Estimates of operator moduli of continuity

In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that | S| − | T | C S −T log 2+log S + T S − T for all bounded operators S and T on Hilbert space. Here |S| def = (S∗S)1/2. Moreover, we show that this inequality is sharp. We prove in this paper that if f is a nondecreasing continuous function on R that vanishes on (−∞, 0] and is concave on [0,∞), then its operator modulus of continuity Ωf admits the estimate Ωf (δ) const ∞ e f (δt)dt t2 log t , δ>0. We also study the problem of sharpness of estimates obtained in Aleksandrov and Peller (2010) [2,3]. We construct a C∞ function f on R such that f L∞ 1, f Lip 1, andIn the last section of the paper we obtain sharp estimates of f (A)−f (B) in the case when the spectrum of A has n points. Moreover, we obtain a more general result in terms of the ε-entropy of the spectrum that also improves the estimate of the operator moduli of continuity of Lipschitz functions on finite intervals, which was obtained in Aleksandrov and Peller (2010) [2]. © 2011 Elsevier Inc. All rights reserved.