A connection between operator topologies, polynomial interpolation, and synthesis of diagonal operators

In this paper we extend the work done by Deters and Seubert in [I. Deters, S. Seubert, Cyclic vectors of diagonal operators on the space of functions analytic on a disk, J. Math. Anal. Appl. 334 (2007) 1209–1219]. In particular, we shall improve upon a theorem in the aforementioned paper and make connections between the strong operator topology on H(B(0,1)), polynomial behavior on a sequence of points, and the synthesis of diagonal operators on H(B(0,1)).