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

Abstract

Let us denote with ${\mathcal{A}}_{\hat{I}}^2»({\mathbb{C}}_{+})$ ($\hat{I}»>-1$) 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 $\mathfrak{T}({\mathcal{A}}_{\hat{I}}^2»({\mathbb{C}}_{+}))$ of bounded translation invariant operators. We prove that every element of $\mathfrak{T}({\mathcal{A}}_{\hat{I}}^2»({\mathbb{C}}_{+}))$ is a Toeplit operator $T_A^{(\hat{I}»)}$ for some translation invariant operator $A$ of the ambient $L^2$-space of ${\mathcal{A}}_{\hat{I}}^2»({\mathbb{C}}_{+})$. Furthermore, we prove that this can be achieved through a commutative von Neumann algebra ${\mathfrak{A}}_{\hat{I}}»$ that yields an assignment $A\mapsto T_A^{(\hat{I}»)}$ that turns out to be a $\ast$-isomorphism of $\ast$-algebras. Our main tool is a Bargmann transform ${\mathcal{B}}_{\hat{I}}»$ for which we establish several operator and representation theoretic properties. We also describe the translation invariant subspaces of ${\mathcal{A}}_{\hat{I}}^2»({\mathbb{C}}_{+})$ 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.