Tauberian-type theorem for \((e)\)-convergent sequences (Q2376616)

The author proves the following theorem which is a Tauberian-type theorem for \((e)\)-convergent sequences. Let \(\mathcal{H}=\mathcal{H}(\Omega)\), \(e=(e_n(z))_{n\geq 0}\) and \(k_{\mathcal{H,\lambda}}\) be the same as in the definition of \((e)\)-convergent sequences. Also, let \(\ell_1^2\) denote the unit sphere of the sequences space \(\ell^2\): \[ \ell_1^2:=\left\{ (x_m) _{m\geq 0}\in \ell ^2:\|(x_m) \|_{\ell^2}=1\right\} . \] Let \((a_n)_{n\geq 0}\) be a bounded sequence of complex numbers such that \((a_n)_{n\geq 0}\) \((e)\)-convergences to \(a\). Denoting by \(k_{\mathcal{H,\lambda}}\) the reproducing kernel of the standard functional Hilbert space \(\mathcal{H}\) at \(\lambda\), we assume that \[ \sum_{m=0}^{+\infty }(a_m-a) \left| \sum_{n=0}^{+\infty } \bar{x}_{mn}e_n(\lambda) \right| ^{2}=\mathrm{o}\left(\left\| k_{\mathcal{H,\lambda }}\right\| ^{2}\right) \] for every double sequence \((x_{mn})_{m,n=0}^{+\infty }\) with \(\|(x_{mn})_m\| =1\) (for all \(n\geq 0\)) and \(\|(x_{mn})_n\|=1\) (for all \(m\geq 0\)), whenever \(\lambda\) tends to a point in the boundary of \(\Omega\). Then \((a_n)_n\) converges to \(a\) in the usual sense.