Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problemsarising in the theory of self-adjoint operators, especially in the perturbationtheory. They allow to give a precise meaning to operations with functionsof two ordered operator-valued non-commuting arguments. In a different language,the theory of double operator integrals turns into the problem of scalarvaluedmultipliers for operator-valued kernels of integral operators.The paper gives a short survey of the main ideas, technical tools andresults of the theory. Proofs are given only in the rare occasions, usually theyare replaced by references to the original papers. Various applications arediscussed.