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

TL;DR

This paper introduces generalized weighted number operators for functionals of discrete-time normal martingales, representing them via creation and annihilation operators, and explores their algebraic relations.

Contribution

It defines a new class of operators called GWN operators on generalized functionals of discrete-time normal martingales, linking them to creation and annihilation operators.

Findings

GWN operators can be expressed using creation and annihilation operators.

Derived formulas for commutation relations between GWN and creation/annihilation operators.

Established representation theorems for GWN operators in the functional space.

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.