# Gaussian analytic functions and operator symbols of Dirichlet type

## TL;DR

This paper explores the correlation structure of Gaussian analytic functions related to the Dirichlet space, introduces the mock-Bloch space, and connects these to operator symbols and nonlinear wave equations.

## Contribution

It establishes bounds on weighted sums of inner products in Hilbert spaces, introduces the mock-Bloch space, and characterizes Dirichlet symbols of operators via a nonlinear wave equation.

## Key findings

- Bound on weighted sums in Hilbert spaces with average sense
- Introduction of the mock-Bloch space as a larger space than Bloch space
- Representation of correlations as Dirichlet symbols of contractions and characterization of Grunsky operators

## Abstract

Let $\calH$ be a separable infinite-dimensional $\C$-linear Hilbert space, with sesquilinear inner product $\langle\cdot,\cdot\rangle_\calH$. Given any two orthonormal systems $x_1,x_2,x_3,\ldots$ and $y_1,y_2,y_3,\ldots$ in $\calH$, we show that the weighted sums \[ S(l):=\sum_{j,k:j+k=l}\bigg(\frac{l}{jk}\bigg)^{\frac12}\, \langle x_j,y_k\rangle_{\calH} \] satisfy $|S(l)|^2\lessapprox2$ holds in an average sense. A construction due to Zachary Chase shows that this would not be true if the number $2$ is replaced by the smaller number $1.72$. In the construction, the system $y_1,y_2,y_3,\ldots$ is a permutation of the system $x_1,x_2,x_3,\ldots$. We interpret our bound in terms of the correlation $\expect \Phi(z)\Psi(z)$ of two copies of a Gaussian analytic function with possibly intricate Gaussian correlation structure between them. The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally M\"obius invariant. The study of the correlations $\expect\Phi(z)\Psi(z)$ leads us to introduce a new space, the \emph{mock-Bloch space}, which is slightly bigger than the standard Bloch space. Our bound has an interpretation in terms of McMullen's asymptotic variance, originally considered for functions in the Bloch space. Finally, we show that the correlations $\expect\Phi(z)\Psi(w)$ may be expressed as Dirichlet symbols of contractions on $L^2(\D)$, and show that the Dirichlet symbols of Grunsky operators associated with univalent functions find a natural characterization in terms of a nonlinear wave equation.