Lema de Cotlar

Sean ${\displaystyle E,\,F}$ dos espacios de Hilbert y sea

${\displaystyle T:\;E\to F}$

un operador lineal. Se asume que

${\displaystyle T=\sum _{j\in \mathbb {Z} }T_{j},}$

donde cada

${\displaystyle T_{j}:\;E\to F,\ j\in \mathbb {Z} }$

es un operador lineal continuo.

Significa

${\displaystyle a_{jk}=\Vert T_{j}T_{k}^{\ast }\Vert _{F\to F}}$, ${\displaystyle b_{jk}=\Vert T_{j}^{\ast }T_{k}\Vert _{E\to E}.}$

Si

${\displaystyle A=\max _{j}\sum _{k}{\sqrt {a_{jk}}}<\infty }$

y

${\displaystyle B=\max _{j}\sum _{k}{\sqrt {b_{jk}}}<\infty ,}$

entonces

${\displaystyle \Vert T\Vert _{E\to F}\leq {\sqrt {AB}}.}$