arXiv subjects

Zouheir Amara

Publications and source records attributed to Zouheir Amara.

3 recordsLinked to original sources

arXiv2026

Spectral and numerical radius preservers on pairs of unitarily similar operators

Let $\mathcal{H}$ be a complex Hilbert space. We characterize the bijective linear maps $Φ:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ that preserve the spectral radius, respectively the numerical radius, on pairs of unitarily similar operators. We also characterize bijective linear maps that transform unitary similarity into similarity. Our results cover both finite and infinite-dimensional Hilbert spaces, with no separability assumption in the latter case.

math.FA↗

arXiv2024

Interpolation theorems for conjugations and applications

Let $\mathcal{H}$ be a separable complex Hilbert space. A conjugate-linear map $C:\mathcal{H}\to \mathcal{H}$ is called a conjugation if it is an involutive isometry. In this paper, we focus on the following interpolation problems: Let $\{x_i\}_{i\in I}$ and $\{y_i\}_{i\in I}$ be orthonormal sets of vectors in $\mathcal{H}$, and let $\{N_k\}_{k\in K}$ be a set of mutually commuting normal operators. We seek to determine under which conditions there exists a conjugation $C$ on $\mathcal{H}$ such that \begin{enumerate}[\rm (a)] \item $Cx_i=y_i$ and $CN_kC=N_k^*$ for all $i\in I$ and $k\in K$; or \item $Cx_i=y_i$ and $CN_kC=-N_k^*$ for all $i\in I$ and $k\in K$. \end{enumerate} We provide complete answers to problems (a) and (b) using the spectral projections of normal operators. Our results are then applied to the study of complex symmetric and skew symmetric operators, as well as to the characterization of hyperinvariant subspaces of normal operators through conjugations.

math.FA↗

arXiv2022

$C$-normality of rank-one perturbations of normal operators

For a separable complex Hilbert space $H$, we say that a bounded linear operator $T$ acting on $H$ is $C$-normal, where $C$ is a conjugation on $H$, if it satisfies $CT^*TC=TT^*$. For a normal operator, we give geometric conditions which guarantee that its rank-one perturbation is a $C$-normal for some conjugation $C$.

math.FA↗