Samy Houache

Sch\"affer's matrix inequality: the exact asymptotic constant

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Sch\"affer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subseque...