Schur test

In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the ${\displaystyle L^{2}\to L^{2}}$ operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).

Here is one version. Let ${\displaystyle X,\,Y}$ be two measurable spaces (such as ${\displaystyle \mathbb {R} ^{n}}$). Let ${\displaystyle \,T}$ be an integral operator with the non-negative Schwartz kernel ${\displaystyle \,K(x,y)}$, ${\displaystyle x\in X}$, ${\displaystyle y\in Y}$:

${\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.}$

If there exist real functions ${\displaystyle \,p(x)>0}$ and ${\displaystyle \,q(y)>0}$ and numbers ${\displaystyle \,\alpha ,\beta >0}$ such that

${\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)}$

for almost all ${\displaystyle \,x}$ and

${\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)}$

for almost all ${\displaystyle \,y}$, then ${\displaystyle \,T}$ extends to a continuous operator ${\displaystyle T:L^{2}\to L^{2}}$ with the operator norm

${\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.}$

Such functions ${\displaystyle \,p(x)}$, ${\displaystyle \,q(y)}$ are called the Schur test functions.

In the original version, ${\displaystyle \,T}$ is a matrix and ${\displaystyle \,\alpha =\beta =1}$.