$\newcommand{\Ker}{\operatorname{Ker}}$
Let $X$ be Hilbert space and let $T:X\to X$ be a bounded operator. Define the operator $S: X/\Ker T \to X/\Ker T$ via $S(x+\Ker T)=Sx+\Ker T$. I can show that $||S||\leq ||T||$, and I am wondering whether the norms are actually equal. Is this true or false in general?
Edit: Considering the answer below, I am adding the conditions that $\Ker(T)$ is finite dimensional and $\operatorname{Im}(T)$ is not (say $T$ is surjective). Could you please show a counterxample in this situation as well.