Powers and Functional Calculus for m-Selfadjoint and Skew m-Selfadjoint Operators
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 Tr and Ts are, respectively, m- and ℓ-selfadjoint, then Tgcd(r,s) is min{m,ℓ}-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.
Related
- Functional Calculus on Noncommutative Tori, II. Complex Powers, Logarithms, and Sectorial Projections
- Antitonicity property of the Moore-Penrose inverse for selfadjoint operators
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Nonlinear SBP operators for stable approximations of skew-symmetric hyperbolic initial boundary value problems
- Crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape
BibTeX
@misc{faghihahmadi2026powers,
title = {{Powers and Functional Calculus for $m$-Selfadjoint and Skew $m$-Selfadjoint Operators}},
author = {M. Faghih-Ahmadi and T. Bermúdez and K. Hedayatian and H. Zaway},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26572},
eprint = {2610.26572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26572},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Powers and Functional Calculus for m-Selfadjoint and Skew m-Selfadjoint Operators
AU - M. Faghih-Ahmadi
AU - T. Bermúdez
AU - K. Hedayatian
AU - H. Zaway
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26572
UR - https://arxiv.org/abs/2610.26572
AB - 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 ℓ-selfadjoint, then T^gcd(r,s) is min{m,ℓ}-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.
ER -