Let $A,B\in M_n$ s.t. $A^*A+B^*B\leq I_n$. Is it possible to get an unitary or invertable $J\in M_n$ s.t. $B=J\sqrt{I-A^*A}$ ?
Comments: I can see existance of such a contraction $J\in M_n$ satisfying $B=J\sqrt{I-A^*A}$ but not the desired J. Moreover I observe that if $A^*A+B^*B= I_n$ then we will get desired J as unitary by polar decomposition of matrices. But can we hope same thing for matrix inequality also?
Any comment regarding the problem is highly appreciated. Thanks in advance.