Indecomposable operators on Form Hilbert Spaces

Abstract

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 $\phi$. As in Hilbert spaces, $\phi$ induces a topology on $E$ making it a complete space. For every $n\in \mathbb{N}$, we describe definite spaces $(E_n,\phi_n)$, with $\dim(E_n)=2^n$ over the base field $K_n=\mathbb{R}((\chi_1,\ldots,\chi_n))$, and we build a family of selfadjoint and indecomposable operators. Later we build an orthomodular definite space $(E,\phi)$ with infinite dimension and we also prove that the sequence of operators in this family induces a bounded, selfadjoint and indecomposable operator in $(E,\phi)$.