Paranormal elements in normed algebra

A. M. Bikchentaev ; S. A. Abed

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2018), pp. 13-19.

Résumé

For a normed algebra $\mathcal{A}$ and natural numbers $k$ we introduce and investigate the $\Vert \cdot \Vert$-closed classes ${\mathcal{P}}_k(\mathcal{A})$. We show that ${\mathcal{P}}_1(\mathcal{A})$ is a subset of ${\mathcal{P}}_k(\mathcal{A})$ for all $k$. If $T$ in ${\mathcal{P}}_1(\mathcal{A})$, then $T^n$ lies in ${\mathcal{P}}_1(\mathcal{A})$ for all natural $n$. If $\mathcal{A}$ is unital, $U,V\in \mathcal{A}$ are such that $\Vert U\Vert =\Vert V\Vert =1$, $VU=I$ and $T$ lies in ${\mathcal{P}}_k(\mathcal{A})$, then $UTV$ lies in ${\mathcal{P}}_k(\mathcal{A})$ for all natural $k$. Let $\mathcal{A}$ be unital, then 1) if an element $T$ in ${\mathcal{P}}_1(\mathcal{A})$ is right invertible, then the right inverse element $T^{-1}$ lies in ${\mathcal{P}}_1(\mathcal{A})$; 2) for $\Vert I\Vert =1$ the class ${\mathcal{P}}_1(\mathcal{A})$ consists of normaloid elements; 3) if the spectrum of an element $T$ in ${\mathcal{P}}_1(\mathcal{A})$ lies on the unit circle, then $\Vert TX\Vert =\Vert X\Vert$ for all $X\in \mathcal{A}$. If $\mathcal{A}=\mathcal{B}(\mathcal{H})$, then the class ${\mathcal{P}}_1(\mathcal{A})$ coincides with the set of all paranormal operators on a Hilbert space $\mathcal{H}$.

Détail

Mots-clés : Hilbert space, $C^{\ast }$-algebra, paranormal operator, quasinilpotent operator, isometry, hyponormal operator, normaloid operator, normed algebra, unital algebra.

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