Power-Series Summability Methods in de Branges–Rovnyak Spaces

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.