Abstract.
The main result of this paper is the following theorem: Given δ, 0 < δ < 1/3 and \(n \in \mathbb{N},\) there exists an (n + 1) × n inner matrix function \(F \in H_{(n + 1)\, \times \,n}^\infty \) such that
but the norm of any left inverse for F is at least \(\left[ {\delta /(1 - \delta )} \right]^{ - n} \geq \left( {\frac{3} {2}\delta } \right)^{ - n} \cdot \) This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper bound \(C \cdot \delta ^{ - n - 1} \log \delta ^{ - 2n} \) obtained recently by T. Trent [Tre]. In particular, both estimates grow exponentially in n; the (only) previously known lower bound \(C\delta ^{ - 2} \log (\delta ^2 n + 1)\) (obtained by the author [Tr1]) grows logarithmically in n. Also, the lower bound is obtained for (n +1) × n matrices, thus giving a negative answer to the so-called “codimension one conjecture.” Another important result is Theorem 2.4 connecting left invertibility in H∞ and co-analytic orthogonal complements.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Treil, S. Lower Bounds in the Matrix Corona Theorem and the Codimension One Conjecture. Geom. funct. anal. 14, 1118–1133 (2004). https://doi.org/10.1007/s00039-004-0485-4
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00039-004-0485-4