Do contraction semigroups inherit some properties of their infinitesimal generator?

Question

Let $H$ be a Hilbert space and $\{S(t)\}_{t\ge 0}$ be a contraction semigroup. Also, let $A$ be the infinitesimal generator of the semigroup $\{S(t)\}_{t\ge 0}$ and $K$ is the kernel of $A$. My questions are as follows.

1. If $Ah \in K^\perp$ for all $h \in H$, then $S(t)h \in K^\perp$ for all $h\in H$? (where $K^\perp$ is its orthogonal complement of $K$.)

2. If a certain subset $K \subset H$ is invariant for $A$, then $K$ is also invariant for $S(t)$? In other words, if $A(K) \subset K$, then $S(t)(K) \subset K$?

Any help would be very much appreciated!