arXiv subjects

Abderrazzak Ech-charyfy

Publications and source records attributed to Abderrazzak Ech-charyfy.

3 recordsLinked to original sources

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↗

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

A Combinatorial Formula for Recursive Operator Sequences and Applications

We study sequences of bounded operators $(T_n)_{n \ge 0}$ on a complex separable Hilbert space $\mathcal{H}$ that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients $A_0, A_1, \dots, A_{r-1}$ are pairwise commuting bounded operators on $\mathcal{H}$. \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for $T_n$. As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

math.FA↗