On dominating sequences in the unit disc
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.
scientific article; zbMATH DE number 3855792
scientific article