Non-Archimedean analogues of orthogonal and symmetric operators

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).