A note on maximal numerical range

Abstract

Let $\mathcal{ B(H)}$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $\mathcal H$. Let $A \in \mathcal{ B(H)}$ be a normaloid compact operator. In this note, we show that there exist $\lambda$ in $\sigma_{p}(A)$ the point spectrum of $A$ and a sequence $(x_{n})$ of unit vectors $x_{n}\in \h$ such that $$ \modu{\lambda}= \nor{A} \quad \text{and} \quad \displaystyle \lim_{n}\langle A^{k}x_{n},x_{n} \rangle=\lambda^{k} \quad (k=1,2, \dots).$$ As a consequence of the obtained result, we show that $$ \lambda^{k} \in \sigma_{p}(A^{k})\cap W_{max}(A^{k}) \quad (k=1,2, \dots), $$ where $W_{max}(T)$ denotes the maximal numerical range of an operator $T \in \mathcal{ B(H)}$.