Unbounded Versions of Two Old Summability Theorems

Abstract

In this note, we obtain extensions of a theorem of Meyer-König and Zeller and a theorem of Wilansky in that the given results do not require a summability matrix to be a bounded operator from the convergent sequences into themselves. The culmination of the results in this note is that a triangle matrix method T with null columns maps a bounded divergent sequence to a null sequence if and only if the range of T is not closed in the null sequences.

Keywords

• unbounded summability matrix
• closed operator
• range closed
• Tauberian theorem