Abstract. The famous Gowers tree space is the first example of a space not containing c0, 1 or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has 2 as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form λI + W where W is a weakly compact (hence strictly singular) operator.

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It is shown that variants of the HI methods could yield objects closely connected to the classical Banach spaces. Thus we present a new c 0 saturated space, denoted as X 0 , with rather tight structure. The space X 0 is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure. Everyone is either of type I, namely, contains an isomorph of X 0 itself or else is isomorphic to a subspace of c 0 (type II). Furthermore for any analytic decomposition of X 0 into two subspaces one is of type I and the other is of type II. The operators of X 0 share common features with those of HI spaces.2000 Mathematics Subject Classification. 46B20, 46B26. Key words and phrases. c 0 saturated Banach spaces, space of operators, saturated norms, hereditarily indecomposable.