Let \(X\) be an \(N(0,1)\) random variable defined on a probability space \((\Omega,{\mathcal F},P)\) and regarded as an element of \(L^2(P)\). The standard Gram-Schmidt orthogonalization process applied to the sequence \(1,X,X^2,X^3,\dots\) in \(L^2(P)\) produces the sequence \(h_0= 1\), \(h_1= X\), \(h_2= X^2-1\), \(h_3= X^3- 3X,\dots\) of Hermite polynomials. For \(p\geq 1\), let \(L^p(X)\) denote the closed linear span over \(\mathbb{C}\) of the \(h_n\) with respect to the \(L^p\)-norm. The \(h_n\) form a basic sequence in this space.
Given \(z\in\mathbb{C}\), define the Mehler transform \(M_z: L^p(X)\to L^q(X)\) by \(M_z\left(\sum^\infty_{n=0} a_nh_n\right)= \sum^\infty_{n= 0} a_nz^n h_n\).
The main result of this paper is a new proof of a theorem by Weissler and Epperson which runs as follows: Let \(1\leq p\leq q<\infty\). Then we have \(\| M_z\|= 1\) if and only if
(i) \(| z|^2\leq p/q\), and
(ii) \((q- 1)| z|^4- (p+q-2)(\text{Re }z)^2- (pq- p-q+ 2)(\text{Im }z)^2+ (p-1)\geq 0\).
Moreover, if these conditions fail, then \(M_z\) is not even a bounded operator.
Using the notion of Wick product, the author then greatly extends this result from a single \(X\) to a (closed) subspace of \(L^2(P)\) consisting of centred Gaussian random variables. Such a space is called a Gaussian-Hilbert space [see his “Gaussian-Hilbert spaces” (Cambridge, 1997), Chapter III], and it is natural to consider two such spaces, \(H\) and \(H'\), connected by a basic operator \(A: H\to H'\). (With the single \(X\), we have \(H= H'=\) linear span of \(X\), and \(A= zI\), where \(I\) is the identity operator on \(H\)).