Summability factors for absolute Riesz weighted summability methods over Banach algebras (Q1281336)

Let \(A\) be a Banach algebra over \({\mathbb{R}}\) or \({\mathbb{C}}\) and \(X\) a left Banach \(A\)-module (which may in addition be a left Banach \(A\)-algebra). If \((\tau_{nk})\) is an infinite matrix over \(A\) defining a matrix transformation \(T_nx=\sum_k\tau_{nk}x_k\) of a series \(\sum_kx_k\) in \(X\) to a sequence \((T_nx)\), then \(\overline{T}_nx=\sum_k\overline {\tau}_{nk}x_k\), where \(\overline{\tau}_{nk}=\tau_{nk}-\tau_{n-1,k}\), is a matrix transformation which transforms a series \(\sum_kx_k\) to a series \(\sum_n\overline{T}_nx\). The series \(\sum_kx_k\) is said to be \(T_A\)-summable if \((T_nx)\) is in \(c(X)\), the space of convergent sequences in \(X\), and is \(| T_A| \)-summable if \((\overline{T}_nx)\) is in \(l(X)\), the space of absolutely summable sequences in \(X\). If \(B_A\) is a method of summability over \(A\), the elements \(\epsilon_n\) of \(A\) are summability factors (a) of \((| T_A| ,B_A)\)-type if the series \(\sum\epsilon_nx_n\) is \(B_A\)-summable for each \(| T_A| \)-summable series \(\sum x_k\) in \(X\), and (b) of \((| T_A| ,| B_A|)\)-type if the series \(\sum\epsilon_nx_n\) is \(| B_A| \)-summable for each \(| T_A| \)-summable series \(\sum x_n\) in \(X\). If \(A\) has a unit \(e_A\) and \((p_n)\) is a sequence in \(A\) for which \(P_n=p_0+\cdots+p_n\) is invertible in \(A\) for each \(n\), the Riesz weighted means summability method \(P_A\) (which transforms a series to a sequence) is defined by the matrix \((\tau_{nk})\), where \[ \tau_{nk}=\begin{cases} e_A-P_n^{-1}P_{k-1}, &\text{if \(k\leq n\);}\\ \theta_A, &\text{if \(k>n\),}\end{cases} \] where \(\theta_A\) is the null element of \(A\). The Riesz weighted means summability method \(\overline{P}_A\) (which transforms a series to a series) is defined by the matrix \((\overline{\tau}_{nk})\), where \(\overline{\tau}_{nk}=P_{n-1}^{-1}p_nP_n^{-1}P_{k-1}\). After giving generalized versions of the Knopp-Lorentz theorem [\textit{K. Knopp} and \textit{G. G. Lorentz}, Beiträge zur absoluten Limitierung, Arch. Math. 2, 10-16 (1949)/(1950; Zbl 0041.18402)] and a theorem of \textit{H. Hahn} [Über Folgen linearer Operationen, Monatsh. Math. Phys. 32, 3-88 (1922; JFM 48.0473.01)], the authors prove theorems characterizing \((| P_A| ,B_A)\)-factors and \((| P_A| ,| B_A|)\)-factors. These generalize some theorems of \textit{G. Kangro} [On summability factors, Tartu Ülik. Toimetised 37, 191-232 (1955) (Russian)].