ABSTRACT
In this section we give acting and boundedness conditions for partial integral operators on the space C = C([a, b] x [e, d) which are both necessary and sufficient. Moreover, we discuss some algebras and ideals of such operators, considered as subspaces of the algebra .c(C) of all bounded linear operators on C. We also study conditions for the strong or norm continuity of operator functions whose vaJ.ues are partial integral operators acting in C. These problems are related to the solvability of the equation
(13.1) z(t,s) = Pz(t,s) + f(t,s), where J : [a, b) x [e, d) - IR is a given function and the operator P is a sum
(13.2) of the four operators
(13.3)
(13.4)
(13.5)
and
(13.6)
Nz(t,s) = 161tl n(t, ..,T,U)Z(T, u) dTdu. The operators (13.4) and (13.5) are partial integral operators, while (13.6) is an ordinary integral operator acting on functions over [a, b) x [e, d). As we have seen in § 11, the properties of the partial integral operator (13.2) heavily depend on the spaces in which it is studied,
and may be very different from those of an ordinary integral operator.
