Absolute Cesàro series spaces and matrix operators
Denote by \(l_k\), \(1\leq k<\infty\), the space of \(k\)-absolutely convergent series. Then, for \(\lambda>-1\) and \(\lambda+\mu\neq -1, -2, \dots\), the authors introduce the generalized Cesàro series space \(|C_{\lambda, \mu}|_k\) as \[ |C_{\lambda, \mu}|_k=(l_k)_T=\{x=(x_n)\in\omega: T(x)\in l_k\}, \] where \(T=(t_{n\upsilon})\) is a triangular matrix given by the following expression \[ t_{n\upsilon} = \begin{cases} 1 & \quad n, \upsilon=0,\\ \frac{A_{n-\upsilon}^{\lambda-1}{A_\upsilon^\mu}\upsilon}{n^{1/k}A_n^{\lambda+\mu}} & \quad 1\leq \upsilon \leq n, \\ 0 & \quad \upsilon>n, \end{cases} \] and \(A_0^\lambda=1\), \(A_n^\lambda=\frac{(\lambda+1)(\lambda+2)\ldots(\lambda+n)}{n!}\), \(A_{-n}^\lambda=0\), \(n\geq 1\). It is shown that the space \(|C_{\lambda, \mu}|_k\) is a \(BK\)-space induced by the norm \(\|x\|_{|C_{\lambda, \mu}|_k}=\|T(x)\|_{l_k}\) (Theorem 2.2) if \(\mu\) and \(\lambda+\mu\) are non-negative integers. The isometric characterization of the space is established in Theorem 2.3, which says \(|C_{\lambda, \mu}|_k\) is linearly isomorphic to the space \(l_k\) for \(1\leq k<\infty\). Further, the \(\beta\)-dual of \(|C_{\lambda, \mu}|_k\) and results related to matrix transformations under certain restrictions on \(\lambda, \mu\) are derived in Theorems 2.1, 2.4 and 2.5, respectively. Several consequences of the established theorems depending on the parameters \(\lambda, \mu\), etc., and the matrix \(A\) are discussed in Section 4. The results obtained in this paper strengthen several earlier known results.
scientific article; zbMATH DE number 6860441
scientific article