Year: 2026
Type: preprint
Abstract: We study powers and functional calculi of higher-order selfadjoint and skew higher-order selfadjoint operators on complex Hilbert spaces. First, we compare the behavior of their powers with the corresponding theory for $m$-isometries. We prove that if $T$ is invertible and two powers $T^r$ and $T^s$ are, respectively, $m$- and $\ell$-selfadjoint, then $T^{\gcd(r,s)}$ is $\min\{m,\ell\}$-selfadjoint. In particular, invertibility yields the expected converse for consecutive powers. We also show that two consecutive skew $m$-selfadjoint powers must be non-invertible; in fact, nilpotent. For polynomials, we obtain coefficient characterizations under a cardinality assumption on the spectrum. Finally, for holomorphic functions we give spectral characterizations according to whether the spectrum is finite, or the domain of analyticity is a region, and we derive consequences for Möbius transformations and $m$-isometries. Several examples illustrate why the spectral hypotheses cannot be omitted.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agencia Estatal de Investigación
Awards: G4998806521, G7673984769
OpenAlex ID: W7223015167