We are concerned with some unbounded linear operators on the so-called $p$-adic Hilbert space ${𝔼}_{\omega }$. Both the Closedness and the self-adjointness of those unbounded linear operators are investigated. As applications, we shall consider the diagonal operator on ${𝔼}_{\omega }$, and the solvability of the equation $Au=v$ where $A$ is a linear operator on ${𝔼}_{\omega }$.

Let K be an algebraically closed field, complete for an ultra- metric absolute value, let D be an infinite subset of K and let H(D) be the set of analytic elements on D. We denote by Mult(H(D), UD) the set of semi-norms Phi of the K-vector space H(D) which are continuous with respect to the topology of uniform convergence on D and which satisfy further Phi(f g)=Phi(f) Phi(g) whenever f,g elements of H(D) such that fg element of H(D). This set is provided with the topology of simple convergence....

Orthomodular spaces are the counterpart of Hilbert spaces for fields other than $ℝ$ or $ℂ$. Both share numerous properties, foremost among them is the validity of the Projection theorem. Nevertheless in the study of bounded linear operators which started in [3], there appeared striking differences with the classical theory. In fact, in this paper we shall construct, on the canonical non-archimedean orthomodular space $E$ of [5], two infinite families of self-adjoint bounded linear operators having no...