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 (on)to Y？

PS: The answder 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?