Generalized weighted number operators on functionals of discrete-time normal martingales

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.

Keywords:

• Discrete-time normal martingale
• Gel'fand triple
• generalized functional
• generalized weighted number operator
• generalized annihilation and creation operator
• commutation relations

2010 Mathematics Subject Classifications:

• Primary: 60H40
• Secondary: 81S25

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).

Additional information

Funding

This work is supported by National Natural Science Foundation of China [gant numbers 11861057 and 12261080].