arXiv subjects

Nizar El Idrissi

Publications and source records attributed to Nizar El Idrissi.

12 recordsLinked to original sources

arXiv2026

Probability measure annihilating all finite-dimensional subspaces

We propose in this short note a prime numbers-based method for constructing probability measures on infinite-dimensional Banach spaces annihilating all finite-dimensional subspaces, supplementing the methods of construction of Gaussian measures and infinite-product-type probability measures. This new method confirms that probability measures with this property are generic amongst probability measures that are not supported on finite-dimensional subspaces. In the process, we show the existence of an uncountable measurable family of independent vectors having the cardinality of the continuum in any infinite-dimensional Banach space.

math.FA↗

arXiv2026

The topology of finite and infinite-dimensional Stiefel manifolds and the relation to Grassmannians

Stiefel manifolds arise naturally as spaces of injective operators and as total spaces of principal bundles over Grassmannians. While their finite-dimensional topology is governed by Bott periodicity, the infinite-dimensional theory exhibits a striking collapse phenomenon stemming from Kuiper's contractibility theorem. In this expository article, we present a unified treatment of finite and infinite-dimensional Stiefel manifolds over real and complex Hilbert spaces, emphasizing three structural principles: (i) the interpretation of Stiefel manifolds as spaces of injective operators, (ii) the polar decomposition as a canonical factorization yielding a homotopy decomposition, (iii) the role or function of Grassmannians as classifying spaces for $\operatorname{GL}_n(\mathbb{F})$ in the stable limit. In particular, we show that the polar decomposition provides a global homeomorphism \[ \operatorname{St}(n,H) \cong \operatorname{St}_{\operatorname{orth}}(n,H) \times \operatorname{P}_n(\mathbb{F}) \] valid in arbitrary Hilbert dimension, and that this factorization isolates the entire difference between finite and infinite-dimensional topology. Then, we discuss the implications for the homotopy type of Stiefel manifolds and the relation of Stiefel manifolds to the theory of classifying spaces and characteristic classes.

math.FA↗

arXiv2026

A general equivalence relation on the global class of morphisms of a category

We introduce a general equivalence relation on the global class of morphisms of a category that subsumes several classical notions of equivalence in mathematics. We show that group-action equivalence relations, pullback equivalence relations, and coarse analysis-operator-based equivalence relations on continuous Bessel families are special cases of this general definition.

math.CT↗

arXiv2026

Intersections of translates of finite-dimensionally valued frame spaces are conditionally slice-full and almost slice-full

In recent work, the topology of frame spaces $\mathcal{F}_{(X,μ),n}$ has been studied via Stiefel manifolds, revealing in particular a connectedness property for intersections of their translates when $\operatorname{span}(\{a_j\}_{j \in J}$ is not too large, in fact when $\operatorname{codim}(\operatorname{span}\{a_j^l\}_{(j,l) \in J \times [\![1,n]\!]}) \geq 3n$, where $\{a_j\}_{j \in J}$ is the translating family \cite{ElIdrissiKabbajMoalige2023}. The investigation of the connectedness of the intersections of translates of the frame space can be extended to questions about the algebro-geometric and measure-theoretic structure of such intersections. The present article addresses these questions by uncovering an almost-linear structure within intersections of translated frame spaces. We show that the set of non-frames in finite-dimensional Hilbert $C^*$-modules inherits the structure of a slice-wise real affine algebraic subvariety. As a consequence, it is a small subset in a precise measure-theoretic sense. In particular, we prove that for any finite-dimensional Hilbert $C^*$-module $\mathcal{H}$ and any countable collection of translates of the frame space $\mathcal{F}_{(X,μ),\mathcal{H}}$, the intersection is conditionally slice-full in $L^2(X,μ;\mathcal{H})$ and almost surely slice-full. We inform the reader that the notions of slice-wise real affine algebraic subvarieties (although related to ind-varieties), conditionally slice-full subsets and slice-full subsets (although related to shy sets) of a Hausdorff topological vector space are, to our knowledge, both new.

math.FA↗

arXiv2026

Two Hornich-Hlawka-type and Gram matrix-based inequalities

We establish two inequalities in real inner product spaces. The first is a multiplicative strengthening of the classical Hornich-Hlawka inequality: for all vectors $x, y, z$ in a real inner product space $H$ \[ \|x\|\,\|y\| + \|z\|\,\|x+y+z\| \;\geq\; \|x+z\|\,\|y+z\|. \] We provide a complete characterization of the equality cases in terms of the linear dependence of $x,y,z$, and explicit conditions on their Gram matrix, showing in particular that equality occurs only in flat (at most two-dimensional) configurations. We also show that this inequality implies the classical Hornich-Hlawka inequality, thereby establishing a strict hierarchy between the two. The second result is a parametric inequality derived from the positive semidefiniteness of Gram matrices: for all $x,y,z \in H$ and $α, β, γ\in \mathbb{R}$, \[ α^2\|x\|^2\langle y,z\rangle^2 + β^2\|y\|^2\langle x,z\rangle^2 + γ^2\|z\|^2\langle x,y\rangle^2 + 2(αβ+ αγ+ βγ)\langle x,y\rangle\langle x,z\rangle\langle y,z\rangle \;\geq\; 0. \] Optimizing over the parameters yields sharp inequalities relating the pairwise inner products and norms of three vectors, which can be viewed as reverse inequalities to the Gram determinant inequality $\det G \geq 0$. As a special case, this recovers and strengthens the classical Cauchy-Schwarz inequality.

math.CA↗

arXiv2026

Frames generated by the functional calculus and function frames of a normal operator

In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for $T$. Here, $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors, $T$ is a bounded normal operator, $f$ and $(f_n)_{n \in \mathbb{N}}$ are functions defined on a neighborhood of the spectrum $σ(T)$, and $g$ is a cyclic vector for $T$. After that, we characterize the frame property of such sequences in terms of the approximate point spectrum of $T^*$. Examples include certain operators (normal operators, compact operators, unilateral shifts, multiplication operators on Hardy spaces, etc.) that either generate only Riesz bases or allow redundancy. Our bridge theorem makes explicit the structural equivalence between frames generated by the functional calculus and function frames.

math.FA↗

arXiv2025

Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all finite-dimensional vector subspaces are considered negligible with respect to the Hilbert space probability measure.

math.FA↗

arXiv2025

Invertibility in partially ordered nonassociative rings

Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, (nonassociative) rings can be endowed with an extra ''structure'' such as order and topology allowing more richness in the theory. The two main theorems of this article are contributions to invertibility in the context of partially ordered nonassociative rings \textit{and} Hausdorff sequentially Cauchy-complete weak-quasi-topological nonassociative rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered nonassociative ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a suitably generalized concept of seminorm from a nonassociative ring to a partially ordered nonassociative ring endowed with Frink's interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. Part of the assumption of the second theorem is that of Hausdorff sequential Cauchy-completeness of the first ring under the topology induced by the seminorm $f$ (which takes values in a partially ordered nonassociative ring endowed with Frink's interval topology). Frink's interval topology is an example of a coarse locally-convex $T_1$ topology. Moreover, to our knowledge, the topology induced by a seminorm into a partially ordered nonassociative ring has never been introduced. Some additional original facts, such as the fact that the topology on a nonassociative ring $R_1$ induced by a norm into a totally ordered associative division ring $R_2$ endowed with Frink's interval topology (or equivalently, with the order topology, since the order of $R_2$ is total) is a Hausdorff locally convex quasi-topological group with an additional separate continuity property of the product, are dealt with in the second section ''Preliminaries''.

math.AC↗

arXiv2023

Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold

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 $\bigcap_{j \in J} \left( U(j) + St(n,H) \right)$ 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 $St(n,H) \cap S$ 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 $St(n,L^2(X,μ;\mathbb{F}))$ is isometric to $\mathcal{F}_{(X,Σ,μ),n}^\mathbb{F}$, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.

math.FA↗

arXiv2023

Independence, infinite dimension, and operators

In [Appl. Comput. Harmon. Anal., 46(3):664-673, 2019], O. Christensen and M. Hasannasab observed that assuming the existence of an operator $T$ sending $e_n$ to $e_{n+1}$ for all $n \in \mathbb{N}$ (where $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors) guarantees that $(e_n)_{n \in \mathbb{N}}$ is linearly independent if and only if $\dim(\text{span}\{e_n\}_{n \in \mathbb{N}}) = \infty$. 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 $i \in I$ where $I$ is countable as a replacement of the previous one, the conclusion will only stay true if $ϕ: I \to I$ is conjugate to the successor function $succ : n \mapsto n+1$ defined on $\mathbb{N}$. We finally prove a tentative generalization of the result, where we replace the condition $T(e_i)=e_{ϕ(i)}$ for all $i \in I$ 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.

math.FA↗

arXiv2023

Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions.

math.DS↗

arXiv2022

Path-connectedness of the intersection of translates of St(n,H)

If $H$ is a Hilbert space, the Stiefel manifold $St(n,H)$ is formed by all the independent $n$-tuples in $H$. In this article, we contribute to the topological study of Stiefel manifolds by proving a path-connectedness result. We prove that the intersection of translates of $St(n,H)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $n$-tuples. We rely on a lemma that we prove for the occasion.

math.GN↗