arXiv subjects

Fadil Chabbabi

Publications and source records attributed to Fadil Chabbabi.

5 recordsLinked to original sources

arXiv2022

On the image of the mean transform

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $T=V|T|$ be the polar decomposition of an operator $T\in B(H)$. The mean transform of $T$ is defined by $M(T)=\frac{T+|T|V}{2}$. In this paper, we discuss several properties related to the spectrum, the kernel, the image, the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators as positive, normal, unitary, hyponormal and co-hyponormal operators.

math.FA↗

arXiv2022

The Aluthge and the mean transforms of $m$-isometries

Let $T\in B(H)$ be a bounded linear operator on a Hilbert space $H$, let $T = V|T|$ be its polar decomposition of $T$ and let $λ\in [0,1]$. The $λ$-Aluthge transform $Δ_λ(T)$ and the mean transforms $M(T)$ are defined respectively by: \[Δ_λ(T):=|T|^λV|T|^{1-λ} \;\; \text{and} \;\; M(T):=\frac12(|T|V+V|T|).\] In this paper, we use several examples of weighted shift operators to prove that the Aluthge and mean transforms do not preserve the class of $m-$isometries in any directions.

math.FA↗

arXiv2022

Commuting maps with the Mean Transform under Jordan product

In this article, we give a complete characterization of the bijective maps which commute with the mean transform under Jordan product. The main result is the following : Let $H,K$ be two complex Hilbert spaces and $Φ:B(H) \to B(K)$ be a bijective map, then $$ \mathcal {M}(Φ(A)\circΦ(B))=Φ(\mathcal{M}(A\circ B)) \;\; \text{for all}\;\; A, B \in B(H)$$ if and only if there exists a unitary or anti-unitary operator $U:H\to K $ such that, $$ Φ(T)= UTU^* \; \text{for all} \;T\in B(H).$$

math.FA↗

arXiv2016

Product commuting maps with the $λ$-Aluthge transform

Let H and K be two Hilbert spaces and B(H) be the algebra of all bounded linear operators from H into itself. The main purpose of this paper is to obtain a characterization of bijective maps $Φ$ : B(H) $\rightarrow$ B(K) satisfying the following condition $Δ$ $λ$ ($Φ$(A)$Φ$(B)) = $Φ$($Δ$ $λ$ (AB)) f orall A, B $\in$ B(H), where $Δ$ $λ$ (T) stands the $λ$-Aluthge transform of the operator T $\in$ B(H). More precisely, we prove that a bijective map $Φ$ satisfies the above condition, if and only , if $Φ$(A) = U AU * for all A $\in$ B(H), for some unitary operator U : H $\rightarrow$ K.

math.FA↗

arXiv2016

New formulas for the spectral radius via Aluthge transform

In this paper we give several expressions of spectral radius of a bounded operator on a Hilbert space, in terms of iterates of Aluthge transformation, numerical radius and the asymptotic behavior of the powers of this operator. Also we obtain several characterizations of normaloid operators.

math.FA↗