Composition Operators on the Space $$\mathcal {H}^2$$ of Dirichlet Series

Composition operators on weighted Hilbert spaces of Dirichlet series

“…The limit is not finite in general for $$a \geqslant 0$$, as we now exemplify. Example Applying the transference principle [24] to the example constructed by Zorboska in [29], we obtain a Dirichlet series $$\varphi$$ such that the $$a$$‐weighted counting function is finite if and only if $$a>\frac{1}{2}$$. More precisely, we consider the Dirichlet series $$\varphi (s)=g(2^{-s})$$, where $$g(z)=e^{-\frac{1+z}{1-z}},\,z\in \mathbb {D}$$.…”

Counting functions for Dirichlet series and compactness of composition operators

“…In our context, we try to characterize compactness of $$C_\varphi$$ from properties of its symbol $$\varphi$$. This has been investigated in many papers (like [1, 3, 4, 7–9, 13]). Following the seminal paper of Shapiro [14] for composition operators on $$H^2(\mathbb {D})$$, a natural way for doing so is to characterize compactness of $$C_\varphi$$ by mean of some counting function related to $$\varphi$$.…”