Second order differential operators on C[0,1] with Wentzell-Robin boundary conditions

In this note we give a simple proof that the operator A defined by (1) A f ≔ ( a f ′ ) ′ , D ( A ) ≔ { f ∈ C 2 [ 0 , 1 ] : ( a f ′ ) ′ ( j ) + β j f ′ ( j ) + γ j f ( j ) = 0 ;   j = 0 , 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq1294.tif"/>

generates an analytic semigroup on C[0,1]. Here 0 < α < α(·) ∈ C ¹[0,1] is a strictly positive differentiable function and β_j, γ_j ∈ ℂ, j = 0, 1, are arbitrary complex numbers. Moreover, we give conditions implying positivity and stability of the generated semigroup.