PreprintCC BY 4.0arXiv
Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on Nη-type Quotient Modules in Polydisc
Abstract
Let η1,⋯,ηd be nonconstant one-variable inner functions satisfying ηi(0)=0 for all 1≤ i≤ d. Denote by Mη the submodule of the Hardy space H2(𝔻d) generated by the function family {ηi(zi)−ηi+1(zi+1)| 1≤ i≤ d−1}. For the corresponding quotient module Nη=H2(𝔻d)ominus Mη, we define the compressed multiplication operators Szi=PNηMzi|Nη, where PNη is the orthogonal projection from H2(𝔻d) onto Nη. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators [Szi∗,Szi] and cross-commutators [Szi∗,Szj] with i≠ j. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.
Related
- Nagy and Pollard-Hilding-Type Stability of Hilbert-Schmidt Frames
- The Pego Theorem for the Hilbert--Schmidt Class
- Compact and Hilbert-Schmidt composition operators on weighted Hardy spaces
- Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost
- On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm
BibTeX
@misc{xu2026hilbertschmidt,
title = {{Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on $N_η$-type Quotient Modules in Polydisc}},
author = {Anjian Xu and Yuchao Xiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02692},
eprint = {2610.02692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02692},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on N_η-type Quotient Modules in Polydisc
AU - Anjian Xu
AU - Yuchao Xiang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02692
UR - https://arxiv.org/abs/2610.02692
AB - Let η_1,⋯,η_d be nonconstant one-variable inner functions satisfying η_i(0)=0 for all 1≤ i≤ d. Denote by M_η the submodule of the Hardy space H^2(𝔻^d) generated by the function family {η_i(z_i)−η_i+1(z_i+1)| 1≤ i≤ d−1}. For the corresponding quotient module N_η=H^2(𝔻^d)ominus M_η, we define the compressed multiplication operators S_z_i=P_N_ηM_z_i|_N_η, where P_N_η is the orthogonal projection from H^2(𝔻^d) onto N_η. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators [S_z_i^∗,S_z_i] and cross-commutators [S_z_i^∗,S_z_j] with i≠ j. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.
ER -