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Quadratic Fock Space Calculus (II): Positivity of the Preservation Operator and Linear Independence of the Quadratic Exponential Vectors

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Abstract

It have been proved in Accardi and Dhahri (J Math Phys 51:2, 2010) that the set of the exponential vectors \(\Phi (g), \; g\in {\mathcal {K}}:=L^2({\mathbb {R}}^d)\cap L^{\infty }({\mathbb {R}}^d)\) associated with different test functions \(g_i\in {\mathcal {K}}\), are linearly independents. Even this result is true, we present an alternative proof that is consistent with the results of this paper. In this paper, we start by a review of some results on the quadratic Fock space obtained in Accardi and Dhahri (J Math Phys 51:2, 2010) and Rebei (J Math Anal Appl 439(1): 135–153, 2016) , then we prove that the number operator is positive for non negative test function from which we deduce that the creation operator is injective. As application of the injectivity, we give an algebraic proof of the linear independence of the quadratic exponential vectors \(\Phi (g)\).

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Acknowledgements

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

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Authors and Affiliations

  1. Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Makkah, Saudi Arabia

    Omar Alzeley & Hafedh Rguigui

  2. Department of Mathematics, College of Science, Qassim University, Po.Box 6666, 51452, Buraidah, Saudi Arabia

    Habib Rebei

  3. High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011, Hammam Sousse, Tunisia

    Hafedh Rguigui

Authors
  1. Omar Alzeley
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  2. Habib Rebei
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  3. Hafedh Rguigui
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All authors contributed in all tasks of this manuscript.

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Correspondence to Hafedh Rguigui.

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Communicated by Palle Jorgensen

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This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory” edited by Marek Bozejko, Palle Jorgensen and Yuri Kondratiev.

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Alzeley, O., Rebei, H. & Rguigui, H. Quadratic Fock Space Calculus (II): Positivity of the Preservation Operator and Linear Independence of the Quadratic Exponential Vectors. Complex Anal. Oper. Theory 18, 66 (2024). https://doi.org/10.1007/s11785-024-01503-7

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  • DOI: https://doi.org/10.1007/s11785-024-01503-7

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