  In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck
operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla
v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where
$v_{\star}:\mathbb{R}^d\rightarrow\mathbb{R}^m$ is a rotating wave profile with
$v_{\star}(x)\to v_{\infty}\in\mathbb{R}^m$ as $|x|\to\infty$,
$f:\mathbb{R}^m\rightarrow\mathbb{R}^m$ is smooth, $A\in\mathbb{R}^{m,m}$ has
eigenvalues with positive real parts and commutes with the limit matrix
$Df(v_{\infty})$. The matrix $S\in\mathbb{R}^{d,d}$ is assumed to be
skew-symmetric with eigenvalues $(\lambda_1,\ldots,\lambda_d)=(\pm
i\sigma_1,\ldots,\pm i \sigma_k,0,\ldots,0)$. The spectra of these linearized
operators are crucial for the nonlinear stability of rotating waves in reaction
diffusion systems. We prove under suitable conditions that every
$\lambda\in\mathbb{C}$ satisfying the dispersion relation $$\det\Big(\lambda
I_m + \eta^2 A - Df(v_{\infty}) + i\langle n,\sigma\rangle
I_m\Big)=0\quad\text{for some $\eta\in\mathbb{R}$ and $n\in\mathbb{Z}^k$}$$
belongs to the essential spectrum $\sigma_{\mathrm{ess}}(\mathcal{L})$ in
$L^p$. For values $\mathrm{Re}\,\lambda$ to the right of the spectral bound for
$Df(v_{\infty})$ we show that the operator $\lambda I-\mathcal{L}$ is Fredholm
of index $0$, solve the identification problem for the adjoint operator
$(\lambda I-\mathcal{L})^*$, and formulate the Fredholm alternative. Moreover,
we show that the set
$$\sigma(S)\cup\{\lambda_i+\lambda_j:\;\lambda_i,\lambda_j\in\sigma(S),\,1\leqslant
i<j\leqslant d\}$$ belongs to the point spectrum
$\sigma_{\mathrm{pt}}(\mathcal{L})$ in $L^p$. We determine their eigenfunctions
and show that they decay exponentially in space. As an application we analyze
spinning soliton solutions which occur in the Ginzburg-Landau equation and
compute their numerical spectra as well as associated eigenfunctions.
