With cedram.org Annales Mathématiques Blaise Pascal Search for an article Search within the site English français Table of contents for this issue | Previous article | Next article Carla Barrios RodríguezTwo Families of Self-adjoint Indecomposable Operators in an Orthomodular SpaceAnnales mathématiques Blaise Pascal, 15 no. 2 (2008), p. 189-209, doi: 10.5802/ambp.247 Article PDF | Reviews MR 2473817 | Zbl 1163.47061 Class. Math.: 46S10, 47L10Keywords: Indecomposable operators, Algebras of bounded operators Résumé - AbstractOrthomodular spaces are the counterpart of Hilbert spaces for fields other than $\mathbb{R}$ or $\mathbb{C}$. 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 $\mathcal{A}$ studied in [3]. Bibliography[1] Carla Barrios Rodríguez, Dos familias de operadores autoadjuntos e indescomponibles en un espacio ortomodular, Technical report, Pontificia Universidad Católica de Chile, 2004 [2] Herbert Gross and Urs-Martin Künzi. On a class of orthomodular quadratic spaces. Enseign. Math. (2), 31(3-4):187-212, 1985.  MR 819350 |  Zbl 0603.46030[3] Hans A. Keller and Hermina Ochsenius A.. Bounded operators on non-Archimedian orthomodular spaces. Math. Slovaca, 45(4):413-434, 1995.  MR 1387058 |  Zbl 0855.46049[4] Hans A. Keller and Herminia Ochsenius A., An algebra of self-adjoint operators on a non-Archimedean orthomodular space, $p$-adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. 192, Dekker, 1997, p. 253–264  MR 1459214 |  Zbl 0892.47074[5] Hans Arwed Keller. Ein nicht-klassischer Hilbertscher Raum. Math. Z., 172(1):41-49, 1980. Article |  MR 576294 |  Zbl 0414.46018[6] Paulo Ribenboim. Théorie des valuations, volume 1964 of Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été. Les Presses de l’Université de Montréal, Montreal, Que., 1968  Zbl 0139.26201 © 2019Annales Mathématiques Blaise Pascal Published by the Laboratoire de mathématiques CNRS - UMR 6620 Université Blaise Pascal de Clermont-Ferrand