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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 6, Pages 3–28 (Mi izv266)  

Non-Archimedean analogues of orthogonal and symmetric operators

S. A. Albeverioa, J. M. Bayod, C. Perez-Garsia, A. Yu. Khrennikov, R. Cianci

a Ruhr-Universität Bochum, Mathematischer Institut

Abstract: We study orthogonal and symmetric operators on non-Archimedean Hilbert spaces in connection with the $p$-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators on $p$-adic Hilbert spaces represent physical observables. We study the spectral properties of one of the most important quantum operators, namely, the position operator (which is represented on $p$-adic Hilbert $L_2$-space with respect to the $p$-adic Gaussian measure). Orthogonal isometric isomorphisms of $p$-adic Hilbert spaces preserve the precision of measurements. We study properties of orthogonal operators. It is proved that every orthogonal operator on non-Archimedean Hilbert space is continuous. However, there are discontinuous operators with dense domain of definition that preserve the inner product. There exist non-isometric orthogonal operators. We describe some classes of orthogonal isometric operators on finite-dimensional spaces. We study some general questions in the theory of non-Archimedean Hilbert spaces (in particular, general connections between the topology, norm and inner product).

DOI: https://doi.org/10.4213/im266

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English version:
Izvestiya: Mathematics, 1999, 63:6, 1063–1087

Bibliographic databases:

MSC: 46S10
Received: 28.10.1997

Citation: S. A. Albeverio, J. M. Bayod, C. Perez-Garsia, A. Yu. Khrennikov, R. Cianci, “Non-Archimedean analogues of orthogonal and symmetric operators”, Izv. RAN. Ser. Mat., 63:6 (1999), 3–28; Izv. Math., 63:6 (1999), 1063–1087

Citation in format AMSBIB
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\vol 63
\issue 6
\pages 3--28
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\pages 1063--1087
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