Let X, Y be Banach spaces and let L(X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on L(X,Y) and apply this theory to obtain commutator estimates of the form∥f(B)S−Sf(A)∥L(X,Y)≤const∥BS−SA∥L(X,Y)for a large class of functions f, where A∈L(X), B∈L(Y) are scalar type operators and S∈L(X,Y). In particular, we establish this estimate for f(t):=|t| and for diagonalizable operators on X=ℓp and Y=ℓq for p