Reflexivity of $L(E, F)$

Abstract:

Let $E$ and $F$ be Banach spaces and denote by $L(E,F)$ (resp., $K(E,F))$ the space of all bounded linear operators (resp., all compact operators) from $E$ to $F$. In this note the following theorem is proved: If $E$ and $F$ are reflexive and one of $E$ and $F$ has the approximation property then the following are equivalent: (i) $L(E,F)$ is reflexive, (ii) $L(E,F) = K(E,F)$, (iii) if $T \ne 0 \in L(E,F)$, then $||T|| = ||Tx||$ for some $x \in E,||x|| = 1$. This result extends a recent result of Ruckle (Proc. Amer. Math. Soc. 34 (1972), 171-174) who showed (i) and (ii) are equivalent when both $E$ and $F$ have the approximation property. Moreover the proof suggests strongly that the assumption of the approximation property may be dropped.