Matrix norm question

Question

Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if

$$||A^*A+AA^*||=||A^*A||$$

does it imply that $AA^*=0$. If not, could you give a counter-example?

Answer 1

For a counterexample, consider $$ A:=\pmatrix{0&1\\0&0}. $$