Products of Cesàro convergent sequences with applications to convex solid sets and integral operators

Abstract:

Let $0\le a_{n}, b_{n}, c_{n}$ such that $a_{n}=b_{n}c_{n}$. If $a=\lim _{n\to \infty }a_{n}$, and $\{b_{n}\}$ and $\{c_{n}\}$ Cesàro converge to $b$, respectively $c$, then $a\le bc$. This implies that if in addition $\{b_{n}\}$ and $\{c_{n}\}$ are similarly ordered, then $a=bc$. As applications we prove that the pointwise product of two convex solid sets closed in measure is again closed in measure and a factorization result for kernels of regular integral operators on $L_{p}$–spaces.