Is $\|e^{zT-zT^*}\|$ bounded for all $z\in \mathbb{C}$?

Question

Let $E$ be an infinite-dimensional complex Hilbert space and $T\in \mathcal{L}(E)$.

Is $\|e^{zT-zT^*}\|$ bounded for all $z\in \mathbb{C}$?

Answer 1

Take $T=iI $. If we write $z=a+ib $, we have $$\|e^{z (T-T^*)}\|=|e^{2iz}|=e^{-2b}, $$ not bounded as we are free to choose $b $.

Note, on the other hand, that $\|e^{zT-\bar zT^*}\|=1$ for all $z $ (as it is a unitary).