<abstract xmlns="http://www.w3.org/1999/xhtml"> 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 ℌW 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 ℌW with ℌ as well ℌ with ℌW. Furthermore, we show that for any Hilbert space with W-metric ℌW, being 0 an element of the spectrum of W (0 ∈σ(W)), has a decomposition ℌW = ⊕n∈ℕ∪{∞}ℌWψn where ℌWψn ≃ L2(σ(W), χdμn(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.</abstract>

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 ℌW 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 ℌW with ℌ as well ℌ with ℌW. Furthermore, we show that for any Hilbert space with W-metric ℌW, being 0 an element of the spectrum of W (0 ∈σ(W)), has a decomposition ℌW = ⊕n∈ℕ∪{∞}ℌWψn where ℌWψn ≃ L2(σ(W), χdμn(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.

Frames of subspaces in Hilbert spaces with W-metrics

Department of Mathematics, Universidad del Atlántico and Intelectual.Co Barranquilla,

Department of Mathematics, Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), Mexico city,

Department of Mathematics, Universidad Surcolombiana, Neiva,

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"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...

Frames of subspaces in Hilbert spaces with W-metrics

Published Online: Apr 22, 2017

Page range: 5 - 22

Received: Jan 01, 2014

Accepted: Mar 01, 2014

DOI: https://doi.org/10.1515/auom-2015-0021

Keywords
frame of subspaces, Dual frame, Fusion frames, frames in Krein spaces, Krein spaces, frames in W-spaces

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Frames of subspaces in Hilbert spaces with W-metrics

Published Online: Apr 22, 2017

Page range: 5 - 22

Received: Jan 01, 2014

Accepted: Mar 01, 2014

DOI: https://doi.org/10.1515/auom-2015-0021

Keywordsframe of subspaces, Dual frame, Fusion frames, frames in Krein spaces, Krein spaces, frames in W-spaces

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
frame of subspaces, Dual frame, Fusion frames, frames in Krein spaces, Krein spaces, frames in W-spaces