Multiple operator integrals in perturbation theory

Abstract

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 H\"older 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 $B_{\infty,1}^1({\Bbb R}^2)$.