Schur test information

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

Here is one version.^[1] Let $X,\,Y$ be two measurable spaces (such as $\mathbb {R} ^{n}$ ). Let $\,T$ be an integral operator with the non-negative Schwartz kernel $\,K(x,y)$ , $x\in X$ , $y\in Y$ :

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

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

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

for almost all $\,x$ and

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

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

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

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

In the original version, $\,T$ is a matrix and $\,\alpha =\beta =1$ .^[2]

^ Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96., Springer-Verlag, Berlin, 1978. Theorem 5.2.
^ I. Schur, Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. reine angew. Math. 140 (1911), 1–28.

[1] Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96., Springer-Verlag, Berlin, 1978. Theorem 5.2.

[2] I. Schur, Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. reine angew. Math. 140 (1911), 1–28.

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