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?



Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply  that $X$ isomorphic onto $Y$?
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"](http://blms.oxfordjournals.org/content/28/3/297.abstract) Bull. London Math. Soc. (1996) 28 (3): 297-304.


