Cantor–Bernstein theorems for certain symmetric bases in Banach spaces

Abstract

We prove the following Cantor–Bernstein type theorem, which applies well to the class of symmetric sequence spaces studied earlier by Altshuler, Casazza, and Lin: Let X and Y be Banach spaces having symmetric bases (xn) and (yn), respectively. If each of the bases (xn) and (yn) is equivalent to a basic sequence generated by one vector of the other, then the spaces X and Y are isomorphic. As a consequence, we obtain the strong equivalence that two Lorentz sequence spaces have the same linear dimension if and only if they are isomorphic.