Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics  Project B3Numerical Analysis of equivariant evolution equations14052 Denny Otten. In this paper we study differential operators of the form \[ \left[\mathcal{L}_{\infty}v\right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle  Bv(x),\,x\in\mathbb{R}^d,\,d\geq 2, \] 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 OrnsteinUhlenbeck operator with an unbounded drift term defined by a skewsymmetric matrix \(S\in\mathbb{R}^{d,d}\). Differential operators such as \(\mathcal{L}_{\infty}\) arise as linearizations at rotating waves in timedependent 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 OrnsteinUhlenbeck semigroup \[ \left[T(t)v\right](x) = \int_{\mathbb{R}^d}H(x,\xi,t)v(\xi)d\xi,\,x\in\mathbb{R}^d,\,t>0, \] in exponentially weighted function spaces. This is used to derive resolvent estimates for \(\mathcal{L}_{\infty}\) in exponentially weighted \(L^p\)spaces \(L^p_{\theta}(\mathbb{R}^d,\mathbb{C}^N)\), \(1\leq p\lt \infty\), as well as in exponentially weighted \(C_{\mathrm{b}}\)spaces \(C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)\).
