Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics  Project B3Numerical Analysis of equivariant evolution equations15042 Denny Otten. In this paper we study perturbed OrnsteinUhlenbeck operators \[ \left[\mathcal{L}_{\infty} v\right](x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangleB v(x),\,x\in\mathbb{R}^d,\,d\geq 2, \] for simultaneously diagonalizable matrices \(A,B\in\mathbb{C}^{N,N}\). The unbounded drift term is defined by a skewsymmetric matrix \(S\in\mathbb{R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in timedependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of \(\mathcal{L}_{\infty}\) in \(L^p(\mathbb{R}^d,\mathbb{C}^N)\), \(1\lt p \lt \infty\), is the following \(L^p\)dissipativity condition \[ z^2\text{Re}\left\langle w,Aw\right\rangle + (p2)\text{Re}\left\langle w,z\right\rangle\text{Re}\left\langle z,Aw\right\rangle\geq \gamma_A z^2w^2\;\forall\,z,w\in\mathbb{C}^N \] for some \(\gamma_A>0\). We prove that the \(L^p\)dissipativity condition is equivalent to a new \(L^p\)antieigenvalue condition \[ A\text{ invertible}\quad\text{and}\quad\mu_1(A)>\frac{p2}{p},\,1\lt p \lt\infty,\,\mu_1(A)\text{ first antieigenvalue of \(A\),} \] which is a lower \(p\)dependent bound of the first antieigenvalue of the diffusion matrix \(A\). This relation provides a complete algebraic characterization and a geometric meaning of \(L^p\)dissipativity for complexvalued OrnsteinUhlenbeck operators in terms of the antieigenvalues of \(A\). The proof is based on the method of Lagrange multipliers. We also discuss several special cases in which the first antieigenvalue can be given explicitly.
