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Cyclicity in Dirichlet Spaces

Part of: Geometric function theory Special classes of linear operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  11 January 2019

Y. Elmadani  and
I. Labghail
Y. Elmadani
Affiliation:
CeReMAR LAMA, Mohammed V University, Faculty of science Rabat, 10 000 Rabat, Morocco Email: [email protected]@gmail.com
I. Labghail
Affiliation:
CeReMAR LAMA, Mohammed V University, Faculty of science Rabat, 10 000 Rabat, Morocco Email: [email protected]@gmail.com
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Abstract

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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 ${\mathcal{D}}(\unicode[STIX]{x1D707})$.

Keywords

Dirichlet-type spacecyclic vectorcapacitystrong-type inequality

MSC classification

Primary: 47B38: Operators on function spaces (general)
Secondary: 30C85: Capacity and harmonic measure in the complex plane 30H05: Bounded analytic functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 247 - 257
Copyright
© Canadian Mathematical Society 2019 

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