Gebelein inequality in a Hilbert space (Q6569315)

Letting \(\mu\) be the standard Gaussian measure and \(f\in L^2(\mu)\) with \(\int_\mathbb{R}f(x)\text{d}\mu(x)=0\), the Gebelein inequality states that \(\lVert P_\rho f\rVert_2\leq|\rho|\lVert f\rVert_2\) for \(-1\leq\rho\leq1\), where \(P_\rho\) is the Ornstein--Uhlenbeck operator defined by\N\[\N(P_\rho f)(y)=\int_\mathbb{R}f\left(\rho y+\sqrt{1-\rho^2}z\right)\,\text{d}\mu(z)\N\]\Nfor \(y\in\mathbb{R}\) and \(f\in L^2(\mu)\). The author generalises this inequality to a real, separable Hilbert space, and discusses applications to establishing strong laws of large numbers for Gaussian functionals.