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?