The main result of this paper is the following theorem: Given delta, 0 < delta; < 1/3 and n is an element of N, there exists an (n + 1) x n inner matrix function F e H-(n+1)(infinity) (x n) such thatI greater than or equal to F*(z)F(z) greater than or equal to delta(2)I, For All z is an element of D,but the norm of any left inverse for F is at least delta/1(1-delta)(-n) greater than or equal to (3/2delta)(-n) This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper bound C . 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 Cdelta(-2) log(delta(2)n + 1) (obtained by the author Tr1) grows logarithmically in n. Also, the lower bound is obtained for (n + 1) x 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-infinity and co-analytic orthogonal complements.
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