A new Lp-antieigenvalue condition for Ornstein–Uhlenbeck operators

In this paper we study perturbed Ornstein–Uhlenbeck operators[L∞v](x)=A△v(x)+〈Sx,∇v(x)〉−Bv(x),x∈Rd,d⩾2, for simultaneously diagonalizable matrices A,B∈CN,NA,B∈CN,N. The unbounded drift term is defined by a skew-symmetric matrix S∈Rd,dS∈Rd,d. Differential operators of this form appear investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of L∞L∞ in Lp(Rd,CN)Lp(Rd,CN), 10γA>0. We prove that the LpLp-dissipativity condition is equivalent to a new LpLp-antieigenvalue conditionA invertibleandμ1(A)>|p−2|p,1