Nizar El Idrissi
@uit.ac.ma
Ibn Tofail University
RESEARCH, TEACHING, or OTHER INTERESTS
Mathematics
Scopus Publications
Scholar Citations
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Scopus Publications
Nizar El Idrissi, Samir Kabbaj, and Brahim Moalige
Universidad Catolica del Norte - Chile
If H is a Hilbert space, the non-compact Stiefel manifold St(n, H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that ∩j∈J (U(j) + St(n, H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n, H)∩S contains all polynomial paths contained in S and passing through a point in St(n, H). As a consequence, we prove that St(n, H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n, L2(X, μ; F)) is isometric to, FF(X, μ), n, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.
Nizar El Idrissi and Samir Kabbaj
Walter de Gruyter GmbH
Abstract In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to e n +1 for all n ∈ ℕ (where (e n ) n ∈ℕ is a sequence of vectors) guarantees that (e n ) n ∈ℕ is linearly independent if and only if dim{e n } n ∈ℕ = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like T(e i ) = e ϕ ( i ) for all i ∈ I where I is countable as a replacement of the previous one, the conclusion will only stay true if ϕ : I → I is conjugate to the successor function succ : n ↦n + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition T(e i ) = e ϕ ( i ) for all i ∈ I where ϕ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.
Nizar El Idrissi, Samir Kabbaj, and Brahim Moalige
Universidad Catolica del Norte - Chile
We propose in this article some characterizations of the notion of frame in ℓ²(I; H). The first one is general, and depends on a procedure of inserting a family of vectors instead of x in the definition of a frame. This allows us to define the analysis, synthesis and frame operator on the space ℓ²(I; H) instead of H. The second one is specific to ℓ² (I; Ck) and relate it to the freeness of the finite set of components of the frame. The third one concerns normalised tight frames in ℓ²(I; Ck). Afterwards, we give an example of a frame in ℓ²(I; C²) using another sufficient condition in dimension 2. We conclude with some topological applications of these characterizations.