A Tauberian theorem for null sequences (Q1330816)

For a (real) sequence \(x= \{x_ n\}_{n\in\mathbb{N}}\) and for \(E\subset\mathbb{N}\) let \(C_ E\) denote the characteristic function of the set \(E\) and let \(C_ E x\) denote the pointwise product of the sequence \(C_ E\) and \(x\). The following Tauberian theorem for null sequences is established: Suppose \(A\not\in (\ell^ \infty, c_ 0)\). If the sequence \(x\) is such that for every subsequence \(y\) of \(x\) every sequence in \(C(y):= \{C_ E y\): \(E\subset \mathbb{N}\}\) is \(A\)-summable to 0, then \(x\) converges to 0. This result is similar to a Tauberian theorem for subsequences given by Maddox. In addition, the author obtains a characterization of the class \((\ell^ \infty, c_ 0)\) which sharpens a result of Natarajan.