Linear integral operators in spaces of continuous and essentially bounded vector functions

M. J. Alves ; E. V. Alves ; Zh. Munembe ; I. V. Nepomnyaschih

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 145, pp. 5-19

Résumé

The well-established criterion for the action and boundedness of a linear integral operator $K$ from the space $L_\infty$ 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 $L_\infty,$ with the norms of $K$ in $C$ and $L_\infty$ 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 $L_\infty,$ but not acting from $L_\infty$ into $C.$ Convenient conditions for checking the boundedness of the operator $K$ in $C$ and $L_\infty$ 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 infinite-dimensionality of $Y,$ they are sufficient but not necessary (as proven). For $\dim Y\infty,$ unimprovable estimates for the norm of the operator $K$ are provided in terms of a $1$-absolutely summing constant $\pi_1(Y),$ 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.

Détail

Keywords: Banach space, linear integral operator, norm of linear operator, $1$-absolutely summing constant