Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space
Carla Barrios Rodríguez
Annales Mathématiques Blaise Pascal, Volume 15 (2008) no. 2, p. 189-209

Orthomodular spaces are the counterpart of Hilbert spaces for fields other than or . 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 : https://doi.org/10.5802/ambp.247
Classification:  46S10,  47L10
Keywords: Indecomposable operators, Algebras of bounded operators
@article{AMBP_2008__15_2_189_0,
     author = {Barrios Rodr\'\i guez, Carla},
     title = {Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space},
     journal = {Annales Math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {15},
     number = {2},
     year = {2008},
     pages = {189-209},
     doi = {10.5802/ambp.247},
     mrnumber = {2473817},
     zbl = {1163.47061},
     language = {en},
     url = {https://ambp.centre-mersenne.org/item/AMBP_2008__15_2_189_0}
}
Barrios Rodríguez, Carla. 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/item/AMBP_2008__15_2_189_0/

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

[2] Herbert Gross; Urs-Martin Künzi On a class of orthomodular quadratic spaces, Enseign. Math. (2), Tome 31 (1985) no. 3-4, pp. 187-212 | MR 819350 | Zbl 0603.46030

[3] Hans A. Keller; Hermina Ochsenius A. Bounded operators on non-Archimedian orthomodular spaces, Math. Slovaca, Tome 45 (1995) no. 4, pp. 413-434 | MR 1387058 | Zbl 0855.46049

[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), Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 192 (1997), pp. 253-264 | MR 1459214 | Zbl 0892.47074

[5] Hans Arwed Keller Ein nicht-klassischer Hilbertscher Raum, Math. Z., Tome 172 (1980) no. 1, pp. 41-49 | Article | MR 576294 | Zbl 0414.46018

[6] Paulo Ribenboim Théorie des valuations, Les Presses de l’Université de Montréal, Montreal, Que., Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, Tome 1964 (1968) | Zbl 0139.26201