Properties of an integral operator preserved under pointwise multiplication of its kernel by a function (Q1341166)

Let \(\mu\) be a \(\sigma\)-finite positive measure defined on some \(\sigma\)- algebra of subsets of a set \(\Omega\) and \(K\) be a function on \(\Omega \times \Omega\). The author considers the linear integral operator \({\mathfrak K}\) with the kernel \(K\) determined by the formula \({\mathfrak K}(f) = \int_ \Omega K(x,t) f(t)d \mu (t)\) on the following spaces: (1) \(L^ p (\Omega)\) for \(1 < p < \infty\), (2) \(L^ \infty (\Omega)\), (3) \(C(\Omega)\). For each case the certain conditions on the kernel \(K\) are determined under which the operator \({\mathfrak K}\) is bounded, compact or weakly compact. The paper investigates conditions under which these properties of \({\mathfrak K}\) also hold for the integral operator with the kernel obtained by multiplication of \(K\) by an essentially bounded measurable function on \(\Omega \times \Omega\). Numerous examples are given.