Partially integral operators (Q909219)

A partially integral operator T is an operator of the form \[ Tf(s)=\int_{X}K(s,t)f(t)d\mu (t), \] where (X,\(\mu)\) is a space with measure. A necessary and sufficient condition for an operator T: \(L_ p(X,\mu)\to M(Y,\nu)\) to be partially integral is derived and it is proved that in separable \(L_ 2(X,\mu)\) every bounded operator T has a representation \(T=AB\) with partially integral operator A and Carleman integral operator B; this assertion without the word ``partially'' is wrong.