The continuity of increasing polynomial operators on ordered topological vector spaces

Abstract

IN RECENT YEARS there have been many investigations of polynomial operators. On the one hand, they are the natural generalization of the classical polynomials of n variables to general vector spaces and hence fundamental to the study of the differential calculus and analytic functions in topological vector spaces (see, e.g., [l] and [13] respectively). Other classical properties also extend to polynomial operators, e.g., the Weierstrass approximation theorem in various versions (see [14] and [19]), Lagrange and Hermite interpolation [17], and solutions to polynomial equations (see, e.g., [16] and [18] for surveys and [S], [6], and [9] for more recent work). On the other hand, polynomial operators are one of the simplest generalizations of linear operators and share many of their basic properties. In this paper the following classical theorems on the continuity of positive linear maps on ordered topological vector spaces are generalized to increasing polynomial operators. The polynomial integral operators with positive kernels defined by the author in [4] give examples of a wide class of increasing polynomial operators. The theorems on linear operators are due to Nachbin, Namioka, and Schaefer and their statement is adapted from Peressini [15, p. 861, whose terminology we shall also follow.