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Estimates of operator moduli of continuity. / Peller, V. V.; Александров, Алексей Борисович.
в: Journal of Functional Analysis, Том 261, № 10, 15.11.2011, стр. 2741-2796.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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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 -
ID: 5210046