An operator on \(L^ p\) without best compact approximation (Q1078492)

The authors show the validity of the following two theorems: Theorem 1. For every \(1\leq p\leq \infty\), \(p\neq 2\), there is an operator on \(L^ p\) whose distance to the compact operators on \(L^ p\) is not attained; (the novelty is for the cases \(1<p<\infty\) with \(p\neq 2)\) and Theorem 2. Fix \(p\neq 1,2,\infty\). For each \(\epsilon >0\) there is \(\delta (\epsilon)>0\) so that if \(K\in K(L^ p)\) and \(\| K\| >\epsilon\), then \(\| I-K\| \geq 1+\delta (\epsilon)\); (also the zero operator on \(L^ p\) is the unique best compact approximation of the identity operator I).