Power-Series Summability Methods in de Branges–Rovnyak Spaces

Abstract

We show that there exists a de Branges–Rovnyak space $${\mathcal {H}}(b)$$ on the unit disk containing a function f with the following property: even though f can be approximated by polynomials in $${\mathcal {H}}(b)$$ , neither the Taylor partial sums of f nor their Cesàro, Abel, Borel or logarithmic means converge to f in $${\mathcal {H}}(b)$$ . A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.