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Concept
Schur test
Summary
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. Let be two measurable spaces (such as ). Let be an integral operator with the non-negative Schwartz kernel , , :
If there exist real functions and and numbers such that
for almost all and
for almost all , then 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 .
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:
if
where satisfies , for some , then the operator extends to a continuous operator , with
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 for any .
Official source
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