Wentzell-Robin boundary conditions on C[0,1]

Given a ∈ C ¹[0,1], a(x) ≥ α > 0, we prove that the second order differential operator on C[0,1] defined by A _W u := (au′)′ with Wentzell-Robin boundary conditions $$left( {au'} right)^prime left( j right) + beta _j u'left( j right) + gamma _j uleft( j right) = 0, j = 0,1,$$ where β _j and γ _j are real numbers, generates a holomorphic C ₀-semigroup on C[0,1].