Doubly commuting invariant and co-invariant subspaces
Abstract
Given complex separable Hilbert spaces ℋi and bounded operators Ti on ℋi, for i=1,…,n, we consider the tensor product space H:=ℋ1⊗⋯⊗ℋn and a tuple (T̃1,…,T̃n) on H where T̃i=Iℋ1⊗⋯⊗ Iℋi−1⊗ Ti⊗ Iℋi+1⊗⋯⊗ Iℋn. A closed jointly (T̃1,…,T̃n)-invariant subspace M of H is said to be doubly commuting if the tuple (T̃1|M,…,T̃n|M) is doubly commuting. In this paper, we show that if the hereditary polynomials in each Ti are dense in ℬ(ℋi) with respect to the weak operator topology for i=1,…,n, then doubly commuting jointly (T̃1,…,T̃n)-invariant subspaces M of H are characterized in the form ℳ1⊗⋯⊗ℳn, where ℳi⊆ℋi is Ti-invariant, for each i=1,…,n. As applications, we provide a unified framework for understanding and characterizing doubly commuting quotient modules of weighted Bergman spaces over the open unit polydisc, doubly commuting quotient modules of tensor products of complete Nevanlinna-Pick spaces, and doubly commuting invariant and co-invariant subspaces of certain operators acting on tensor products of de Branges-Rovnyak spaces H(b) associated with both extreme and nonextreme points b. This recovers, through a different approach, the known characterizations of doubly commuting quotient modules of the Hardy space over the open unit polydisc and doubly commuting submodules of Jordan blocks of Hardy space over the open unit polydisc.
Related
- Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Projections and minimal invariant subspaces in the Hardy space over the bidisk
- Invariant subspaces for free linearizations of Lipschitz maps
- Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
BibTeX
@misc{barik2026doubly,
title = {{Doubly commuting invariant and co-invariant subspaces}},
author = {Sibaprasad Barik and Shubham Jain and Samir Panja},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18403},
eprint = {2610.18403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18403},
}
RIS
TY - UNPB
TI - Doubly commuting invariant and co-invariant subspaces
AU - Sibaprasad Barik
AU - Shubham Jain
AU - Samir Panja
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18403
UR - https://arxiv.org/abs/2610.18403
AB - Given complex separable Hilbert spaces ℋ_i and bounded operators T_i on ℋ_i, for i=1,…,n, we consider the tensor product space H:=ℋ_1⊗⋯⊗ℋ_n and a tuple (T̃_1,…,T̃_n) on H where T̃_i=I_ℋ_1⊗⋯⊗ I_ℋ_i−1⊗ T_i⊗ I_ℋ_i+1⊗⋯⊗ I_ℋ_n. A closed jointly (T̃_1,…,T̃_n)-invariant subspace M of H is said to be doubly commuting if the tuple (T̃_1|_M,…,T̃_n|_M) is doubly commuting. In this paper, we show that if the hereditary polynomials in each T_i are dense in ℬ(ℋ_i) with respect to the weak operator topology for i=1,…,n, then doubly commuting jointly (T̃_1,…,T̃_n)-invariant subspaces M of H are characterized in the form ℳ_1⊗⋯⊗ℳ_n, where ℳ_i⊆ℋ_i is T_i-invariant, for each i=1,…,n. As applications, we provide a unified framework for understanding and characterizing doubly commuting quotient modules of weighted Bergman spaces over the open unit polydisc, doubly commuting quotient modules of tensor products of complete Nevanlinna-Pick spaces, and doubly commuting invariant and co-invariant subspaces of certain operators acting on tensor products of de Branges-Rovnyak spaces H(b) associated with both extreme and nonextreme points b. This recovers, through a different approach, the known characterizations of doubly commuting quotient modules of the Hardy space over the open unit polydisc and doubly commuting submodules of Jordan blocks of Hardy space over the open unit polydisc.
ER -