Линейные интегральные операторы в пространствах непрерывных и существенно ограниченных вектор-функций

Abstract

The well-established criterion for the action and boundedness of a linear integral operator K from the space L_∞ of essentially bounded functions to the space C of functions continuous on a compact set is extended to the case of functions taking values in Banach spaces. The study further shows that if the operator K is active and bounded in the space C, it is also active and bounded in the space L_∞, with the norms of K in C and L_∞ being identical. A precise expression for the general value of the norm of the operator K in these spaces, expressed in terms of its operator kernel, is provided. Addicionally, an example of an integral operator (for scalar functions) is given, active and bounded in each of the spaces C and L_∞, but not acting from L_∞ into C. Convenient conditions for checking the boundedness of the operator K in C and L_∞ are discussed. In the case of the Banach space Y of the image function values of K being finite-dimensional, these conditions are both necessary and sufficient. In the case of infinitedimensionality of Y, they are sufficient but not necessary (as proven). For "dim" Y<∞, unimprovable estimates for the norm of the operator K are provided in terms of a 1-absolutely summing constant π_1 (Y), determined by the geometric properties of the norm in Y. Specifically, it is defined as the supremum over finite sets of nonzero elements of Y of the ratio of the sum of the norms of these elements to the supremum (over functionals with unit norm) of the sums of absolute values of the functional on these elements.