Linear operators in the space of regulated functions. (English) Zbl 0790.47023

Schwabik, Štefan

Math. Bohem. 117, No. 1, 79-92 (1992).

A real-valued function \(f:[a,b] \to \mathbb{R}\) is said to be regulated if it has two-sided limits at all points. Let \(G(a,b)\) be the space of all regulated functions and let \(G_ L(a,b)\) be the space of all regulated functions on \([a,b]\) which are left continuous. \(G(a,b)\) [respectively, \(G_ L(a,b)]\) is a Banach space under the supnorm. The author gives integral representations for the bounded and compact linear operations from \(G_ L(a,b)\) into \(G(a,b)\).

MSC:

47B38	Linear operators on function spaces (general)
47B07	Linear operators defined by compactness properties
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A25	Integration with respect to measures and other set functions
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

space of all regulated functions; integral representations for the bounded and compact linear operations