We introduce the concept of essentially incomparable Banach spaces, and give some examples. Then, for two essentially incomparable Banach spaces X and Y, we prove that a complemented subspace of the product X x Y is isomorphic to the product of a complemented subspace of X and a complemented subspace of Y. If, additionally, X and Y are isomorphic to their respective hyperplanes, then the group of invertible operators in X x Y is not connected. The results can be applied to some classical Banach spaces.
@article{urn:eudml:doc:39935,
title = {On essentially incomparable Banach spaces.},
journal = {Extracta Mathematicae},
volume = {6},
year = {1991},
pages = {135-138},
mrnumber = {MR1185360},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39935}
}
González, Manuel. On essentially incomparable Banach spaces.. Extracta Mathematicae, Tome 6 (1991) pp. 135-138. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39935/