Linear discrete operators on the disk algebra (Q2718960)

Let \(\mathbb{T}= \{z\in\mathbb{C}:|z|=1\}\) be the unit circle and let \(A\) be the disk-algebra, i.e., the set of all functions \(f\) analytic inside and continuous on the boundary of \(\mathbb{D}= \{z\in\mathbb{C}:|z|\leq 1\}\). In this paper the following question is considered:NEWLINENEWLINENEWLINEUnder what conditions on the points \(z_{k,n}\in\mathbb{D}\) do there exist sequences of finite-dimensional operators \(L_n: A\to A\), \(n\geq 1\) of the form \(L_nf= \sum^{m_n}_{k=1} f(z_{k,n})\ell_{k,n}\), \(f,\ell_{k,n}\in A\) which strongly convergence to identity generator \(I\)?NEWLINENEWLINENEWLINEThe authors show that if \(z_{k,n}\) satisfy the Carleson condition, then there exists a function \(f\in A\) such that \(L_nf\nrightarrow f\), \(n\to\infty\).NEWLINENEWLINENEWLINEThe second result of this paper shows that if \(L_n\), \(n\geq 1\) are projections, then for any choice of \(z_{k,n}\) the operators \(L_n\), \(n\geq 1\) do not convergence to the \(I\).