### A discrete fixed point theorem of Eilenberg as a particular case of the contraction principle.

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This paper is mainly concerned with extensions of the so-called Vishik functional calculus for analytic bounded linear operators to a class of unbounded linear operators on ${c}_{0}$. For that, our first task consists of introducing a new class of linear operators denoted $W\left({c}_{0}(J,\omega ,\mathbb{K})\right)$ and next we make extensive use of such a new class along with the concept of convergence in the sense of resolvents to construct a functional calculus for a large class of unbounded linear operators.

The aim of this paper is the study of a certain class of compact-like sets within some spaces of continuous functions over non-Archimedean ground fields. As a result, some p-adic Ascoli theorems are obtained.

The paper considers the representation of non-degenerate bilinear forms on the non-Archimedean Hilbert space ${\mathbb{E}}_{\omega}\times {\mathbb{E}}_{\omega}$ by linear operators. More precisely, upon making some suitable assumptions we prove that if $\varphi $ is a non-degenerate bilinear form on ${\mathbb{E}}_{\omega}\times {\mathbb{E}}_{\omega}$, then $\varphi $ is representable by a unique linear operator $A$ whose adjoint operator ${A}^{*}$ exists.

The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if $\phi $ is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then $\phi $ is representable by a unique self-adjoint (possibly unbounded) operator $A$.

The paper is concerned with the spectral analysis for the class of linear operators $A={D}_{\lambda}+X\otimes Y$ in non-archimedean Hilbert space, where ${D}_{\lambda}$ is a diagonal operator and $X\otimes Y$ is a rank one operator. The results of this paper turn out to be a generalization of those results obtained by Diarra.