Schur test

In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the

operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).

Here is one version.

be two measurable spaces (such as

be an integral operator with the non-negative Schwartz kernel

: If there exist real functions

and numbers

α , β > 0

extends to a continuous operator

with the operator norm Such functions

are called the Schur test functions.

In the original version,

is a matrix and

α = β = 1

[2] A common usage of the Schur test is to take

Then we get: This inequality is valid no matter whether the Schwartz kernel

is non-negative or not.

A similar statement about

operator norms is known as Young's inequality for integral operators:[3] if where

satisfies

, then the operator

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

extends to a continuous operator

Using the Cauchy–Schwarz inequality and inequality (1), we get: Integrating the above relation in

, using Fubini's Theorem, and applying inequality (2), we get: It follows that

α β