On essentially incomparable Banach spaces (Q1323471)

We introduce the concept of essentially incomparable Banach spaces, and give several characterizations and some examples. Then, for two essentially incomparable Banach spaces \(X\) and \(Y\), we prove that a complemented subspace of the product \(X\times Y\) 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 its respective hyperplanes, then the group of invertible operators in \(X\times Y\) is not connected. The results can be applied to describe the complemented subspaces of classical Banach spaces.