Interpolating Blaschke products and the left spectrum of multiplication operators on the Bergman space
The paper presents material towards an answer to the following well known problem: Can every Blaschke product be uniformly approximated by interpolating Blaschke products? In the first part, we encounter D. Marshall's and D. Sarason's result that whenever the zeros of a Blaschke product \(B\) lie in a closed convex subset \(C\) of \(\mathbb{D}\cup\{1\}\), then \((B-\lambda)/(1-\bar\lambda B)\) is an interpolating Blaschke product provided that \(\lambda\) is not a cluster point of \(B\mid_ C\) at 1 and \(\lambda\not\in\{B(a)\): \(B'(a)=0\}\). Using the result that interpolating Blaschke products are dense in the set of finite products of interpolating Blaschke products, the author then derives the fact that every Blaschke product whose zeros lie on finitely many radii can be uniformly approximated by interpolating Blaschke products. In the second part, rather trivial corollaries are derived from a theorem of McDonald and Sundberg on the connection between the left spectrum of the multiplication operator on the Bergman space \(L_ a^ 2(\mathbb{D})\) and the one point Gleason parts of the spectrum of \(H^ \infty\).
scientific article; zbMATH DE number 78113