Indecomposable operators on form Hilbert spaces. (English) Zbl 1135.46043

Summary: The class of orthomodular spaces described by Gross and Künzi based on H. Keller’s work is a generalization of classic Hilbert spaces. Let \(E\) be an orthomodular space in this class, endowed with a positive form \(\varphi\). As in Hilbert spaces, \(\varphi\) induces a topology on \(E\) making it a complete space. For every \(n\in \mathbb N\), we describe definite spaces \((E_n,\varphi_n)\), with \(\dim(E_n)=2^n\) over the base field \(K_n=\mathbb R((\chi_1,\cdots,\chi_n))\), and we build a family of selfadjoint and indecomposable operators. Later we build an orthomodular definite space \((E,\varphi)\) with infinite dimension and we also prove that the sequence of operators in this family induces a bounded, selfadjoint and indecomposable operator in \((E,\varphi)\).


46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
Full Text: Euclid