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

The main result of this paper is the following theorem: Given δ, 0 < δ < 1/3 and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgI % Gioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xf % H4Kaaiilaaaa!43C1! $$n \in \mathbb{N},$$ there exists an (n + 1) × n inner matrix function % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabgI % GiolaadIeadaqhaaWcbaGaaiikaiaad6gacqGHRaWkcaaIXaGaaiyk % aiaaykW7cqGHxdaTcaaMc8UaamOBaaqaaiabg6HiLcaaaaa!44B1! $$F \in H_{(n + 1)\, \times \,n}^\infty $$ such that $$ I \geq F^* (z)F(z) \geq \delta ^2 I,\quad \forall z \in \mathbb{D}, $$ but the norm of any left inverse for F is at least % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq % aH0oazcaGGVaGaaiikaiaaigdacqGHsislcqaH0oazcaGGPaaacaGL % BbGaayzxaaWaaWbaaSqabeaacqGHsislcaWGUbaaaOGaeyyzIm7aae % WaaeaadaWcaaqaaiaaiodaaeaacaaIYaaaaiabes7aKbGaayjkaiaa % wMcaamaaCaaaleqabaGaeyOeI0IaamOBaaaakiabgwSixdaa!4BD3! $$\left[ {\delta /(1 - \delta )} \right]^{ - n} \geq \left( {\frac{3} {2}\delta } \right)^{ - n} \cdot $$ This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper bound % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabgw % Sixlabes7aKnaaCaaaleqabaGaeyOeI0IaamOBaiabgkHiTiaaigda % aaGcciGGSbGaai4BaiaacEgacqaH0oazdaahaaWcbeqaaiabgkHiTi % aaikdacaWGUbaaaaaa!45A2! $$C \cdot \delta ^{ - n - 1} \log \delta ^{ - 2n} $$ obtained recently by T. Trent [Tre]. In particular, both estimates grow exponentially in n; the (only) previously known lower bound % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabes % 7aKnaaCaaaleqabaGaeyOeI0IaaGOmaaaakiGacYgacaGGVbGaai4z % aiaacIcacqaH0oazdaahaaWcbeqaaiaaikdaaaGccaWGUbGaey4kaS % IaaGymaiaacMcaaaa!438C! $$C\delta ^{ - 2} \log (\delta ^2 n + 1)$$ (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.