Frames of subspaces in Hilbert spaces with W-metrics

In this paper we start considering a sesquilinear form 〈W·,·〉 defined over a Hilbert space (ℌ,〈·,·〉) where W is bounded (W* = W ∈ Ɓ(ℌ)) and ker W = {0}. We study the dynamic of frame of subspaces over the completion of (ℌ, 〈W·,·〉) which is denoted by ℌ_W and is called Hilbert space with W-metric or simply W-space. The sense of dynamics studied here refers to the behavior of frame of subspaces comparing ℌ_W with ℌ as well ℌ with ℌ_W. Furthermore, we show that for any Hilbert space with W-metric ℌ_W, being 0 an element of the spectrum of W (0 ∈σ(W)), has a decomposition ℌ_W = ⊕_{n∈ℕ∪{∞}}ℌ^W_ψn where ℌ^W_ψn ≃ L₂(σ(W), χdμ_n(χ)) for all n ∈ ℕ ∪ {∞}, L₂ denotes a Hilbert space square integrable and μ a Lebesgue measure. Finally, the case when W is unbounded also considered.