ABSTRACT
L~ :::> L~ =L1 in case p =00. In the latter case, it is more natural to consider, rather than the adjoint A· of the operator (3.1) on L~, the restriction A' of A· on the subspace L~ = L1 i A' is sometimes called the Kothe adjoint (or a8sociate operator) of A. Thus, if A is defined by (3.1), we are interested in the existence of its Kothe adjoint A', and also in conditions which allow us to represent A' again in the form (3.1). In ZABREJKO [1968] it is shown that, if X is a linear integral operator with kernel k = k(8, u) in L" (1 ~ p < (0), then the Kothe adjoint X' always exists, but is not necessarily an integral operator. However, for every y E L q (q = pl(p - 1» which belongs to the domain of definition of the linear integral operator
X#y(s) = 1" k#(s, u)y(u) du generated by the kernel k# = k#(s, u) = k(u, s), one has X'lI = X#y. In particular, if the operator X is regular then the operator K# is defined for each y E Lq, and thus X' = X# on the whole space Lq• A similar result holds for the operator (3.1), as was shown in KALITVIN-ZABREJKO [1991]:
Lemma 3.3. Suppose that the linear operator (3.1) acts in the space L, (1 ~ p ~ (0). Then the operator (3.1) admits the Kothe adjoint
(3.6) A'y(s) = A#lI(s),
where
Chapter I: Equations of Barbashin Type
(3.7) A*y(s) = e(s)y(s) +1" k#(s, u)y(u) du for any function y E L9 for which the right-hand side of (3.6) makes sense. In particular, if the operator (3.1) is regular, then (3.6) holds for all y E Lq• In connection with the results given above, the problem arises how to find simple conditions (at least sufficient) on the functions c = c(s) and k = k(8, u) under which the operator (3.1) is continuous or regular in some L" space (1 ~ p ~ 00). Such conditions may be found by combining analogous condit~ons for each of the two components of (3.1), i.e. the multiplication operator (3.8) and the integral operator
(;z(s) = c(s)z(s)
(3.9) Kz(s) =1"k(s, u)z(u) du. By means of Kantorovich's regularity theorems for bounded linear operators in Loo , one may conclude from Lemma 3.2 and Lemma 3.3 that, in case p = 1 or p = 00, the operator (3.1) acts in L" if and only if both components (3.8) and (3.9) act in L". In case 1 < p < 00, however, an analogous result is not known. On the one hand, a necessary and sufficient condition for the linear multiplication operator (3.8) to act in L" is, of course, that c E Loo• On the other hand, a necessary and sufficient condition on the kernel k = k(8, u) for the linear integral operator (3.9) to act in L" is known only in case p = lor p = 00. Some 8uffidentconditions on k, however, which guarantee that the operator (3.9) acts in L" for 1 ~ p ~ 00 will be given in the next subsections of this chapter.
