Cyclicity in Dirichlet Spaces

Elmadani, Y. ; Labghail, I.

Canadian mathematical bulletin, Tome 62 (2019) no. 2, pp. 247-257

Résumé

Let $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$-capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$), $f$ vanishes on $E$, and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$.

Détail

DOI : 10.4153/CMB-2018-039-3

Mots-clés : Dirichlet-type space, cyclic vector, capacity, strong-type inequality