In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlar and Elias Stein. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another when the operator can be decomposed into almost orthogonal pieces. The original version of this lemma (for self-adjoint and mutually commuting operators) was proved by Mischa Cotlar in 1955[1] and allowed him to conclude that the Hilbert transform is a continuous linear operator in without using the Fourier transform. A more general version was proved by Elias Stein.[2]
Let be two Hilbert spaces. Consider a family of operators , , with each a continuous 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
Here is an example of an orthogonal family of operators. Consider the inifite-dimensional matrices
and also
Then for each , hence the series does not converge in the uniform operator topology.
Yet, since and for , the Cotlar–Stein almost orthogonality lemma tells us that
converges in the strong operator topology and is bounded by 1.