Abstract

A Hilbert space operator $T$ is called $n$-paranormal and $*$-$n$-paranormal if $\|Tx\|^n \leq \|T^nx\| \cdot \|x\|^{n-1}$ and $\|T^*x\|^n \leq \|T^nx\| \cdot \|x\|^{n-1}$, respectively. Let $\mathfrak{P}(n)$ and $\mathfrak{S}(n)$ be the sets of all $n$-paranormal operators and $*$-$n$-paranormal operators, respectively. In this paper we study and discuss the relationship between these two sets of operators and especially show $\displaystyle \bigcap_{n=3}^{\infty} \mathfrak{P}(n) = \mathfrak{P}(3) \bigcap \mathfrak{P}(4)$. Finally we introduce $*$-$n$-paranormality for an operator on a Banach space and give some spectral properties.

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