A Note on Complementary Subspaces in c 0
Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 537-540
Voir la notice de l'article provenant de la source Cambridge University Press
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 Tco 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].
Berg, I. D. A Note on Complementary Subspaces in c 0. Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 537-540. doi: 10.4153/CJM-1972-046-x
@article{10_4153_CJM_1972_046_x,
author = {Berg, I. D.},
title = {A {Note} on {Complementary} {Subspaces} in c 0},
journal = {Canadian journal of mathematics},
pages = {537--540},
year = {1972},
volume = {24},
number = {3},
doi = {10.4153/CJM-1972-046-x},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-046-x/}
}
[1] 1. Berg, I. D., Extensions of certain maps to automorphisms of m, Can J. Math. 22 (1970), 308–316. Google Scholar
[2] 2. Peɫcyński, A., Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. Google Scholar
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