arXiv subjects

El Hassan Benabdi

Publications and source records attributed to El Hassan Benabdi.

4 recordsLinked to original sources

arXiv2023

A note on the generalized maximal numerical range of operators

The paper considers some new properties of the so-called $A$-maximal numerical range of operators, denoted by $W_{\max}^A(\cdot)$, where $A$ is a positive bounded linear operator acting on a complex Hilbert space $\mathcal{H}$. Some characterizations of $A$-normaloid operators are also given. In particular, we extend a recent recent by Spitkovsky in [Oper. Matrices, 13, 3(2019)]. Namely, it is shown that an $A$-bounded linear operator $T$ acting on $\mathcal{H}$ is $A$-normaloid if and only if $W_{\max}^A(T)\cap \partial W_A(T)\neq\varnothing$. Here $\partial W_A(T)$ stands for the boundary of $A$-numerical range of $T$. Some new $A$-numerical radius inequalities generalizing and improving earlier well-known results are also given.

math.FA↗

arXiv2021

Drazin invertibility of linear operators on quaternionic Banach spaces

Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$ where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.

math.FA↗

arXiv2020

The Spectral Theorem for Quaternionic Normal Operators

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a bounded normal right quaternionic linear operator on $\mathcal{H}$. In this paper, we prove that there exists a unique spectral measure $E$ in $\mathcal{H}$ such that $$T=\int_{σ_S(T)}λdE_λ,$$ where $σ_S(T)$ denotes the spherical spectrum of $T$.

math.FA↗

arXiv2020

A new approach to the S-functional calculus

In this paper, we first prove that the S-spectrum of a bounded right quaternionic linear operator on a two-sided quaternionic Banach space is a union of the spectrum of some bounded linear operators on a complex Banach space. Furthermore, we show that the S-functional calculus is obtained by the Riesz-Dunford functional calculus for complex linear operators. We also give simple proofs of some already existing results.

math.FA↗