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

Submission history

From: Raul Quiroga-Barranco [view email]
[v1] Tue, 23 Jun 2026 22:06:46 UTC (20 KB)