Aurel Stan
The Ohio State University
Title
Necessary conditions for a linear operator to be number operator.
Abstract
We review first the definitions of the joint creation, preservation, annihilation, and number operators generated by a finite family of random variables having finite moments of all orders. The main topic of this talk is to try answer the following question: which linear operators, from the algebra of polynomials, of finitely many variables, to itself, are number operators of finite families of random variables, in which the variables of the polynomials have been identified with the random variables?
It turns out that there are two types of conditions: algebraic, described in terms of commutators involving the number operator and the multiplication by each variable operator, and analytic, described in terms of some matrices being semi-positive definite. While we have not been able to address the analytic conditions, we believe that we found all the algebraic conditions.
This is a joint work with G. Popa and R. Dutta