IUMJ

Title: Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities

Authors: A. B. Aleksandrov and V. V. Peller

Issue: Volume 59 (2010), Issue 4, 1451-1490

Abstract:

This is a continuation of our papers [A.B. Aleksandrov and V.V. Peller, \emph{Operator H\"older-Zygmund functions}, Adv. Math. \textbf{224} (2010), 910--966] and [A.B. Aleksandrov and V.V. Peller, \emph{Functions of operators under perturbations of class $\mathbf{S}_{p}$}, J. Funct. Anal. \textbf{258} (2010), 3675--3724]. In those papers we obtained estimates for finite differences $(\Delta_{K}f)(A) = f(A+K) - f(A)$ of the order $1$ and \[ (\Delta_{K}^{m}f)(A) \stackrel{\mathrm{def}}{=} \sum_{j=0}^{m} (-1)^{m-j} {m \choose j} f(A+jK) \] of the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operators. In this paper we extend results of [the works cited above] to the case of unbounded self-adjoint operators $A$. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of [\emph{Operator H\"older-Zygmund functions}, cited above]. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers [cited above] were announced in [A.B. Aleksandrov and V.V. Peller, \emph{Functions of perturbed operators}, C. R. Math. Acad. Sci. Paris \textbf{347} (2009), 483--488. (English, with English and French summaries)].