Estimates of operator moduli of continuity

• Published in: Journal of functional analysis Vol. 261; no. 10; pp. 2741 - 2796
• Main Authors: Aleksandrov, A.B., Peller, V.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.11.2011
• Subjects: Commutators; Operator Lipschitz function; Operator modulus of continuity; Self-adjoint operator; Self-adjoint operator; Operator Lipschitz function; Operator modulus of continuity; Commutators
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2011.07.009

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 Ω f admits the estimate Ω f ( δ ) ⩽ const ∫ e ∞ f ( δ t ) d t t 2 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 , and Ω f ( δ ) ⩾ const δ log 2 δ , δ ∈ ( 0 , 1 ] . 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-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].