Abstract
We present a Banach space X \mathfrak X with a Schauder basis of length Ï 1 \omega _1 which is saturated by copies of c 0 c_0 and such that for every closed decomposition of a closed subspace X = X 0 â X 1 X=X_0\oplus X_1 , either X 0 X_0 or X 1 X_1 has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of X \mathfrak X have âfew operatorsâ in the sense that every bounded operator T : X â X T:X \rightarrow \mathfrak {X} from a subspace X X of X \mathfrak {X} into X \mathfrak {X} is the sum of a multiple of the inclusion and a Ï 1 \omega _1 -singular operator, i.e., an operator S S which is not an isomorphism on any non-separable subspace of X X . We also show that while X \mathfrak {X} is not distortable (being c 0 c_0 -saturated), it is arbitrarily Ï 1 \omega _{1} -distortable in the sense that for every λ > 1 \lambda >1 there is an equivalent norm â | â â | \| |\cdot \| | on X \mathfrak {X} such that for every non-separable subspace X X of X \mathfrak {X} there exist x , y â S X x,y\in S_X such that â | x â | / â | y â | ℠λ \| |x\| |/\| |y\| |\ge \lambda .
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