Notes on the Codimension One Conjecture in the Operator Corona Theorem
- Authors: Gamal’ M.F.1
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Affiliations:
- St.Petersburg Department of the Steklov Mathematical Institute
- Issue: Vol 229, No 5 (2018)
- Pages: 506-517
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/240473
- DOI: https://doi.org/10.1007/s10958-018-3693-4
- ID: 240473
Cite item
Abstract
Answering a question of S. R. Treil (2004), for every δ, 0 < δ < 1, we construct examples of contractions whose characteristic function F ∈ H∞(ε →ε∗) satisfies the conditions ‖F(z)x‖ ≥ δ‖x‖ and dim ε∗ ⊝ F(z)ε = 1 for every z ∈ ????, x ∈ ε, but is not left invertible. In addition, we show that the condition where is the trace class of operators, which is sufficient for the left invertibility of an operatorvalued function F satisfying the estimate ‖F(z)x‖ ≥ δ‖x‖ for every z ∈ ????, x ∈ ε, with some δ > 0 (Treil, 2004), is necessary for the left invertibility of an inner function F such that dim ε∗ ⊝ F(z)ε < ∞ for some z ∈ ????.
About the authors
M. F. Gamal’
St.Petersburg Department of the Steklov Mathematical Institute
Author for correspondence.
Email: gamal@pdmi.ras.ru
Russian Federation, St.Petersburg