2013/05/15 by Sergej A. Choroszavin, Choroszavin, Sergej A.
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #math.DS #math.FA
paper · pdf · doi:10.48550/arxiv.1305.3505
Misprints/mistakes corrected. Exposition changed
arxiv created 2013/05/28 · arxiv updated 2013/05/29
Given a lineal H0 and x0∈ H0 and a linear injective operator U0: H0 → H0 such that all U0N, N ∈ \bf Z exist and all U0N x0, N ∈ \bf Z are linearly independent, anyone can define on spanU0N x0 | N ∈ \bf Z a (pre)hilbert scalar product such that U0 becomes a unitary operator. The problem under consideration is: Suppose there is specified an indefinite inner product ,0 on H0 and U0 is a ,0-unitary operator. Can one introduce a (pre)hilbert topology on Lx0 so that after completion and possible extension the resulting ,ext is continuous, the resulting Uext is ,ext-unitary and there exists a pair L+, L- mutually ,ext-orthogonal, maximal strictly positive and respectively negative subspaces, so that they are Uext-invariant? More generally, can one construct a sequence (chain) of transformations of the type "restrict, change topology, make completion, extend, restrict,..." with the same result? (And, as a result, after some transformations, which are natural in the field of indefinite inner product spaces, Uext will become usual Hilbert space unitary operator). For a relatively wide class of pairs "operator, inner product" positive solutions proposed.