Lower Bounds in the Matrix Corona Theorem and the Codimension One Conjecture

Main result of this paper is the following theorem: given δ, 0 < δ < 1/3 and n ∈ N there exists an (n + 1) × n inner matrix function F ∈ H∞ (n+1)×n such that I ≥ F ∗(z)F (z) ≥ δI ∀z ∈ D, but the norm of any left inverse for F is at least [δ/(1−δ)]−n ≥ (2δ). This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper b bound C · δ−n−1 log δ−2n obtained recently by T. Trent. In particular, both estimates grow exponentially in n; the (only) previously known lower bound Cδ−2 log(δ2n + 1) (obtained by the author) grew logarithmically in n. Also, the lower bound is obtained for (n + 1) × n matrices, thus giving the negative answer to the so-called “codimension one conjecture.” Another important result is Theorem 2.4 connecting left invertiblity in H∞ and co-analytic orthogonal complements.