Definite kernel

The kernel $ K ( P , Q ) $ of a linear integral Fredholm operator which satisfies the relation

$$ \int\limits _ { P } \int\limits _ { Q } K ( P , Q ) \phi ( P) \overline{ {\phi ( Q) }}\; d P d Q \geq 0 \ ( \leq 0 ) , $$

where $ P , Q $ are points in a Euclidean space, $ \phi $ is an arbitrary square-integrable function, and $ \overline \phi \; $ is its complex conjugate function. Depending on the sign of the inequality, the kernel is said to be, respectively, non-negative (non-negative definite) or non-positive (non-positive definite).

Non-negative (non-positive) is sometimes the name given to a kernel which satisfies, in the domain of integration, the inequality $ K ( P , Q ) \geq 0 $( $ \leq 0 $).

References

[a1]	A.C. Zaanen, "Linear analysis" , North-Holland (1956)
[a2]	K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)