Concept

Schur test

Summary
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. 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: :T f(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\le\alpha p(x) for almost all ,x and : (2)\qquad \int_X p(x)K(x,y),dx\le\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} \
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