PROBABILITY MEASURES IN TERMS OF CREATION, ANNIHILATION, AND NEUTRAL OPERATORS

Abstract

Let u be a probability measure on Rd with finite moments of all orders. Then we can define the creation operator a+(j), the annihilation operator a-(j), and the neutral operator a0(j) for each coordinate 1 < = j < = d. We use the neutral operators a0(i) and the commutators [a-(j), a+(k)] to characterize polynomially symmetric, polynomially factorizable, and moment-equal probability measures. We also present some results for probability measures on the real line with finite support, infinite support, and compact support. 1. Creation, annihilation, and neutral operators Let u be a probability measure on Rd with finite moments of all orders, namely, for any nonnegative integers i1, i2,..., id,Z Rd |x i11 xi22 * * * xid d | du(x) < 1, 1 April 20, 2004 10:2 Proceedings Trim Size: 9in x 6in acks-levico 2 where x = (x1, x2,..., xd) 2 Rd. Let F0 = R and for n> = 1 let Fn be the vector space of all polynomials in x1, x2,..., xd of degree < = n. Then we have the inclusion chai