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

It have been proved in Accardi and Dhahri (J Math Phys 51:2, 2010) that the set of the exponential vectors \(\Phi (g), \; g\in {\mathcal {K}}:=L^2({\mathbb {R}}^d)\cap L^{\infty }({\mathbb {R}}^d)\) $\mathrm{\Phi }(g),g\in \mathcal{K}:=L^2({\mathbb{R}}^d)\cap L^{\infty }({\mathbb{R}}^d)$ associated with different test functions \(g_i\in {\mathcal {K}}\) $g_i\in \mathcal{K}$ , 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 \(\Phi (g)\) $\mathrm{\Phi }(g)$ .