## $$C^*$$-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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