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

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

Operator and commutator moduli of continuity for normal operators

We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if 0<\a<1 and $f$ is a H\"older function of order \a, then for normal operators $N_1$ and $N_2$, \|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{\L_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}. In the last section we obtain lower estimates for constants in operator H\"older estimates.Comment: 33 page