Cotlar-Stein Almost Orthogonality Lemma

When deriving the estimates on integral operators one often uses the Almost Orthogonality principle of M. Cotlar and E.M. Stein, first proved by M. Cotlar in [Cot55]. This result is classical; our excuse for formulating it once again is a need to have its weighted form which sometimes allows to reduce the number of integrations by parts in half (hereby weakening smoothness requirements), and also to state explicitly the convergence of the series ∑i Ti in the strong operator topology. Let E and F be the Hilbert spaces, and let T be a linear operator which acts from E to F . An often situation is that one can decompose the operator T into an infinite sum of operators T = ∑i Ti, which satisfy certain estimates, and the question is, under which assumptions on Ti one can deduce an adequate estimate on T .