An expression of spectral radius via Aluthge transformation

Abstract:

For an operator $T\in B(H)$, the Aluthge transformation of $T$ is defined by $\widetilde {T}=|T|^{\frac {1}{2}}U|T|^{\frac {1}{2}}$. And also for a natural number $n$, the $n$-th Aluthge transformation of $T$ is defined by $\widetilde {T_{n}}=\widetilde {(\widetilde {T_{n-1}})}$ and $\widetilde {T_{1}}=\widetilde {T}$. In this paper, we shall show \begin{equation*} \lim _{n\to \infty }\|\widetilde {T_{n}}\|=r(T),\end{equation*} where $r(T)$ is the spectral radius.