On dominating sequences in the unit disc
type: Item
instance of: scholarly article
title: On dominating sequences in the unit disc
zbMATH Open document ID: 0538.47002
DOI: 10.1007/BF01163870
author: H. Bercovici
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173516
review text: A subset A of a region G in the complex plane is dominating if \(\sup \{u(\lambda):\lambda \in G\}=\sup \{u(\lambda):\lambda \in A\}\) for \(u\in H^{\infty}(G).\) We show that every dominating subset A of the unit disc contains a dominating sequence \(\{\lambda_ n\}_{n\geq 0}\) such that \(\lim_{n\neq m,\quad n,m\to \infty}\rho(\lambda_ n,\lambda_ m)=1,\) where \(\rho\) denotes the pseudohyperbolic metric. Moreover, the sequence \(\{\lambda_ n\}_{n\geq 0}\) can be chosen such that \(\{| \lambda_ n| \}_{n\geq 0}\) converges to 1 very fast. As a consequence we show that every dominating subset of the unit disc can be partitioned into a sequence of pairwise disjoint dominating subsets. We also show that the pseudohyperbolic distance is connected with the relative positions of the eigenvalues of a contraction operator on a Hilbert space, and this points to possible applications of dominating sequences to operator theory.
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 3855792
zbMATH Keywords: dominating sequence
zbMATH Keywords: pseudohyperbolic metric
zbMATH Keywords: dominating subset of the unit disc
zbMATH Keywords: contraction operator on a Hilbert space
MaRDI profile type: Publication
cites work: Q3955945
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description: scientific article; zbMATH DE number 3855792