arXiv subjects
Publications and source records attributed to Vladimir Peller.
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in the trace norm for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$.
The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions coincides with the class of operator Lipschitz functions.
We study Schatten--von Neumann properties of multiple operator integrals with integrands in the Haagerup tensor product of $L^\infty$ spaces. We obtain sharp, best possible estimates. This allowed us to obtain sharp Schatten--von Neumann estimates in the case of {\it Haagerup-like} tensor products.
The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz functions on closed subsets of the plane as well as commutator Lipschitz functions on such subsets. Am important role is played by double operator integrals and Schur multipliers.
We start with the Birman--Solomyak approach to define double operator integrals and consider applications in estimating operator differences $f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the Birman--Solomyak approach to the Lifshits--Krein trace formula that is based on double operator integrals. We study the class of operator Lipschitz functions, operator differentiable functions, operator Hölder functions, obtain Schatten--von Neumann estimates for operator differences. Finally, we consider in Chapter 1 estimates of functions of normal operators and functions of $d$-tuples of commuting self-adjoint operators. In Chapter 2 we define multiple operator integrals with integrands in the integral projective tensor product of $L^\infty$ spaces. We consider applications of such multiple operator integrals to the problem of the existence of higher operator derivatives and to the problem of estimating higher operator differences. We also consider connections with trace formulae for functions of operators under perturbations of class $\boldsymbol{S}_m$, $m\ge2$. In the last chapter we define Haagerup-like tensor products of the first kind and of the second kind and we use them to study functions of noncommuting self-adjoint operators under perturbation. We show that for functions $f$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ and for $p\in[1,2]$ we have a Lipschitz type estimate in the Schatten--von Neumann norm $\boldsymbol{S}_p$ for functions of pairs of noncommuting self-adjoint operators, but there is no such a Lipschitz type estimate in the norm of $\boldsymbol{S}_p$ with $p>2$ as well as in the operator norm. We also use triple operator integrals to estimate the trace norms of commutators of functions of almost commuting self-adjoint operators and extend the Helton--Howe trace formula for arbitrary functions in the Besov space $B_{\infty,1}^1({\Bbb R}^2)$.
Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $φ\mapstoφ(A,B)$ for functions $φ$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$. This functional calculus is linear, the operators $φ(A,B)$ and $ψ(A,B)$ almost commute for $φ,\,ψ\in B_{\infty,1}^1({\Bbb R}^2)$, and $φ(A,B)=u(A)v(B)$ whenever $φ(s,t)=u(s)v(t)$. We extend the Helton--Howe trace formula for arbitrary functions in $B_{\infty,1}^1({\Bbb R}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of $L^\infty$ spaces.
We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const(\|A_1-A_2\|_{\bS_p}+\|B_1-B_2\|_{\bS_p})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in $\bS_p$ with $p>2$. The main tool is Schatten--von Neumann norm estimates for triple operator integrals.
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 $ψ(A,B)$ almost commute for $\f,\,ψ\in B_{\be,1}^1(\R^2)$, $\f(A,B)=u(A)v(B)$ whenever $\f(s,t)=u(s)v(t)$, and the Helton--Howe trace formula holds. The main tool is triple operator integrals.
We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in the operator norm.
Let $(A_1,\cdots,A_n)$ and $(B_1,\cdots,B_n)$ be $n$-tuples of commuting self-adjoint operators on Hilbert space. For functions $f$ on $\R^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of $f(A_1,\cdots,A_n)-f(B_1,\cdots,B_n)$ in terms of the corresponding norms of $A_j-B_j$, $1\le j\le n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for $n\ge3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary $n$. We also get sharp estimates for quasicommutators $f(A_1,\cdots,A_n)R-Rf(B_1,\cdots,B_n)$ in terms of norms of $A_jR-RB_j$, $1\le j\le n$, for a bounded linear operator $R$.
We generalize earlier results of Peller, Aleksandrov - Peller, Aleksandrov - Peller - Potapov - Sukochev to the case of functions of $n$-tuples of commuting self-adjoint operators. In particular, we prove that if a function $f$ belongs to the Besov space $B_{\be,1}^1(\R^n)$, then $f$ is operator Lipschitz and we show that if $f$ satisfies a Hölder condition of order $\a$, then $\|f(A_1...,A_n)-f(B_1,...,B_n)\|\le\const\max_{1\le j\le n}\|A_j-B_j\|^\a$ for all $n$-tuples of commuting self-adjoint operators $(A_1,...,A_n)$ and $(B_1,...,B_n)$. We also consider the case of arbitrary moduli of continuity and the case when the operators $A_j-B_j$ belong to the Schatten--von Neumann class $\bS_p$.
We discuss the process of deterioration of quality of mathematical conferences caused by computer presentations.
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 Hölder function of order $\a$, then for normal operators $N_1$ and $N_2$, $$ \|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{Ł_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}. $$ In the last section we obtain lower estimates for constants in operator Hölder estimates.
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 $Ø_f$ admits the estimate $$ Ø_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. $$ 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 $$ Ø_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(0,1]. $$ 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}.
We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a Hölder function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator Hölder of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Finally, we obtain sharp estimates in the case of perturbations of Schatten-von Neumann class $\bS_p$ and obtain analogs of all the results for commutators and quasicommutators. Note that the proofs in the case of dissipative operators are considerably more complicated than the proofs of the corresponding results for self-adjoint operators, unitary operators, and contractions that were obtained earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}.
We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained. This formula includes the Livshits--Krein trace formula in the case $m=1$ and the Koplienko trace formula in the case $m=2$. We establish most general trace formulae in the case of perturbation of Schatten--von Neumann class $\bS_m$. We also improve the trace formula obtained in \cite{PSS} for operator Taylor polynomials and prove it for arbitrary functions in he Besov space $B_{\be1}^m(\R)$. We consider several other special cases of our general trace formulae. In particular, we establish a trace formula for $m$th order operator differences.
In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the Hölder class $Ł_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{Ł_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $Ł_ø(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\constø(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $ø$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\inŁ_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.
In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this note we extend those results to the case of functions of normal operators. We show that if $f$ belongs to the Hölder class $Ł_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{Ł_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $Ł_ø(\R^2)=\big\{f: |f(\z_1)-f(\z_2)|\le\constø(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $ø$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\inŁ_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.