An extension of the Koplienko-Neidhardt trace formulae

Koplienko [Ko] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class \bS_2. A similar formula in the case of unitary operators was obtained by Neidhardt [N]. In this paper we improve their results and obtain sharp conditions under which the Koplienko--Neidhardt trace formulae hold.Comment: 21 page

Multiple operator integrals and higher operator derivatives

In this paper we consider the problem of the existence of higher derivatives of the function t\mapsto\f(A+tK), where \f is a function on the real line, $A$ is a self-adjoint operator, and $K$ is a bounded self-adjoint operator. We improve earlier results by Sten'kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten'kin. We also consider a similar problem for unitary operators.Comment: 24 page

On S. Mazur's problems 8 and 88 from the Scottish Book

The paper discusses Problems 8 and 88 posed by Stanislaw Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Section 5 of my paper "Estimates of functions of power bounded operators on Hilbert spaces", J. Operator Theory 7 (1982), 341-372. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper by Pelczynski and Sukochev.Comment: 8 page

Almost commuting functions of almost commuting self-adjoint operators

Let $A$ and $B$ be almost commuting (i.e, AB-BA\in\bS_1) self-adjoint operators. We construct a functional calculus \f\mapsto\f(A,B) for \f in the Besov class B_{\be,1}^1(\R^2). This functional calculus is linear, the operators \f(A,B) and $\psi(A,B)$ almost commute for \f,\,\psi\in B_{\be,1}^1(\R^2), \f(A,B)=u(A)v(B) whenever \f(s,t)=u(s)v(t), and the Helton--Howe trace formula holds. The main tool is triple operator integrals.Comment: 6 page

Estimates of operator moduli of continuity

In \cite{AP2} 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 \cite{AP2} for certain special classes of functions. In particular, we improve estimates of Kato \cite{Ka} and show that $$\big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|})$$ 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 \O_f admits the estimate \O_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. We also study the problem of sharpness of estimates obtained in \cite{AP2} and \cite{AP4}. We construct a C^\be function $f$ on $\R$ such that \|f\|_{L^\be}\le1, \|f\|_{\Li}\le1, and \O_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(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 \cite{AP2}.Comment: 50 page