Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs
Abstract
We consider composition operators in L2(μ), where μ is a discrete measure. Our main result shows that, for every integer p≥2, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length p and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the ``moment property implies subnormality'' problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.
Related
- The Subnormality of 𝕂-Homogeneous Multiplication Operators on Bounded Symmetric Domains
- Antitonicity property of the Moore-Penrose inverse for selfadjoint operators
- Characterizing the Daugavet property by squares of rank-one operators
- An Arithmetic Property of Intertwining Operators for Real Reductive Groups
- The Kato square root property for fractional elliptic operators in divergence form
BibTeX
@misc{budzynski2026solutions,
title = {{Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs}},
author = {Piotr Budzyński and Zenon Jan Jabłoński and Il Bong Jung and Jan Stochel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31173},
eprint = {2609.31173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31173},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs
AU - Piotr Budzyński
AU - Zenon Jan Jabłoński
AU - Il Bong Jung
AU - Jan Stochel
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31173
UR - https://arxiv.org/abs/2609.31173
AB - We consider composition operators in L^2(μ), where μ is a discrete measure. Our main result shows that, for every integer p≥2, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length p and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the ``moment property implies subnormality'' problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.
ER -