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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. Mathematics Subject Classification (2000). Primary: 47B49, 47A55.

Abstract. Matrix periodic differential operators (DO's)are considered. The operators are assumed to admit a factorization of the form A = X * X , where X is a homogeneous first order DO. Let A ε = A(ε −1 x, D), ε > 0. The behavior of the solutions u ε (x, τ) of the Cauchy problem for the Schrödinger equation i∂ τ u ε = A ε u ε , and also the behavior of those for the hyperbolic equation ∂ 2 τ u ε = −A ε u ε , is studied as ε → 0. Let u 0 be the solution of the corresponding homogenized problem. Estimates of order ε are obtained for theThe estimates are uniform with respect to the norm of initial data in the Sobolev space H s (R d ; C n ), where s = 3 in the case of the Schrödinger equation and s = 2 in the case of the hyperbolic equation. The dependence of the constants in estimates on the time τ is traced, which makes it possible to obtain qualified error estimates for small ε and large |τ | = O(ε −α ) with appropriate α < 1. §0. Introduction 0.1. The class of operators. The present paper is a continuation of the authors' investigations [BSu1, BSu2, BSu3, BSu4, BSu5, Su1, Su2] in homogenization theory for a class of (matrix) periodic differential operators (DO's) acting in the space L 2 (R d ; C n ). This class is rather wide and includes many operators of mathematical physics.We consider matrix elliptic positive second order DO's in L 2 (R d ; C n ) that admit a factorization of the formHere b(D) is a homogeneous matrix first order DO with constant coefficients. Its symbol b(ξ) is an (m × n)-matrix of rank n (we assume that m ≥ n). It is assumed that the matrix-valued functions f (x) (of size n × n) and g(x) (of size m × m) are periodic with respect to some lattice Γ in R d and thatFor a more precise description of the operators (0.1), see Subsection 4.1. It is convenient to start with a narrower class of operators of the formand accordingly, to accept the "two-level"order of exposition.

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