Optimal Weyl inequality in Banach spaces

Abstract:

A well-known multiplicative Weyl inequality states that the sequence of eigenvalues $(\lambda _k(T))$ and the sequence of approximation numbers $(a_k(T))$ of any compact operator $T$ in a Banach space satisfy \[ \prod _{k=1}^n |\lambda _k(T)| \le n^{n/2} \prod _{k=1}^n a_k(T)\] for all $n$. We prove here that the constant $n^{n/2}$ is optimal, which solves a longstanding problem.