A Note on Complementary Subspaces in c₀

Extract

A well known result of A. Pelcynski [2] states that each subspace of c₀ which is isomorphic to c₀ and of infinite deficiency has a complementary subspace which is itself isomorphic to c₀. We are concerned here with the question of when there exists R, a subset of the integers, such that the complementary subspace X can actually be taken to be C₀(R). That is, we are concerned with determining when the basis vectors for X can be chosen as a subset of the usual basis vectors for c₀. If T: C₀ → C₀ is norm increasing and ‖T‖ < 2, it is not hard to see, as we shall show, that Tc_o admits a complement of the form C₀(R). However, this bound cannot be improved; indeed, it is possible to construct norm increasing T: C₀ → C₀ such that ‖T‖ = 2 and yet Tc₀ admits no such complement. The construction of such a T is the main point of this note. This construction also enables us to dispose of a speculation of ours in [1].