Estimates in the Carleson corona theorem, ideals of the algebra $H^\infty$, a problem of S.-Nagy

Let $E_1,E_2$ be Hilbert spaces, $H^\infty(E_1,E_2)$ be the space of functions, bounded and analytic in the disk $\mathbb D$, with values in the space of bounded linear operators from $E_1$ to $E_2$. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function $F$, $F\in H^\infty(E_1,E_2)$. For $\dim E_1=1$ this problem is a generalization of the corona problem. Let $C_n(\delta)=\sup\{\|G\|_\infty\colon F\in H^\infty(E_1,E_2),\,\dim E_1=n,\,\|F\|_\infty\le1,\,\|F(z)a\|_2\ge\delta\|a\|_2\ (z\in\mathbb D,\,a\in E_1 );\ G\in H^\infty(E_2,E_1)\ \text{is a~function of minimal norm for which}\ GF=I_{E_1}\}$. Then
$$ \frac1{\sqrt2\delta^2}\le C_1(\delta)\le\frac{20(\log 1/\delta+1)^{3/2}}{\delta^2},\qquad c_n\delta^{-(n-1)}\le C_n(\delta)\le a_n\delta^{-(2n+1)}, $$
where $a_n,c_n$ are constants depending only on $n$. The behavior of the function $C_1$ as $\delta\to1$ is described. Other results are obtained also.