On left ideals of sets of partially integral operators (Q1920774)

The operators of the title take the form \(Tf(s)=\int_X K(s,t)f(t) d\mu(t)\) where \((X,\mu)\) is a finite measure space which is not purely atomic and \(K\) is a complex--valued measurable kernel defined on \(X\times X\). The author presents two sufficient conditions for such an operator to map \(L^2(X,\mu)\) into an ellipsoid, i.e., for the existence of a square summable sequence \(\{z_n\}\) in \(L^2\) such that for each \(f\in L^2\), it is possible to express \(Tf=\sum\alpha_nz_n\) with square summable (scalar) coefficients.