School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
8-2025
Abstract
In this paper, we find and study explicit connections of invariant (hyperinvariant) subspaces among the linear pencil, a pair of operators, and their operator transforms. We show that for two given operators, a family of linear combinations has an invariant subspace if and only if the pair of the two operators has a joint invariant subspace. Next, we observe that if the two operators are commuting, then the results for invariant subspaces can be extended to hyperinvariant subspaces. We also remove the commuting condition of the pair of operators in previous results in this topic, as the commuting condition is not required to define a joint invariant subspace of a pair of operators. To accomplish this, we employ a technique adopting a horizontal asymptote of the linear combination of two operators. Next, we investigate the relations of invariant and hyperinvariant subspaces of operators by replacing the condition of dense ranges for the pair of operators with the condition of bounded belowness of a column operator, and vice versa. Finally, as an application of the results given above, we describe when a 2 by 2 operator matrix has a nontrivial invariant subspace. Our work has potential applications to studying PDEs and optimal control problems.
Creative Commons License
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Publication Title
Communications in Nonlinear Science and Numerical Simulation
DOI
10.1016/j.cnsns.2025.108818
Comments
Original published version available at https://doi.org/10.1016/j.cnsns.2025.108818