Ideals of sets of integral and partially integral operators (Q803520)

The authors shows that (i) any right (two-sided) ideal of the set of all partially integral operators in \(L_ 2\), or of the set of all integral operators in \(L_ 2\), is contained in the set C \((C_ 2)\) of all Carleman (Hilbert- Schmidt) integral oprators in \(L_ 2;\) (ii) any left ideal of the set of all partially integral operators in \(L_ 2\) is contained in the left ideal of this set consisting of all continuous linear operators in \(L_ 2\) that map the unit ball in \(L_{\infty}\) into an ellipsoid in \(L_ 2;\) (iii) any left ideal of the set of all integral operators in \(L_ 2\) is contained in the left ideal of this set consisting of all regular integral operators in \(C_ 2\).