Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L $ where $L\in \mathcal B(\mathbb H)$.

Question 1. Is there a reference for this result?

Question 2. Is there a Banach space version?

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L $ where $L\in \mathcal B(\mathbb H)$.

Question 1. Is there a reference for this result?

Question 2. Is there a Banach space version?