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Some properties of the matrix class (ℓα, ℓα), 0 < α ⩽ 1

Natarajan Pinnangudi N.

Commentationes Mathematicae

2020

Vol. 60, No. 1/2

23--36

Throughout the present paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The class (ℓα, ℓα), 0 < α ⩽ 1, of all infinite matrices transforming sequences in ℓα to sequences in ℓα is characterized. The structure of (ℓα, ℓα), 0 < α ⩽ 1, is then discussed. Following Fridy [Properties of absolute summability matrices, Proc. Amer. Math. Soc. 24 (1970), 583-585], a Steinhaus type result involving the class (ℓα, ℓα) is also proved.

Cauchy multiplication of Euler summable series in ultrametric fields

Deepa R., Natarajan P. N., Srinivasan V.

Commentationes Mathematicae

2013

Vol 53, No. 1

73--79

Euler summability method in a complete, non-trivially valued, ultrametric field of the characteristic zero was introduced by Natarajan in [7]. Some properties of the Euler summability method in such fields were studied in [2] and [7]. The purpose of the present note is to continue the study and to prove a pair of theorems on the Cauchy product of Euler summable sequences and series.