Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space

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 invariant closed subspaces other than the trivial ones. Spectrums of such operators contain exactly one point which, therefore, is not an eigenvalue. We also study relations between the subalgebras of bounded linear operators of $E$, which are the commutant of each of these operators, and the algebra $𝒜$ studied in [3].