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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 $W\left(c_0(J,\omega ,𝕂)\right)$ 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 paper considers the representation of non-degenerate bilinear forms on the non-Archimedean Hilbert space ${𝔼}_{\omega }\times {𝔼}_{\omega }$ by linear operators. More precisely, upon making some suitable assumptions we prove that if $\varphi$ is a non-degenerate bilinear form on ${𝔼}_{\omega }\times {𝔼}_{\omega }$, then $\varphi$ is representable by a unique linear operator $A$ whose adjoint operator $A^{*}$ exists.