O.A.M. Sid Ahmed

In this paper, we introduce the notion of $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal operators on a Hilbert space $\mathscr{H}$ as : An operator $\mathcal{L}$ is called $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal $(0\leq \alpha \leq 1 \leq \beta)$ if \begin{align*} \alpha^{2}\mathcal{L}^{m*}(\mathcal{L}-\lambda)^{*}(\mathcal{L}-\lambda )^{n}\mathcal{L}^{m}& \leq \mathcal{L}^{m*}(\mathcal{L}-\lambda)^{n}(\mathcal{L}-\lambda)^{*}\mathcal{L}^{m}\\ &\leq \beta^{2} \mathcal{L}^{m*}(\mathcal{L}-\lambda)^{*}(\mathcal{L}-\lambda )^{n}\mathcal{L}^{m} \end{align*} for natural numbers $m$ and $n$ and for all $\lambda \in \mathbb{C}$. This paper aims to study several properties of $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal operators.