Let $\mathcal{H}$ be a Hilbert space and let $E\subset \mathcal{H}$ be a closed subspace$.$
If we have a bounded invertible operator $A:E\rightarrow \mathcal{H}$, how to chose a complementary subspace $F$ of $E$ in order to minimize :
$\underset{x\in E,y\in F}{\max }\frac{\left\Vert Ax\right\Vert }{\left\Vert x+y\right\Vert }$
wich is the norm of an operator having $F$ as kernel and extending $A$.
Thank you !