arXiv subjects

Kaissar Idrissi

Publications and source records attributed to Kaissar Idrissi.

5 recordsLinked to original sources

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↗

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.

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arXiv2016

Complex moment problem and recursive relations

We introduce a new strategy in solving the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed \emph{moment} sequences is provided. As a simple application, we obtain a computable solution to the complex moment problem for cubic harmonic characteristic polynomials of the form $z^3+az+b\overline{z}$, where $a$ and $b$ are arbitrary real numbers. We also recapture a recent result due to Curto-Yoo given for cubic column relations in $M(3)$ of the form $Z^3=itZ+u\overline{Z}$ with $t,u$ real numbers satisfying some suitable inequalities. Furthermore, we solve the truncated complex moment problem with column dependence relations of the form $Z^{k+1}= \sum\limits_{0\leq n+ m \leq k} a_{nm} \overline{Z}^n Z^m$ ($a_{nm} \in \mathbb{C}$).

math.FA↗

arXiv2016

The Multivariable moment problems and recursive relations

Let $β\equiv \{ β_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d}$ be a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the infinite moment matrix $M(β)$ is finite-rank positive semidefinite, then $β$ has a unique representing measure, which is $rank M(β)$-atomic. Further, let $β^{(2n)} \equiv \{ β_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d, \mid \mathbf{i} \mid \leq 2n}$ be a given truncated multisequence, with associated moment matrix $M(n)$ and $rank M(n)=r$, then $β^{(2n)}$ has an $r$-atomic representing measure $μ$ supported in the semi-algebraic set $K=\{ (t_1, \ldots, t_d) \in \mathbb{R}^d : q_j(t_1, \ldots, t_d) \geq 0, 1\leq j\leq m \}$, where $q_j \in \mathbb{R}[t_1, \ldots, t_d]$, if $M(n)$ admits a positive rank-preserving extension $M(n+1)$ and the localizing matrices $M_{q_j}(n +[\frac{°q_j +1}{2}])$ are positive semidefinite; moreover, $μ$ has precisely $rank M(n) - rank M_{q_j}(n +[\frac{°q_j +1}{2}])$ atoms in $\mathcal{Z}(q_j) \equiv \{ t\in \mathbb{R}^d: q_j(t)=0 \}$. In this paper, we show that every truncated moment sequence $β^{(2n)}$ is a subsequence of an infinite recursively generated multisequence, we investigate such sequences to give an alternative proof of Curto-Fialkow's results and also to obtain a new interesting results.

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