Kernel Operators

Abstract

This is a brief report on some recent results about kernel operators, the domain of which is an order ideal in a space of real measurable functions and the range of which is contained in an order ideal of the same type. A simple proof (due to A.R. Schep) is indicated of the theorem that any positive linear operator majorized by a kernel operator is itself a kernel operator. It follows easily that the kernel operators form a band in the Riesz space of all order bounded linear operators. Another important theorem is due to A.V. Buhvalov, stating a simple necessary and sufficient condition for an order bounded linear operator to be a kernel operator. One of the corollaries in Schep’s approach is the theorem that any continuous linear operator from L¹ to L^P (1<p≤∞) is a kernel operator (for the special case of Lebesgue measure in the real line due to N. Dunford, 1936); in Buhvalov’s approach this corollary is proved first and the other abovementioned results are derived from it.