Some Schroeder-Bernstein type theorems for Banach spaces (Q2473914)

Let \(X\) and \(Y\) be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. As is well-known [\textit{W. T. Gowers}, Bull.\ Lond.\ Math.\ Soc.\ 28, No. 3, 297--304 (1996; Zbl 0863.46006)], then \(X\) is not necessarily isomorphic to \(Y\). The author continues his investigation of additional conditions which yield the isomorphism of \(X\) and \(Y\) [see, e.g., \textit{E. M. Galego}, Glasg.\ Math.\ J.\ 47, No. 3, 489--500 (2005; Zbl 1101.46005)]. Let \((p,q,r)\) be a triple in \(\mathbb{N}\). The author says that \(Y\) is \((p,q,r)\)-complemented in \(X\) if there exists a subspace \(A\) of \(X\) such that \(X\) is isomorphic to \(Y\oplus A\) and \(X^p\) is isomorphic to \(X^q\oplus A^r\). Then, given a couple of triples \((p,q,r)\) and \((s,t,u)\) and putting \(\Lambda= (q+r-p) (t+u-s)\), the author proves partially the following conjecture: For every pair of Banach spaces \(X\) and \(Y\) such that \(X\) is \((p,q,r)\)-complemented in \(Y\) and \(Y\) is \((s,t,u)\)-complemented in \(X\), we have that \(X\) is isomorphic to \(Y\) if and only if one of the following conditions holds: (a) \(\Lambda\neq 0\), \(\Lambda\) divides \(p-q\) and \(s-t\), \(p=1\) or \(q=1\) or \(s=1\) or \(t=1\). (b) \(p=q=s=t=1\) and the greatest common divisor gcd\((r,u)=1\).