Linear discrete operators on the disk algebra
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Let A be the disk algebra. In this paper we address the following question: Under what conditions on the points zk,n D do there exist operators Ln : A A such that Lnf = k=1mn f(zk,n)lk,n, f, lk,n A, and Lnf f, n , for every f isin; A? Here the convergence is understood in the sense of sup norm in A. Our first result shows that if zk,n satisfy Carleson condition, then there exists a function f A such that Lnf f, n . 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 Ln are required to be projections, then for any choice of zk,n the operators Ln 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. 2000 American Mathematical Society.