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Minimality of Tz*-Invariant Subspaces of H2(𝔻2)

  • João Marcos R. do Carmo
  • Marcos S. Ferreira
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Abstract

In this paper, we establish necessary conditions for the minimality of infinite-dimensional Tz*-invariant subspaces of H2(𝔻2). More precisely, we prove that if M⊂ H2(𝔻2) is an infinite-dimensional, minimal Tz*-invariant subspace, then, for every n∈ℕ0, either Pn(M)̅=H2(𝔻) or Pn(M)̅⊥ =ϕa,θH2(𝔻) where ϕa,θ is an atomic singular inner function and Pn denotes the orthogonal projection of H2(𝔻2) onto H2(z)wn. As Tz* is a universal operator on H2(𝔻2), this result reduces the Invariant Subspace Problem to the analysis of these two alternatives for infinite-dimensional minimal Tz*-invariant subspaces.

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Cite

BibTeXRIS
BibTeX
@misc{carmo2026minimality,
  title = {{Minimality of $T_z^*$-Invariant Subspaces of $H^2(\mathbb D^2)$}},
  author = {João Marcos R. do Carmo and Marcos S. Ferreira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24327},
  eprint = {2610.24327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24327},
}
RIS
TY  - UNPB
TI  - Minimality of T_z^*-Invariant Subspaces of H^2(𝔻^2)
AU  - João Marcos R. do Carmo
AU  - Marcos S. Ferreira
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24327
UR  - https://arxiv.org/abs/2610.24327
AB  - In this paper, we establish necessary conditions for the minimality of infinite-dimensional T_z^*-invariant subspaces of H^2(𝔻^2). More precisely, we prove that if M⊂ H^2(𝔻^2) is an infinite-dimensional, minimal T_z^*-invariant subspace, then, for every n∈ℕ_0, either P_n(M)̅=H^2(𝔻) or P_n(M)̅^⊥ =ϕ_a,θH^2(𝔻) where ϕ_a,θ is an atomic singular inner function and P_n denotes the orthogonal projection of H^2(𝔻^2) onto H^2(z)w^n. As T_z^* is a universal operator on H^2(𝔻^2), this result reduces the Invariant Subspace Problem to the analysis of these two alternatives for infinite-dimensional minimal T_z^*-invariant subspaces.
ER  -