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arXiv · 2211.09289

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

Jing Zhang·Caishi Wang·Lixia Zhang·Lu Zhang
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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.

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Jing Zhang, Caishi Wang, Lixia Zhang, Lu Zhang. 2022-11-17. Generalized weighted number operators on functionals of discrete-time normal martingales. https://arxiv.org/abs/2211.09289

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