2011/04/18 by Aleksei Aleksandrov, Aleksandrov, Aleksei, Vladimir Peller +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #math.CA #math.CV #math.FA #math.SP
paper · pdf · doi:10.48550/arxiv.1104.3553
50 pages
arxiv created 2011/04/18 · arxiv updated 2011/04/19
In \citeAP2 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 \citeAP2 for certain special classes of functions. In particular, we improve estimates of Kato \citeKa and show that ‖ |S|-|T| ‖≤ C‖S-T‖log(2+log(‖S‖+‖T‖)/(‖S-T‖)) for every bounded operators S and T on Hilbert space. Here |S|\df(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 (-\be,0] and is concave on [0,\be), then its operator modulus of continuity Øf admits the estimate Øf(\d)≤\const∫e^\be(f(\d t) dt)/(t2log t), \d>0. We also study the problem of sharpness of estimates obtained in \citeAP2 and \citeAP4. We construct a C^\be function f on \R such that ‖f‖L^\be≤1, ‖f‖\Li≤1, and Øf(\d)≥\const \d√(log\frac2\d), \d∈(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 \e-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 \citeAP2.