if $AA^*=BB^*$ what are the relations between A and B

Question

I'm wondering if we have two linear operators $A, B \in \ell(V)$. and we know that $AA^*=BB^*$. then what informations can this give to us about relationships between $A$ and $B$?

I think they have some strong relations.

Answer 1

in finite dimension by polar decomposition, $A = \sqrt{AA^*}U_a, B = \sqrt{BB^*}U_b$, where the $U_a$ and $U_b$ are unitary operators. so we heve:

$\sqrt{BB^*} = \sqrt{AA^*} = AU^*_a = BU^*_b \Rightarrow A = BU^*_bU_a$

$U^*_bU_a(U^*_bU_a)^* = U^*_bU_aU^*_aU_b = I \Rightarrow U = U^*_bU_a$ is unitary

$\Rightarrow A = BU$ where $U$ is an unitary operator.

notice that if $A = BU$ then $A^* = U^*B^*$ so if we had $A^*A = B^*B$ then $A = U'B$ where $U'$ is an unitary operator.