Linear discrete operators on the disk algebra

Abstract:

Let $\mathcal {A}$ be the disk algebra. In this paper we address the following question: Under what conditions on the points $z_{k,n} \in \mathbf { D}$ do there exist operators $L_n :\mathcal {A} \to \mathcal {A}$ such that \[ L_n f = \sum ^{m_n}_{k=1} f(z_{k,n})l_{k,n},\qquad f, l_{k,n} \in \mathcal {A}, \] and $L_nf \to f$, $n \to \infty$, for every $f \in \mathcal {A}$? Here the convergence is understood in the sense of $sup$ norm in $\mathcal {A}$. Our first result shows that if $z_{k,n}$ satisfy Carleson condition, then there exists a function $f \in \mathcal {A}$ such that $L_nf \not \to f$, $n \to \infty$. This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if $L_n$ are required to be projections, then for any choice of $z_{k,n}$ the operators $L_n$ do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.