Cotlar–Stein lemma

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 and allowed him to conclude that the Hilbert transform is a continuous linear operator in ${\displaystyle L^{2}}$ without using the Fourier transform. A more general version was proved by Elias Stein.

Let ${\displaystyle E,\,F}$ be two Hilbert spaces. Consider a family of operators ${\displaystyle T_{j}}$, ${\displaystyle j\geq 1}$, with each ${\displaystyle T_{j}}$ a bounded linear operator from ${\displaystyle E}$ to ${\displaystyle F}$.

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