Year: 2026
Type: preprint
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)
Source: arXiv (Cornell University)
Authors: Ji, Youqing, Yang, Bangyuan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G4347032124
OpenAlex ID: W7204155847