The arithmetic of certain semigroups of positive operators

Abstract

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that