Quasi-parabolic Composition Operators on Weighted Bergman Spaces

Abstract

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on $A_{\alpha }^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi (z)=$ $z+\psi (z)$, where $\psi \in$ $H^{\infty }({\mathbb {H}})$ and $\mathfrak {I}(\psi (z))> \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A similar method used in Gül (J Math Anal Appl 377:771–791, 2011) for Hardy spaces is used to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.