On some properties of domains of integral operators (Q797794)

Let \(L^ 0\) denote the space of measurable functions on a measure space and let A be a topological vector subspace of \(L^ 0\). For any \(f\in L^ 0 A_ f\) denotes the set \(\{\) \(g\in A\); \(| g|\leq | f|\) a.e.\(\}\) and \(A^{\#}\) is the space of all \(f\in L^ 0\) for which the sets \(A_ f\) are bounded in A. It is shown that for a nonsingular integral operator K \(D_ K^{\#}=D_ K\) and \(\tilde D_ K^{\#}=\tilde D_ K\), where \(D_ K\), \(\tilde D_ K\) denote the proper and the extended domains of K in the sense of \textit{N. Aronszajn} and the author [Math. Ann. 163, 127-154 (1966; Zbl 0171.124)]. The notion of enlargement \(^{\#}\) was introduced independently by \textit{Yu. A. Abramovich} [Vestnik Leningr. Univ. 25, No.1 (Mat. Mekh. Astron. No.1), 7-17 (1970; Zbl 0203.117)] for normed spaces and by \textit{W. Wnuk} [Commentat. Math. 24, No.1, 167-172 (1984)].