Spaceability in sets of operators on C(K)

We show that when C(K) does not have few operators – in the sense of Koszmider (2004) [8] – the set of operators which are not weak multipliers is spaceable. That is, there exists an infinite-dimensional space of operators on C(K), each nonzero element of which is not a weak multiplier. This contrasts to what happens in general Banach spaces that do not have few operators.In addition, we show that there exists a C(K) space on which each operator has the form gI+hJ+S, where g,h∈C(K) and S is strictly singular, in connection to a result by Ferenczi (2007) [6].