PreprintCC BY 4.0arXiv
Joint Moments of Bilinear Resolvents
Abstract
Given A,B∈ B(H)+, we consider the existence and uniqueness of a positive operator C∈ B(H)+ such that (I+mA+nB+mnC)−1=∫[0,1]2smtndE(s,t), m,n≥0, for a positive operator-valued measure E. This is a joint Hausdorff moment problem. Without C, it holds if and only if AB=BA. In dimension two we give a complete criterion for noncommuting pairs. In higher finite dimensions, nonexistence is generic. Under certain spectral and cyclicity assumptions, we obtain a necessary and sufficient condition on arbitrary Hilbert spaces. The completion is unique in this case. We also classify these pairs up to unitary equivalence. The class includes irreducible infinite-dimensional examples, even with continuous spectrum.
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BibTeX
@misc{tian2026joint,
title = {{Joint Moments of Bilinear Resolvents}},
author = {James Tian},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03081},
eprint = {2611.03081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Joint Moments of Bilinear Resolvents
AU - James Tian
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03081
UR - https://arxiv.org/abs/2611.03081
AB - Given A,B∈ B(H)_+, we consider the existence and uniqueness of a positive operator C∈ B(H)_+ such that (I+mA+nB+mnC)^−1=∫_[0,1]^2s^mt^ndE(s,t), m,n≥0, for a positive operator-valued measure E. This is a joint Hausdorff moment problem. Without C, it holds if and only if AB=BA. In dimension two we give a complete criterion for noncommuting pairs. In higher finite dimensions, nonexistence is generic. Under certain spectral and cyclicity assumptions, we obtain a necessary and sufficient condition on arbitrary Hilbert spaces. The completion is unique in this case. We also classify these pairs up to unitary equivalence. The class includes irreducible infinite-dimensional examples, even with continuous spectrum.
ER -