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\(C^*\)-algebra of angular Toeplitz operators on Bergman spaces over the upper half-plane. (English) Zbl 1327.30063

Let \(\mathcal A^{2}(\Pi)\) be the Bergman space of all square integrable functions in the upper half-plane \(\Pi\subset\mathbb C\) equipped with the measure \(d\mu=r\,dr\,d\theta\) and \(P\) be the orthogonal projection from \(L_{2}(\Pi,d\mu)\) onto \(\mathcal A^{2}(\Pi)\). The Toeplitz operator on \(\mathcal A^{2}(\Pi)\) with the symbol \(a\in L_{\infty}(\Pi)\) is defined by the standard formula \(T_{a}f=P(af)\), \(f\in \mathcal A^{2}(\Pi)\). Denote by \(\mathcal A_{\infty}\) the \(C^{\ast}\)-subalgebra of \(L_{\infty}(\Pi)\) which consists of all functions \(a\) representable in the form \(a(z)=b(\arg z)\), \(z\in\Pi\), for some \(b\in L_{\infty}(0,\pi)\). Let \({\mathcal T}({\mathcal A}_{\infty})\) be the operator algebra generated by all operators \(T_{a}\), \(a\in {\mathcal A}_{\infty}\). For \(a\in {\mathcal A}_{\infty}\) denote \[ \gamma_{a}(\lambda) =\frac{2x}{1-e^{-2x\pi}} \int_{0}^{\pi} a(\theta)e^{-2x\theta}\,d\theta, \quad x\in{\mathbb R}. \] It is known that the map \(T_{a}\mapsto\gamma_{a}\), \(a\in {\mathcal A}_{\infty}\), generates an isometrical isomorphism between \({\mathcal T}({\mathcal A}_{\infty})\) and the \(C^{\ast}\)-subalgebra of \(L_{\infty}({\mathbb R})\) generated by all functions \(\gamma_{a}\). In the paper under review, an explicit and intrinsic characterization of the latter \(C^{\ast}\)-subalgebra is given. It is proved that this subalgebra consists of all very slowly oscillating functions on \(\mathbb R\). That means that these functions are bounded on \(\mathbb R\), and their compositions with the function \(\sinh\) are uniformly continuous with respect to the usual metric.

MSC:

30H20 Bergman spaces and Fock spaces
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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Full Text: arXiv Euclid