Are rank of $T$ and $T^*$ equal? Let $H$ be a infinite dimensional Hilbert space and $T:H\to H$ be an operator on $H$.Is it true that $\operatorname{rank}T=\operatorname{rank}T^*$. We know that this is true for finite dimension. Is it true for infinite dimensional Hilbert space or is there any example of $T$ and $T^*$ that their rank are different?
Thanks!

The operator $Tv \mapsto T^*v$ is actually a partial isometry. The closure of the ranges are Murray-von Neumann equivalent.
