Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂ Ω, the Dirichlet-to-Neumann operator D_λ is defined on L²(Γ) for any real λ. We prove a close relationship between the eigenvalues of D_λ and those of the Robin Laplacian Δ_μ, i.e. the Laplacian with Robin boundary conditions ∂_ν u =μ u. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, λNₖ₊₁ ≤ λDₖ, k ∈ N, and to sharpen the inequality to be strict, whenever Ω is a Lipschitz domain in Rᵈ. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by -D_λ, for λ sufficiently small or negative, is irreducible.