Double Operator Integrals

Abstract

The concept of a double operator integral on \({\mathcal {B}}({\mathcal {H}})\) was first introduced by Daletskii and Krein (Trudy Sem Functsion Anal Voronezh Gos Univ 1:81–105, 1956). They launched this theory in order to compute the derivative of the function t↦f(A(t)), where {A(t)}t is a family of bounded self-adjoint operators depending on the parameter t. Although the initial construction allowed to handle a limited class of functions f and produced bounds that depended on the spectra of operators, it created new conceptual and technical opportunities. Further development of perturbation theory and its applications stimulated extension and refinement of double operator integral constructions and methods, with ground breaking contributions made in Birman and Solomyak (Double Stieltjes Operator Integrals. Problems of Mathematical Physics, Izdat. Leningrad University, Leningrad, pp. 33–67, 1966; Double Stieltjes Operator Integrals. Problems of Mathematical Physics, Izdat. Leningrad University, Leningrad, pp. 26–30, 1967; Double Stieltjes Operator Integrals III. Problems of Mathematical Physics. Leningrad University, Leningrad, pp. 27–53, 1973), Peller (Funktsional Anal i Prilozhen 19(2):37–51, 1985), Potapov and Sukochev (Acta Math 207(2):375–389, 2011), Caspers et al. (Am J Math 141(3):593–610, 2019). There are also generalizations of double operator integrals to multilinear transformations, which are considered in the next section. In this chapter we discuss the main constructions and properties of double operator integrals that have found important applications.