  In this paper we study differential operators of the form \begin{align*}
\left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle
Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2,
\end{align*} for matrices $A,B\in\mathbb{C}^{N,N}$, where the eigenvalues of
$A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx,
\nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an
unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$.
Differential operators such as $\mathcal{L}_{\infty}$ arise as linearizations
at rotating waves in time-dependent reaction diffusion systems. The results of
this paper serve as foundation for proving exponential decay of such waves.
Under the assumption that $A$ and $B$ can be diagonalized simultaneously we
construct a heat kernel matrix $H(x,\xi,t)$ of $\mathcal{L}_{\infty}$ that
solves the evolution equation $v_t=\mathcal{L}_{\infty}v$. In the following we
study the Ornstein-Uhlenbeck semigroup \begin{align*} \left[ T(t)v\right](x) =
\int_{\mathbb{R}^d} H(x,\xi,t) v(\xi) d\xi,\,x \in \mathbb{R}^d,\, t>0,
\end{align*} in exponentially weighted function spaces. This is used to derive
resolvent estimates for $\mathcal{L}_{\infty}$ in exponentially weighted
$L^p$-spaces $L^p_{\theta} (\mathbb{R}^d,\mathbb{C}^N)$, $1\leqslant p<\infty$,
as well as in exponentially weighted $C_{\mathrm{b}}$-spaces
$C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$.
