arXiv subjects
Publications and source records attributed to Mohamed Mabrouk.
For a positive element $A$ of a $C^*$-algebra $\mathfrak{A}$, let ${\|X\|}_{A}$ denote the $A$-operator semi-norm of $X\in\mathfrak{A}$. In this paper, we aim to introduce and study the notion of $A$-spectrum for $X$ such that ${\|X\|}_{A}<\infty$. In particular when $A$ is well-supported, we establish an $A$-spectral permanence property for $C^*$-algebras.
Let $a$ be a positive element in a unital $C^*$-algebra $\mathfrak{A}$. We define a semi-norm on $\mathfrak{A}$, which generalizes the $a$-operator semi-norm and the $a$-numerical radius. We investigate basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the $a$-numerical radii of elements in $\mathfrak{A}$. Some other related results are also discussed.
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric polynomials and we derive expansion formulae corresponding to all the families within the $q$-Askey scheme.