Spectrum of $H + H^*$ in Hilbert space

Question

Consider a bounded operator $H$ on a Hilbert space $V$. Let ${\rm Re\ sp} (H)$ be the real parts of all points in the spectrum of $H$. Is it ture that ${\rm sp}( H + H^*) = 2 {\rm Re\ sp}(H)?$ Or in other words, if ${\rm Re\ sp}(H) \subset (0,\infty)$, does it follow that the operator $H+H^*$ is positive definite?

Answer 1

If $H = \pmatrix{0 & 1 \cr 0 & 0}$, then the spectrum of $H$ is $\{0\}$ but $H+H^*$ is not positive since its determinant is -1. To get a counter example with $\text{Re sp}(H)\subset(0,\infty)$ it is enough to add $\epsilon I$ to $H$.