Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space

Carla Barrios Rodríguez[1]

  • [1] Facultad de Matemáticas, Pontificia Universidad Católica de Chile. Casilla 306, Correo 22. Santiago, Chile

Annales mathématiques Blaise Pascal (2008)

  • Volume: 15, Issue: 2, page 189-209
  • ISSN: 1259-1734

Abstract

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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].

How to cite

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Barrios Rodríguez, Carla. "Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space." Annales mathématiques Blaise Pascal 15.2 (2008): 189-209. <http://eudml.org/doc/10560>.

@article{BarriosRodríguez2008,
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].},
affiliation = {Facultad de Matemáticas, Pontificia Universidad Católica de Chile. Casilla 306, Correo 22. Santiago, Chile},
author = {Barrios Rodríguez, Carla},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Indecomposable operators; Algebras of bounded operators; indecomposable operators; algebras of bounded operators},
language = {eng},
month = {7},
number = {2},
pages = {189-209},
publisher = {Annales mathématiques Blaise Pascal},
title = {Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space},
url = {http://eudml.org/doc/10560},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Barrios Rodríguez, Carla
TI - Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space
JO - Annales mathématiques Blaise Pascal
DA - 2008/7//
PB - Annales mathématiques Blaise Pascal
VL - 15
IS - 2
SP - 189
EP - 209
AB - 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].
LA - eng
KW - Indecomposable operators; Algebras of bounded operators; indecomposable operators; algebras of bounded operators
UR - http://eudml.org/doc/10560
ER -

References

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  1. Carla Barrios Rodríguez, Dos familias de operadores autoadjuntos e indescomponibles en un espacio ortomodular, (2004) 
  2. Herbert Gross, Urs-Martin Künzi, On a class of orthomodular quadratic spaces, Enseign. Math. (2) 31 (1985), 187-212 Zbl0603.46030MR819350
  3. Hans A. Keller, Hermina Ochsenius A., Bounded operators on non-Archimedian orthomodular spaces, Math. Slovaca 45 (1995), 413-434 Zbl0855.46049MR1387058
  4. Hans A. Keller, Herminia Ochsenius A., An algebra of self-adjoint operators on a non-Archimedean orthomodular space, -adic functional analysis (Nijmegen, 1996) 192 (1997), 253-264, Dekker, New York Zbl0892.47074MR1459214
  5. Hans Arwed Keller, Ein nicht-klassischer Hilbertscher Raum, Math. Z. 172 (1980), 41-49 Zbl0414.46018MR576294
  6. Paulo Ribenboim, Théorie des valuations, 1964 (1968), Les Presses de l’Université de Montréal, Montreal, Que. Zbl0139.26201

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