# Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

Preprint (PDF), 27 January 2013
Positivity 18 (2014), 235–256
Original version at doi:10.1007/s11117-013-0243-7
By analysing some explicit examples we investigate the positivity and non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator $$\Delta +\lambda I$$ as $$\lambda$$ varies. It is known that the semigroup is positive if $$\lambda<\lambda_1$$, where $$\lambda_1$$ is the principal eigenvalue of $$-\Delta$$ with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on $$\lambda>\lambda_1$$.