Asymptotic Numerical Ranges and Invariant Subspaces of Operators
Abstract
For a bounded linear operator T on a complex separable Hilbert space ℋ and a vector x∈ ℋ, let Wa(T,x) be the set of cluster points of the sequence {⟨|Tn|1/nx,x⟩}n=1∞. We define the asymptotic numerical range and the asymptotic numerical radius of T, respectively, by Wa(T):=⋃|x|=1Wa(T,x) and wa(T):=sup Wa(T). We prove that Wa(T,x) is a compact interval for every x∈ℋ and that Wa(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 x0∈ ℋ with r(T,x0)<wa(T), where r(T,x0) denotes the local spectral radius of T at x0, then the subspaces {x∈ℋ:r(T,x)≤ r(T,x0)}̅ and span{Tn x0:n≥0}̅, which are well known to be hyperinvariant and invariant for T, respectively, are both nontrivial. We also prove that wa(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 {|Tn|1/n}n=1∞.
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BibTeX
@misc{ji2026asymptotic,
title = {{Asymptotic Numerical Ranges and Invariant Subspaces of Operators}},
author = {Youqing Ji and Bangyuan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21553},
eprint = {2608.21553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21553},
}
RIS
TY - UNPB
TI - Asymptotic Numerical Ranges and Invariant Subspaces of Operators
AU - Youqing Ji
AU - Bangyuan Yang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.21553
UR - https://arxiv.org/abs/2608.21553
AB - For a bounded linear operator T on a complex separable Hilbert space ℋ and a vector x∈ ℋ, let W_a(T,x) be the set of cluster points of the sequence {⟨|T^n|^1/nx,x⟩}_n=1^∞. We define the asymptotic numerical range and the asymptotic numerical radius of T, respectively, by W_a(T):=⋃_|x|=1W_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∈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∈ 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 {x∈ℋ:r(T,x)≤ r(T,x_0)}̅ and span{T^n x_0:n≥0}̅, 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^∞.
ER -