This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λ α (R) and 1 < p < ∞, the operator f (A) − f (B) belongs to S p/α , whenever A and B are self-adjoint operators with A − B ∈ S p . We also obtain sharp estimates for the Schatten-von Neumann norms f (A) − f (B) S p/α in terms of A − B S p and establish similar results for other operator ideals. We also estimate Schattenvon Neumann norms of higher order differences m j =0 (−1) m−j m j f (A + jK). We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) − f (B) to belong to S q under the assumption that A − B ∈ S p . We also obtain Schatten-von Neumann estimates for quasicommutators f (A)R − Rf (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A − K) − 2f (A) + f (A + K).

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Abstract. We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class S p for 0 < p < 1. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of S p . We also give a characterization of the Hankel-Schur multipliers of S p whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of S p . Finally, we return to Hankel-Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle.

We describe the maximal class of functions f on the real line, for which the Lifshitz-Krein trace formula trace(f (A) − f (B)) = R f ′ (s)ξ(s) ds holds for arbitrary self-adjoint operators A and B with A − B in the trace class S1. We prove that this class of functions coincide with the class of operator Lipschitz functions.