A Bargmann transform for translation invariant operators on weighted Bergman spaces of the complex half-plane

Abstract

Let us denote with $\mathcal{A}^2_λ(\mathbb{C}_+)$ ($λ> -1$) a weighted Bergman space over the right half-plane $\mathbb{C}_+$, which admits a unitary representation of $\mathbb{R}$ given by the (imaginary) translations of $\mathbb{C}_+$. We study the von Neumann algebra $\mathfrak{T}\big(\mathcal{A}^2_λ(\mathbb{C}_+)\big)$ of bounded translation invariant operators. We prove that every element of $\mathfrak{T}\big(\mathcal{A}^2_λ(\mathbb{C}_+)\big)$ is a Toeplit operator $T^{(λ)}_A$ for some translation invariant operator $A$ of the ambient $L^2$-space of $\mathcal{A}^2_λ(\mathbb{C}_+)$. Furthermore, we prove that this can be achieved through a commutative von Neumann algebra $\mathfrak{A}_λ$ that yields an assignment $A \mapsto T^{(λ)}_A$ that turns out to be a $*$-isomorphism of $*$-algebras. Our main tool is a Bargmann transform $\mathcal{B}_λ$ for which we establish several operator and representation theoretic properties. We also describe the translation invariant subspaces of $\mathcal{A}^2_λ(\mathbb{C}_+)$ and obtain formulas for the diagonalizing spectral functions for translation invariant Toeplitz operators whose symbols are translation invariant operators. The latter generalize previously known results for function symbols.