Arveson–Douglas essential normality for homogeneous ideals
Let $d \ge 1$ and let $H_d^2$ be the Drury–Arveson space on the unit ball of $\mathbb{C}^d$, the reproducing kernel Hilbert space with kernel $K(z,w) = (1 - \langle z, w \rangle)^{-1}$. For a proper homogeneous ideal $I \subseteq \mathbb{C}[z_1, \ldots, z_d]$, let $\overline{I}$ be its closure in $H_d^2$, let $Q = H_d^2 \ominus \overline{I}$, and let $S_j = P_Q M_{z_j}|_Q$ be the compressions of the coordinate multipliers, for $1 \le j \le d$. Let $V(I) \subseteq \mathbb{C}^d$ be the zero set of $I$ and $\mathcal{S}_p$ the Schatten class of compact operators whose singular values are $p$-summable. The inclusion below is known when $I$ is generated by monomials, when $I$ is principal, when $\dim_{\mathbb{C}} V(I) \le 1$, when $d \le 3$, and when $I$ is the ideal of a union of subspaces (see Hartz and Shalit's survey; Arveson; Guo and Wang; Kennedy and Shalit). Engliš and Eschmeier, and independently Douglas, Tang and Yu, proved it when $I$ is the ideal of all polynomials vanishing on a homogeneous variety that is smooth away from the origin.
Problem. Prove or disprove that $$[S_i, S_j^*] = S_i S_j^* - S_j^* S_i \in \mathcal{S}_p \qquad (1 \le i, j \le d,\ p > \dim_{\mathbb{C}} V(I))$$ for every $d \ge 1$ and every proper homogeneous ideal $I$.
Source: Arveson (2005), for every p > d; refined to p > dim V(I) by Douglas (2006). See Engliš and Eschmeier (2015) and Hartz and Shalit's survey, Conjectures 10.0.1 and 10.0.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- OA
- Keywords
- AIM Open Applied Problems, Drury–Arveson space, essential normality, Schatten classes
References
@article{arveson2005,
author = {Arveson, William},
title = {$p$-summable commutators in dimension $d$},
journal = {J. Operator Theory},
volume = {54},
number = {1},
year = {2005},
pages = {101--117},
eprint = {math/0308104},
archivePrefix = {arXiv},
}
@incollection{douglas2006,
author = {Douglas, Ronald G.},
title = {A new kind of index theorem},
booktitle = {Analysis, Geometry and Topology of Elliptic Operators},
publisher = {World Scientific},
address = {Hackensack, NJ},
year = {2006},
pages = {369--382},
doi = {10.1142/9789812773609_0014},
}
@incollection{hartz2025,
author = {Hartz, Michael and Shalit, Orr Moshe},
title = {Operator theory and function theory in {D}rury--{A}rveson space and its quotients},
booktitle = {Operator Theory},
editor = {Alpay, Daniel},
publisher = {Springer},
address = {Basel},
year = {2025},
pages = {1--80},
doi = {10.1007/978-3-0348-0692-3_60-2},
eprint = {1308.1081},
archivePrefix = {arXiv},
note = {Revised edition of O. M. Shalit's chapter (2015), doi 10.1007/978-3-0348-0667-1_60},
}
@article{englis2015,
author = {Engli{\v{s}}, Miroslav and Eschmeier, J{\"o}rg},
title = {Geometric {A}rveson--{D}ouglas conjecture},
journal = {Adv. Math.},
volume = {274},
year = {2015},
pages = {606--630},
doi = {10.1016/j.aim.2014.11.026},
eprint = {1312.6777},
archivePrefix = {arXiv},
}
@article{douglas2016,
author = {Douglas, Ronald G. and Tang, Xiang and Yu, Guoliang},
title = {An analytic {G}rothendieck {R}iemann {R}och theorem},
journal = {Adv. Math.},
volume = {294},
year = {2016},
pages = {307--331},
doi = {10.1016/j.aim.2016.02.031},
eprint = {1404.4396},
archivePrefix = {arXiv},
}
@article{guo2008,
author = {Guo, Kunyu and Wang, Kai},
title = {Essentially normal {H}ilbert modules and {K}-homology},
journal = {Math. Ann.},
volume = {340},
number = {4},
year = {2008},
pages = {907--934},
doi = {10.1007/s00208-007-0175-2},
}
@article{kennedy2012,
author = {Kennedy, Matthew and Shalit, Orr Moshe},
title = {Essential normality and the decomposability of algebraic varieties},
journal = {New York J. Math.},
volume = {18},
year = {2012},
pages = {877--890},
eprint = {1207.2808},
archivePrefix = {arXiv},
}
@misc{aim-arveson-douglas,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Arveson–Douglas essential normality for homogeneous ideals}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/036-arveson-douglas.md},
note = {Entry 036 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00327, "Arveson–Douglas essential normality for homogeneous ideals", clopen.solutions.