Finite-orbit obstructions for multipliers on the polydisk
Abstract
We study the closed spans of finitely many joint orbits of holomorphic multipliers on the Hardy space H2(𝔻d) of the polydisk. If fewer than d symbols are used, every such span has infinite codimension. The symbols may be unbounded, provided that all orbit vectors belong to the Hardy space. We give two elementary proofs. The first uses common level sets and independent point evaluations; for bounded symbols, these evaluations yield joint adjoint eigenvectors. The second uses finite Taylor sections and the nilpotent structure of truncated convolution, also known as the Jury product. For r generators and q<d symbols, the orbit dimension on a cube of side n is O(nq), whereas the ambient dimension is nd. In the bidisk we obtain the explicit codimension bound MN−r(M+N−1), which is sharp for a single orbit. The coordinate multipliers attain the parameter threshold. We also explain how these obstructions relate to kernel interpolation and the established representation of dynamical frames by compressed shifts on model spaces.
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BibTeX
@misc{krishtal2026finiteorbit,
title = {{Finite-orbit obstructions for multipliers on the polydisk}},
author = {Ilya Krishtal and Javad Mashreghi and Prateek Kumar Vishwakarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26755},
eprint = {2609.26755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26755},
}
RIS
TY - UNPB
TI - Finite-orbit obstructions for multipliers on the polydisk
AU - Ilya Krishtal
AU - Javad Mashreghi
AU - Prateek Kumar Vishwakarma
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26755
UR - https://arxiv.org/abs/2609.26755
AB - We study the closed spans of finitely many joint orbits of holomorphic multipliers on the Hardy space H^2(𝔻^d) of the polydisk. If fewer than d symbols are used, every such span has infinite codimension. The symbols may be unbounded, provided that all orbit vectors belong to the Hardy space. We give two elementary proofs. The first uses common level sets and independent point evaluations; for bounded symbols, these evaluations yield joint adjoint eigenvectors. The second uses finite Taylor sections and the nilpotent structure of truncated convolution, also known as the Jury product. For r generators and q<d symbols, the orbit dimension on a cube of side n is O(n^q), whereas the ambient dimension is n^d. In the bidisk we obtain the explicit codimension bound MN−r(M+N−1), which is sharp for a single orbit. The coordinate multipliers attain the parameter threshold. We also explain how these obstructions relate to kernel interpolation and the established representation of dynamical frames by compressed shifts on model spaces.
ER -