Spectral properties of operators that characterize $\ell^{(n)}_\infty$

Abstract

It is well known that the identity is an operator with the following property: if the operator, initially defined on an $n$-dimensional Banach space $V$, can be extended to any Banach space with norm $1$, then $V$ is isometric to ${\ell }_{\infty }^{\left(n\right)}$. We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for complex spaces.