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 $\|\cdot \|$-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 $\|U\|=\|V\|=1$, $VU=I$ 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 $\|I\|=1$ 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 $\|TX\|=\|X\|$ 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

Keywords: Hilbert space, $C^*$-algebra, paranormal operator, quasinilpotent operator, isometry, hyponormal operator, normaloid operator, normed algebra, unital algebra.

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