Asymptotic Numerical Ranges and Invariant Subspaces of Operators

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 $\{⟨\mathrm{\mid }{T}^n{\mathrm{\mid }}^{1\mathrm{/}n}x,x⟩{\}}_{n=1}^{\mathrm{\infty }}$. We define the asymptotic numerical range and the asymptotic numerical radius of $T$, respectively, by $W_a(T):={\bigcup }_{\mathrm{\parallel }x\mathrm{\parallel }=1}{W}_a(T,x)$ 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 $\overline{\mathrm{span}\{T^n{x}_0:n\ge 0\}}$ 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 $\{\mathrm{\mid }{T}^n{\mathrm{\mid }}^{1\mathrm{/}n}{\}}_{n=1}^{\mathrm{\infty }}$ in the weak operator topology (WOT). Then $W_a(T,x)$ is the image of $\mathfrak{C}$ under the WOT-continuous map $A\mapsto ⟨Ax,x⟩$. 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.