Let A ∈ B(X) be a spectral operator on a non-archimedean Banach space over Cp. In this paper, we give a necessary and sufficient condition on the resolvent of A so that the discrete semigroup consisting of powers of A is contractions.

In this paper, we introduce new classes of linear operators so called [Formula: see text]-groups, [Formula: see text]-groups and cosine families of bounded linear operators on non-archimedean Banach spaces over non-archimedean complete valued field [Formula: see text]. We show some results about it.

In this work, we are dealing with the natural topology in a perfect sequence space and the transfert of topologies of a locally K− convex space E with a Schauder basis (e i ) i to such Space. We are also interested with the compatible topologies on E for which the basis(e i ) i is equicontinuous, and the weak basis problem. Finally, we give some applications to barrelled Spaces and G−Spaces.

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