Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space
Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 2, pp. 189-209.

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].

DOI: 10.5802/ambp.247
Classification: 46S10,  47L10
Keywords: Indecomposable operators, Algebras of bounded operators
Carla Barrios Rodríguez 1

1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile. Casilla 306, Correo 22. Santiago, Chile
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Carla Barrios Rodríguez. Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space. Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 2, pp. 189-209. doi : 10.5802/ambp.247. https://ambp.centre-mersenne.org/articles/10.5802/ambp.247/

[1] Carla Barrios Rodríguez Dos familias de operadores autoadjuntos e indescomponibles en un espacio ortomodular (2004) Tesis de Magister en Ciencias Exactas (Matemáticas)

[2] Herbert Gross; Urs-Martin Künzi On a class of orthomodular quadratic spaces, Enseign. Math. (2), Volume 31 (1985) no. 3-4, pp. 187-212 | MR | Zbl

[3] Hans A. Keller; Hermina Ochsenius A. Bounded operators on non-Archimedian orthomodular spaces, Math. Slovaca, Volume 45 (1995) no. 4, pp. 413-434 | MR | Zbl

[4] Hans A. Keller; 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.), Volume 192, Dekker, New York, 1997, pp. 253-264 | MR | Zbl

[5] Hans Arwed Keller Ein nicht-klassischer Hilbertscher Raum, Math. Z., Volume 172 (1980) no. 1, pp. 41-49 | DOI | MR | Zbl

[6] Paulo Ribenboim Théorie des valuations, Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 | Zbl

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