EIGENVALUES OF COMPOSITION COMBINED WITH DIFFERENTIATION

Abstract

Let $\varphi$ be an analytic self-map of the open unit disk $\mathbb{D}$ in the complex plane. Such a map induces through composition the linear composition operator $C_{\varphi }:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right)$. The eigenvalues and the spectrum of such an operator acting on different spaces of analytic functions have been investigated in several articles, see e.g. [1], [8], [16], [28] and [29]. In this article we continue this line of research by combining the composition operator with the differentiation $D:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\to f\prime$. Then we obtain two linear operators $D{C}_{\varphi }:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\mapsto \varphi \prime \left(f\prime \circ \varphi \right)$ and $C_{\varphi }D:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\mapsto f\prime \circ \varphi$. Now, we calculate the eigenvalues of the operators $D{C}_{\varphi }$ and $C_{\varphi }D$.