×

Summability methods in valued fields. (English) Zbl 0719.47056

The paper deals with p-adic summability methods. The basic definition is the following. Let K be a complete non-archimedean valued field. Let A be an infinite matrix whose coefficients \(a_{nk}\) are in K. If for each convergent sequence \(s_ 1,s_ 2,..\). in K (with limit s) the sum \(t_ n:=\sum^{\infty}_{k=0}a_{nk}s_ k\) exists for each n and also \(t:=\lim_{n\to \infty}t_ n\) exists, then A is called convergence- preserving. If in addition we always have \(t=s\), A is called regular.
Criteria for regularity and convergence presevation are stated. Also one can formulate conditions on the coefficients \(a_{nk}\) in order that A maps \(\ell^{\infty}\) to \(c_ 0\), \(\ell_{\alpha}\) into \(\ell_{\alpha}\), etc. Some special summability methods are discussed. There is a comparison with the classical theory over \({\mathbb{R}}\) or \({\mathbb{C}}\). No proofs are given, as the paper is of the survey type (but there is an extensive list of references) making it into a stepping stone for anyone who wants to get to know the ‘state of the art’ in p-adic summability.

MSC:

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
40C05 Matrix methods for summability
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A45 Sequence spaces (including Köthe sequence spaces)
12J10 Valued fields
PDFBibTeX XMLCite