TY - JOUR

T1 - Estimates of operator moduli of continuity

AU - Peller, V. V.

AU - Александров, Алексей Борисович

N1 - Funding Information: ✩ The first author is partially supported by RFBR grant 11-01-00526-a; the second author is partially supported by NSF grant DMS 1001844. * Corresponding author. E-mail address: peller@math.msu.edu (V.V. Peller).

PY - 2011/11/15

Y1 - 2011/11/15

N2 - 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(delta;)≤const ∫e∞ f(δt)dt/t2logt, 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, and. In 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-ntropy 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].

AB - 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(delta;)≤const ∫e∞ f(δt)dt/t2logt, 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, and. In 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-ntropy 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].

KW - Commutators

KW - Operator lipschitz function

KW - Operator modulus of continuity

KW - Self-adjoint operator

UR - http://www.scopus.com/inward/record.url?scp=80052426661&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2011.07.009

DO - 10.1016/j.jfa.2011.07.009

M3 - Article

VL - 261

SP - 2741

EP - 2796

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 10

ER -