Vertical Toeplitz Operators on the Upper Half-Plane and Very Slowly Oscillating Functions

Abstract

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend on the imaginary part of the argument only. Such algebra is known to be commutative, and is isometrically isomorphic to an algebra of bounded complex-valued functions on the positive half-line. In the paper we prove that the latter algebra consists of all bounded functions f that are very slowly oscillating in the sense that the composition of f with the exponential function is uniformly continuous or, in other words, $$\lim_{\frac{x}{y} \to 1} \left|f(x) - f(y)\right| = 0.$$