arXiv subjects

Mohamed Barraa

Publications and source records attributed to Mohamed Barraa.

4 recordsLinked to original sources

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↗

arXiv2015

A formula for the numerical range of an elementary operator on C*-algebra

Let $\A$ be a $C^*-$algebra with unit element $1$ and unitary group $U.$ Let $a=(a_1, ..., a_k)$ and $b=(b_1, ..., b_k)$ two $k-$tuples of elements in $\A.$ The elementary operator associated to $a$ and $b$ is defined by $ R_{a, b} (x)= \sum_{i=1}^ {k} a_ixb_i.$ In this paper we prove the following formula for the numerical range of $R_{a,b}$: $$V(R_{a, b}, B(\A))= [\cup \{ V(\sum_{i=1}^ {k}u^*a_iub_i, \A): u\in U\} ]^-.$$ This formulat solves the problem 4.5 of \cite{Fi}.

math.OA↗