Summability and duality (Q6589789)

Let \(X\) be a Banach space over the field of real or complex numbers, and let \(X^{\ast}\) be its dual. Let \(\left( e_{k}\right) _{k\geq0}\subset X\) and \(\left( \psi_{i}\right) _{i\geq0}\subset X^{\ast}\) be sequences such that \(\left\langle e_{k},\psi_{i}\right\rangle \neq0\) if and only if \(k=i.\) Denote by \(Z\) the norm closure of span\(\left\{ \psi_{k}\mid k\geq0\right\} \) and \(c_{k}:=\left\langle e_{k},\psi_{k}\right\rangle ^{-1}.\)\N\NFor every \(k,\) define the operator \(P_{k}\colon X\rightarrow X\) by \(P_{k}\left( x\right) :=c_{k}\left\langle x,\psi_{k}\right\rangle e_{k}\); then \(\left\Vert P_{k}\right\Vert =\left\vert c_{k}\right\vert \left\Vert e_{k}\right\Vert \left\Vert \psi_{k}\right\Vert \) and the adjoint operator \(P_{k}^{\ast}\colon X^{\ast}\rightarrow X^{\ast}\) is given by \(P_{k}^{\ast }\left( \phi\right) :=c_{k}\left\langle \phi,e_{k}\right\rangle \psi_{k}.\)\N\NThe authors of the paper under review consider summability matrices \(A=\left( a_{nk}\right) _{n,k\geq0}\) with \N\[\N\sum_{k=0}^{\infty}\left\vert a_{nk}\right\vert \left\Vert P_{k}\right\Vert <\infty\tag{1}\N\]\Nand define\N\[\NS_{n}^{A}\left( x\right) :=\sum_{k=0}^{\infty}a_{nk}P_{k}\left( x\right) \text{ \ }\left( x\in X\right)\N\]\Nand\N\[\N\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) :=\sum_{k=0}^{\infty }a_{nk}P_{k}^{\ast}\left( \phi\right) \text{ \ }\left( \phi\in X^{\ast }\right)\N\]\Nfor each \(n.\) The following main results are proved.\N\NTheorem 3.1 (Equivalence theorem). If there exists a matrix \(A^{0}\) (satisfying (1)) such that \(S_{n}^{A^{0}}\left( x\right) \rightarrow x\) in \(X\) for each \(x\in X,\) then the following statements are equivalent for every \(A\) with (1):\N\begin{itemize}\N\item[(i)] \N\(S_{n}^{A}\left( x\right) \rightarrow x\) weakly in \(X\) for each \(x\in X;\)\N\item[(ii)] \N\(S_{n}^{A}\left( x\right) \rightarrow x\) in \(X\) for each \(x\in X;\)\N\item[(iii)]\N\(\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) \rightarrow\phi\) weakly\(^{\ast}\) in \(X^{\ast}\) for each \(\phi\in X^{\ast};\)\N\item[(iv)] \N\(\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) \rightarrow\phi\) weakly\(^{\ast}\) in \(X^{\ast}\) for each \(\phi\in Z;\) \N\item[(v)] \N\(\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) \rightarrow\phi\) in \(X^{\ast}\) for each \(\phi\in Z.\)\N\end{itemize}\NIf \(X\) is reflexive, then the conditions (i)--(v) are equivalent to\N\begin{itemize}\N\item[(vi)] \N \(\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) \rightarrow\phi\) in \(X^{\ast}\) for each \(\phi\in X^{\ast}.\)\N\end{itemize}\N\NTheorem 3.2 (Limitation theorem). If the matrix \(A\) admits a left inverse \(B=\left( b_{in}\right) \) and \(S_{n}^{A}\left( x\right) \rightarrow x\) in \(X\) for each \(x\in X,\) then \(\left\Vert e_{i}\right\Vert \left\Vert \psi _{i}\right\Vert =O\left( \left\vert c_{i}\right\vert \sum_{n=0}^{\infty }\left\vert b_{in}\right\vert \right) \) \ \(\left( i\rightarrow\infty\right) .\)\N\NThe proof of the equivalence theorem is based on a general operator-theoretic Theorem\ 2.1 of the authors. The significiance and applicability of this result are demonstrated in various function spaces, among them are spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, de Branges-Rovnyak spaces.