Abstract
Let M be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of M, one can construct generalized functionals of M. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of M, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of M). We also examine commutation relations between a GWN operator and a generalized annihilation (or creation) operator, and obtain several formulas expressing such commutation relations.
2010 Mathematics Subject Classifications:
Acknowledgements
The authors are extremely grateful to the referees for their valuable comments and suggestions on improvement of the first version of the present paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).