Nörlund and weighted mean matrices as operators on \(\ell_ p\)
Let f be a logarithmic-exponential function defined on \([x_ 0,\infty)\) for some \(x_ 0>n\), and let \(\{a_ n\}\) be a sequence of positive numbers with \(a_ n=f(n)\) for sufficiently large n. Write \(A_ n=\sum^{n}_{k=0}a_ k\). Define the Nörlund matrix \(N_ a=(a_{nk})\) by \[ a_{nk}=a_{n-k}/A_ n\text{ for } 0\leq k\leq n,\text{ and } =0\text{ for } k>n, \] and the weighted mean matrix \(W_ a=(a_{nk})\) by \[ a_{nk}=a_ k/A_ n\text{ for } 0\leq k\leq n,\text{ and } =0\text{ for } k>n. \] For \(1<p<\infty\), let \(\ell_ p\) be the BK- space of all sequences \(\{x_ k\}\) such that \(\sum | x_ k|^ p\) converges. Denote by \(B(\ell_ p)\) the Banach algebra of all bounded linear operators on \(\ell_ p\) with the norm \(\| T\|_ p\) for T in \(B(\ell_ p)\). The authors have established that (i) \(Na\in B(\ell_ p)\Leftrightarrow \lim_{n\to \infty}(na_ n/A_ n)<\infty;\) (ii) \(W_ a\in B(\ell_ p)\Leftrightarrow \lim_{n\to \infty}(A_ n/na_ n)<p.\) Asymptotic properties of the norms of the operators are also obtained.
Asymptotic properties of the norms of the operators
scientific article; zbMATH DE number 4171754