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 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 $\overline{\{x\in\mathcal{H}:r(T,x)\le r(T,x_0)\}}$ is a nontrivial hyperinvariant subspace for $T$, and $\overline{\operatorname{span}\{T^n x_0:n\ge0\}}$ is a nontrivial invariant subspace for $T$. We also prove that $w_a(T)\le r(T)$ in general, where $r(T)$ denotes the spectral radius of $T$, and that $w_a(T)=r(T)$ for every hyponormal operator. Moreover, for every $x\in \mathcal{H}$, the set $W_a(T,x)$ is a compact interval and $W_a(T)$ is a bounded interval, where intervals are allowed to be degenerate. Let $\mathfrak{C}$ denote the set of cluster points of the operator sequence $\{|T^n|^{1/n}\}_{n=1}^{\infty}$ in the weak operator topology (WOT). Then $W_a(T,x)$ is the image of $\mathfrak{C}$ under the WOT-continuous map $A\mapsto\langle Ax,x\rangle$. Although every such image is an interval, $\mathfrak{C}$ itself need not be connected: it is connected whenever $T$ is bounded below, but may be disconnected in general.