Remarks on n-normal operators

Muneo Ch\=o, Ji Eun Lee, Kotaro Tanahashi, Atsushi Uchiyama


Let $T$ be a bounded linear operator on a complex Hilbert space and $n,m \in {\mathbb N}$. Then $T$ is said to be {\it $n$-normal} if $T^*T^n = T^n T^*$ and {\it $(n,m)$-normal} if
$T^{*m}T^n = T^n T^{*m}$. In this paper, we study several  properties of $n$-normal, $(n,m)$-normal  operators.
In particular, we prove that if $T$ is $2$-normal with
$ \sigma(T)  \, \bigcap \, (- \sigma(T)) \subset \{ 0 \},$ then $T$ is polarloid.  Moreover, we study subscalarity of $n$-normal operators. Also, we prove that if $T$ is $(n,m)$-normal, then $T$ is decomposable and Weyl's theorem holds for $f(T)$, where $f$ is an analytic function on $\sigma(T)$ which is not constant on each of the components of its domain.


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