Two Applications of the Nordgren-Rosenthal Theorem

Abstract

In this article, a concept of (Ber)-convergence for the multiple sequences (a k 1,…, k N ) of complex numbers is introduced and a Tauberian theorem for such a summability method is proved. We also solve an open question for a bounded single sequence (a n ) n≥0 posed by Zorboska in [10] in connection with the compactness problem for so-called radial operators on the classical Bergman space over the unit disc 𝔻 = {z ∈ ℂ: |z| <1}. Our proofs essentially use the Berezin symbols technique of operator theory in the reproducing kernel Hilbert spaces. Namely, we apply the Nordgren-Rosenthal theorem regarding compact operators on a reproducing kernels Hilbert space.