A representation theorem for operators on a space of interval functions

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 T H 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 T H = ( I ) ∫ a b f H d α + ∑ i = 1 ∞ [ H ( x i − 1 , x i − 1 + ) − H ( x i − 1 + , x i − 1 + ) ] β ( x i − 1 ) + ∑ i = 1 ∞ [ H ( x i − , x i ) − H ( x i − , x i − ) ] Θ ( x i − 1 , x i ) 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, f H is a function from R to N depending on H and { x i } 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.