= Positivity-preserving linear operator on L1
{title2=$u\ge0\Longrightarrow Tu\ge0$}
= Positivity-preserving operator
{synonym}
A bounded <linear operator> $T:L^1(\Omega)\to L^1(\Omega)$ preserves positivity when it takes nonnegative functions to nonnegative functions. By linearity it also preserves the pointwise order: $u\le v$ implies $Tu\le Tv$. For example, integration against a nonnegative kernel preserves positivity, as does the identity operator. This is order positivity, which differs from a positive quadratic form on a <Hilbert space>.
Back to article page
