Norm attaining operators on some classical Banach spaces (original) (raw)

We construct an operator from L ι [0, 1] to C[0,1] which may not be approximated by norm attaining operators with respect to the operator norm. This solves a question raised by J. Johnson and J. Wolfe and furnishes the first example of a pair of classical Banach spaces such that the norm attaining operators are not dense. C[0,1] is the first example of a classical Banach space which does not have property B. On the other hand, we show that a weakly compact operator from C(K) into a Banach space X may be approximated in norm by norm attaining operators. This shows in particular that the norm attaining operators are dense in B(C(K), L ι [0,1]) and B(C(K), I 2), thus solving two questions raised by Johnson and Wolfe.