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Blaise Pascal
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Carla Barrios Rodríguez
Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space
Annales 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, 47L10
Keywords: Indecomposable operators, Algebras of bounded operators

Résumé - Abstract

Orthomodular 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].


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[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