Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply  that $X$ isomorphic onto $Y$? Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq  Y$, if there exists a left invertible operator $J:X \rightarrow Y$. My question is the following；
If $X$ embed in $Y$, and $Y$ embed in $X$, then $X$ isomorphic onto $Y$？ 
PS: The answer is positive when $X$ and $Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?
 A: This is called the Schroeder-Bernstein problem, and for Banach spaces there are constructions of nonisomorphic Banach spaces which embed into each other.
W. T. Gowers, "A Solution to the Schroeder-Bernstein Problem for Banach Spaces" Bull. London Math. Soc. (1996) 28 (3): 297-304.
A: The Schroeder-Bernstein problem asked about complemented subspaces. It is a much stronger property for a subspace to be complemented. The only spaces in which every subspace is complemented are isomorphic to Hilbert space (Lindenstrauss and Pelczynski). Also, Hilbert space (and its isomorphs) is the only space isomorphic to all of its subspaces. Therefore, if you consider any space which is not isomorphic to Hilbert space but embedds into its subspaces (e.g. $\ell_p $ for $p \not= 2$; in general, these are called minimal spaces), any non-isomorphic subspace gives a counterexample.
Edit: I guess I was confused by the term left invertible. I'll leave my answer up in case someone doesn't know how to do it in this weaker case. 
