×

Some more Steinhaus type theorems over valued fields. (English) Zbl 0943.47058

Let \(A\) be an infinite matrix with entries from a complete non-trivially valued non-Archimedean field. The author finds necessary and sufficient conditions for \(A\) to define a bounded operator between various sequence spaces: \(\ell\to c_0\), \(c\to c_0\), \(\ell_\infty \to c_0\), and also for the relation \[ \sum\limits_{n=0}^\infty (Ax)_n=\sum\limits_{k=0}^\infty x_k,\quad x=(x_k)\in \ell. \] The latter problem is studied also for \(K=\mathbb R\) or \(\mathbb C\).

MSC:

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
40H05 Functional analytic methods in summability
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Bachman, G., Introduction to p-adic numbers and valuation theory, Academic Press, 1964. · Zbl 0192.40103
[2] Escassut, A., Analytic elements in p-adic analysis, World Scientific Publishing Co., 1995. · Zbl 0933.30030
[3] Maddox, I.J., On theorems of Steinhaus type, J. London Math. Soc.42 (1967), 239-244. · Zbl 0145.28802
[4] Natarajan, P.N., The Steinhaus theorem for Toeplitz matrices in non-archimedean fields, Comment. Math. Prace Mat.20 (1978), 417-422. · Zbl 0401.40005
[5] Natarajan, P.N., Some Steinhaus type theorems over valued fields, Ann. Math. Blaise Pascal3 (1996), 183-188. · Zbl 0897.46062
[6] Stieglitz, M., Tietz, H., Matrix transformationen von Folgenräumen Eine Ergebnisübersicht, Math. Z.154 (1977), 1-16. · Zbl 0331.40005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.