα(R2) and N1âN2 belongs to the Schatten-von Neumann class Sp.