Asymptotic Numerical Ranges and Invariant Subspaces of Operators

Abstract

For a bounded linear operator $T$ on a complex separable Hilbert space $\mathcal{H}$ and a vector $x\in \mathcal{H}$, let $W_a(T,x)$ be the set of cluster points of the sequence $\{\langle|T^n|^{1/n}x,x\rangle\}_{n=1}^{\infty}$. We define the asymptotic numerical range and the asymptotic numerical radius of $T$, respectively, by $W_a(T):=\bigcup_{\|x\|=1}W_a(T,x)$ and $w_a(T):=\sup W_a(T)$. We prove that $W_a(T,x)$ is a compact interval for every $x\in\mathcal{H}$ and that $W_a(T)$ is a bounded interval, where intervals are allowed to be degenerate. Using the asymptotic numerical radius, we show that if there is a nonzero vector $x_0\in \mathcal{H}$ with $r(T,x_0)<w_a(T)$, where $r(T,x_0)$ denotes the local spectral radius of $T$ at $x_0$, then the subspaces $\overline{\{x\in\mathcal{H}:r(T,x)\le r(T,x_0)\}}$ and $\overline{\operatorname{span}\{T^n x_0:n\ge0\}}$, which are well known to be hyperinvariant and invariant for $T$, respectively, are both nontrivial. We also prove that $w_a(T)=r(T)$ for every hyponormal operator $T$, where $r(T)$ denotes the spectral radius of $T$. Finally, we investigate the connectedness of the set of WOT cluster points of the sequence $\{|T^n|^{1/n}\}_{n=1}^\infty$.