  We consider the top Lyapunov exponent associated to a dissipative linear
evolution equation posed on a separable Hilbert or Banach space. In many
applications in partial differential equations, such equations are often posed
on a scale of nonequivalent spaces mitigating, e.g., integrability ($L^p$) or
differentiability ($W^{s, p}$). In contrast to finite dimensions, the Lyapunov
exponent could apriori depend on the choice of norm used. In this paper we show
that under quite general conditions, the Lyapunov exponent of a cocycle of
compact linear operators is independent of the norm used. We apply this result
to two important problems from fluid mechanics: the enhanced dissipation rate
for the advection diffusion equation with ergodic velocity field; and the
Lyapunov exponent for the 2d Navier-Stokes equations with stochastic or
periodic forcing.
