Non-Archimedean analogues of orthogonal and symmetric operators. (English. Russian original) Zbl 0974.47054
Izv. Math. 63, No. 6, 1063-1087 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 3-28 (1999).
Authors’ 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).
Reviewer: S.L.Singh (Rishikesh)
MSC:
47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |
46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
47N50 | Applications of operator theory in the physical sciences |
81Q99 | General mathematical topics and methods in quantum theory |
47B99 | Special classes of linear operators |