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Linear discrete operators on the disk algebra

Authors: Ivan V. Ivanov and Boris Shekhtman
Journal: Proc. Amer. Math. Soc. 129 (2001), 1987-1993
MSC (2000): Primary 46-XX
Published electronically: December 13, 2000
MathSciNet review: 1825906
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Let ${\cal A}$ 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 $L_n :{\cal A} \to {\cal A}$ such that

\begin{displaymath}L_n f = \sum^{m_n}_{k=1} f(z_{k,n})l_{k,n},\qquad f, l_{k,n} \in {\cal A}, \end{displaymath}

and $L_nf \to f$, $n \to \infty$, for every $f \in {\cal A}$? Here the convergence is understood in the sense of $sup$ norm in $\cal A$. Our first result shows that if $z_{k,n}$ satisfy Carleson condition, then there exists a function $f \in {\cal A}$ 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.

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Additional Information

Ivan V. Ivanov
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843

Boris Shekhtman
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620

Received by editor(s): November 30, 1998
Received by editor(s) in revised form: October 15, 1999
Published electronically: December 13, 2000
Additional Notes: The results in this paper are part of the Ph.D. thesis of the first author.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society