On the integral representation of linear operators

A. V. Bukhvalov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 5-14

Résumé

Let $(T_i,\mu_i)$ ($i=1,2$) be two $\sigma$-finite measure spaces and let $(T,\mu)$ be their product-space. Let $E$ be an ideal in the space of measurable functions $S(T_1,\mu_1)$ (i.e. ($|e_1|\leq|e_2|$, $e_1\in S$, $e_2\in E)\Rightarrow e_1\in E$). Theorem 2. {\it Let $U$ – be a linear operator from $E$ to $S(T_2,\mu_2)$. The following assertions are equivalent: $$ {\rm 1)} (Ue)(s)=\int K(t,s)e(t)\,d\mu_1(t)\quad (e\in E),\quad\textwhere K(t,s) is a~$\mu$-measurable kernel; $$ 2) if $0\leq e_n\leq e\in E$, $n=1,2,\dots$, and $e_n\to0$ in measure on every set of finite measure, then $(Ue_n)(s)\to0$ $\mu_2$-a.e.}