Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators. The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.