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If the Vlasov-Poisson or Einstein-Vlasov system is linearized about anisotropic steady state, a linear operator arises the properties of which arerelevant in the linear as well as nonlinear stability analysis of the givensteady state. We prove that when defined on a suitable Hilbert space andequipped with the proper domain of definition this transport operator$\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In theVlasov-Poisson case we also determine the kernel of this operator.

On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states