Measure-compact operators, almost compact operators, and linear functional equations in $L_p$

Under study are the measure-compact operators and almost compact operators in $L_p$. We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in $L_p$, we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in $L_p$ with almost compact or measure-compact operators to equivalent linear integral equations in $L_p$ with quasidegenerate Carleman kernels.