Spaceability in sets of operators on $C(K)$

Details Spaceability in sets of operators on $C(K)$ Spaceability in sets of operators on $C(K)$

2012-03-30

arxiv.org/abs/1203.6855v1

Mathematics

Functional Analysis

Scientific paper

We prove that if $K$ is a compact Hausdorff space satisfying either condition $K$ contains a nontrivial convergent sequence, or $C(K)$ is isomorphic to its square, then there exists an infinite-dimensional closed subspace of the space of operators on $C(K)$, each nonzero element of which does \emph{not} have the form $gI+S$, where $g\in C(K)$, $S$ is weakly compact and $I$ is the identity operator. This comes in contrast with what happens in $C(K)$ spaces with \emph{few operators} in the sense of Koszmider [P. Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 300 (2004), no. 1, 151 - 183.], which are precisely $C(K)$ spaces where \emph{every} operator is of the form $gI+S$. In addition we show that, in case $C(K)$ has few operators, there is an opertator $J$ on $C(K\times\{0,1\})=C(K)^2$ such that each operator on $C(K\times\{0,1\})$ is of the form $gI+hJ+S$, where $g,h\in C(K\times\{0,1\})$ and $S$ is strictly singular, in connection to a result by Ferenczi [V. Ferenczi,Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213 (2007), no. 1, 462 - 488.].

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