Gaussian analytic functions and operator symbols of Dirichlet type

Abstract

Let M={mj,k}j,k=1+∞ be an infinite complex-valued matrix which acts contractively on ℓ2. For the weighted short diagonal sumsSM(l):=∑j,k:j+k=l(ljk)12mj,k, we obtain the estimate∑l=2+∞sll|SM(l)|2≤2slog⁡e1−s,0≤s<1. Expressed more vaguely, |SM(l)|2⪅2 holds in the sense of averages. Concerning the optimality of the above bound, a construction due to Zachary Chase shows that the statement does not hold if the number 2 is replaced by the smaller number 1.72. In the construction, M is a permutation matrix. We interpret our bound in terms of the correlation EΦ(z)Ψ(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 Möbius invariant. The study of the correlations EΦ(z)Ψ(z) leads us to introduce a new space, the mock-Bloch space (or Blochish 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 EΦ(z)Ψ(w) may be expressed as Dirichlet symbols of contractions on L2(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.