Frames of subspaces in Hilbert spaces with W-metrics

الأصل · EN

If (,·,·) is a Hilbert space and on it we consider the sesquilinear form W·,· so-called W-metric, where W*=W∈, and W={0}, then the space (,W·,·) is called Hilbert space with W-metric or simply W-space. In this paper we investigate the dynamic of frames of subspace on these spaces, where the sense of dynamics refers to the behavior of frames of subspace in (the completion of (,W·,·)) comparing with and vice versa. This work is based on the study made in KEFER,GMMM on frames in Krein spaces. In a similar way, Casazza and Kutyniok obtained some results in the context of Hilbert spaces, see CG. We take tools of theory of C*-algebra, and properties of, to show that every Hilbert space with W-metric with 0∈σ(W) has a decomposition =ₙ∈∪{∞}ψₙʷ, where ψₙʷ≃ (σ(W),xdμₙ(x)) are Krein spaces, for every n∈∪{∞}. Moreover, we investigate the dynamics of frames of subspace when the self-adjoint operator W is unbounded.