Solid extensions of the Cesàro operator on the Hardy space H2(D)

Abstract

We introduce and study the largest Banach space of analytic functions on the unit disc which is solid for the coefficient-wise order and to which the classical Ces`aro operator $\mathcal{C}:H^2 \to H^2$ can be continuously extended, while still maintaining its values in $H^2$. Properties of this Banach space $\mathcal{H}(ces_2)$ are presented as well as a characterization of individual analytic functions which belong to $\mathcal{H}(ces_2)$. In addition, both the multiplier space of $\mathcal{H}(ces_2)$ and the spectrum of $\mathcal{C}:\mathcal{H}(ces_2) \to \mathcal{H}(ces_2)$ are determined.