A representation theorem for operators on a space ofinterval functions
J. A. Chatfield · International Journal of Mathematics and Mathematical Sciences · 1978
Suppose N is a Banach space of norm |•| and R is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of TH where H is a function from R×R to N such that H(p+,p+) , H(p,p+) , H(p−,p−) , and H(p−,p) each exist for each p and T is a bounded linear operator on the space of all such functions H . In particular we show that TH=(I)∫abfHdα+∑i=1∞[H(xi−1,xi−1+)−H(xi−1+,xi−1+)]β(xi−1)+∑i=1∞[H(xi−,xi)−H(xi−,xi−)]Θ(xi−1,xi) where each of α , β , and Θ depend only on T , α is of bounded variation, β and Θ are 0 except at a countable number of points, fH is a function from R to N depending on H and {xi}i=1∞ denotes the points P in [a,b] . for which [H(p,p+)−H(p+,p+)]≠0 or [H(p−,p)−H(p−,p−)]≠0 . We also define an interior interval function integral and give a relationship between it and the standard interval function integral.