Let be two Hilbert spaces. Consider a family of operators , , with each a bounded linear operator from to .
Denote
The family of operators , is almost orthogonal if
The Cotlar–Stein lemma states that if are almost orthogonal, then the series converges in the strong operator topology, ↵and that
If T1, …, Tn is a finite collection of bounded operators, then[3]
So under the hypotheses of the lemma,
It follows that
and that
Hence, the partial sums
form a Cauchy sequence.
The sum is therefore absolutely convergent with the limit satisfying the stated inequality.
To prove the inequality above set
with |aij| ≤ 1 chosen so that
Then
Hence
Taking 2mth roots and letting m tend to ∞,
which immediately implies the inequality.
There is a generalization of the Cotlar–Stein lemma, with sums replaced by integrals.[4][5] Let X be a locally compact space and μ a Borel measure on X. Let T(x) be a map from X into bounded operators from E to F which is uniformly bounded and continuous in the strong operator topology. If
are finite, then the function T(x)v is integrable for each v in E with
The result can be proved by replacing sums by integrals in the previous proof, or by using Riemann sums to approximate the integrals.
- Cotlar, Mischa (1955), "A combinatorial inequality and its application to L2 spaces", Math. Cuyana, 1: 41–55
- Hörmander, Lars (1994), Analysis of Partial Differential Operators III: Pseudodifferential Operators (2nd ed.), Springer-Verlag, pp. 165–166, ISBN 978-3-540-49937-4
- Knapp, Anthony W.; Stein, Elias (1971), "Intertwining operators for semisimple Lie groups", Ann. Math., 93: 489–579, doi:10.2307/1970887, JSTOR 1970887
- Stein, Elias (1993), Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, ISBN 0-691-03216-5