arXiv subjects

Hamza El Azhar

Publications and source records attributed to Hamza El Azhar.

5 recordsLinked to original sources

arXiv2026

The Local Operator Moment Problem on $\mathbb{R}$

We study the connections between operator moment sequences ${\mathcal T}=\displaystyle(T_n)_{n\in\mathbb{Z}_+}$ of self-adjoint operators on a complex Hilbert space $\mathcal{H}$ and the local moment sequences $\langle{\mathcal T}x,x\rangle = (\langle T_nx,x\rangle)_{n\in\mathbb{Z}_+}$ for arbitrary $x\in \mathcal{H}$. We provide necessary and sufficient conditions for solving the operator moment problem on $\mathbb{R}$, and we show that these criteria are automatically valid on compact subsets of $\mathbb{R}$. Applications of the compact case are used to study subnormal operator weighted shifts. A Stampfli-type propagation theorem for subnormal operator weighted shifts is also established. In addition, we discuss the validity of Tchakaloff's Theorem for operator moment sequences with compact support. In the case of a recursively generated sequence of self-adjoint operators, necessary and sufficient conditions for an affirmative answer to the operator recursive moment problem are provided, and the support of the associated representing operator-valued measure is described.

math.FA↗

arXiv2026

Propagation Phenomena for Operator-Valued Weighted Shifts

This paper is devoted to the study of propagation phenomena for $2$--hyponormal, quadratically hyponormal, and cubically hyponormal operator-valued weighted shifts. \ First, we show that every {\it quadratically} hyponormal matrix-valued weighted shift with two equal weights ({\it excluding the initial weight}) is flat. \ Second, we show that a {\it cubically} hyponormal operator-valued weighted shift with two equal weights ({\it possibly including the initial weight}) is flat. \ Next, we introduce a {\it local flatness} notion for matrix-valued weighted shifts. \ We prove that $2$--hyponormal (in particular, subnormal) matrix-valued weighted shifts satisfy this stronger propagation phenomenon. \ As a result, we prove a {\it structural decomposition theorem} for $2$--hyponormal matrix-valued weighted shifts.

math.FA↗

arXiv2023

The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.

math.FA↗

arXiv2021

Square root problem and Subnormal Aluthge transforms

For a non negative measure $μ$ with $p$ atoms, we study the relation between the Square Root Problem of $μ$ and the problem of subnormality of ${\tilde W_μ}$ the Aluthge transform of the associated unilateral weighted shift. We use an approach based on uniquely represented elements in the support of $μ*μ$. We first show that if ${\tilde W_μ}$ is subnormal, then $2p-1\le card(supp(μ*μ))\le [\frac{(p-1)^2+6}{2}]$. We rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms.

math.FA↗

arXiv2019

The quintic complex moment problem

Let $γ^{(m)} \equiv \{ γ_{ij} \}_{0 \leq i +j \leq m}$ be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure $μ$ on $\mathbb{C}$ (called a representing measure for $γ^{(m)}$) such that $γ_{ij} = \int \overline{z}^i z^j dμ$ for $0 \leq i +j \leq m$. The TCMP has been completely solved only when $m= 1, 2, 3, 4$. We provide in this paper a concrete solution to the quintic TCMP (that is, when $m = 5$). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.

math.FA↗