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 RR that vanishes on (−∞,0](−∞,0] and is concave on [0,∞)[0,∞), then its operator modulus of continuity ΩfΩf admits the estimateΩf(δ)⩽const∫e∞f(δt)dtt2logt,δ>0. We also study the problem of sharpness of estimates obtained in Aleksandrov and Peller (2010) [2] and [3]. We construct a C∞C∞ function f   on RR such that ‖f‖L∞⩽1‖f‖L∞⩽1, ‖f‖Lip⩽1‖f‖Lip⩽1, andΩf(δ)⩾constδlog2δ,δ∈(0,1]. In the last section of the paper we obtain sharp estimates of ‖f(A)−f(B)‖‖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].