Numerical radius attaining compact linear operators (Q313502)

Let \(X\) be a complex Banach space. An operator \(T\) on \(X\) is said to be numerical radius attaining if \(\sup\{x^\ast(T(x)): x^\ast(x)=1=\|x\|=\|x^\ast\|\}\) is attained. Examples of Banach spaces for which this collection of operators is not dense in the space of operators are known. In this interesting paper, the authors exhibit a compact operator which cannot be approximated by numerical radius attaining operators. The example is constructed by taking the \(\ell^\infty\)-sum of any quotient space \(Y\) of an \(\ell^p\) (\(1 < p<2\)) space without the approximation property and an appropriate closed subspace \(Z\) of \(c_0\) as the underlying Banach space. It is known that the hypothesis implies that there is a compact operator \(S: Z \rightarrow Y\) which cannot be approximated by compact operators. The operator \(y+z \rightarrow S(z)\) provides the required counterexample.