Isometries of vector-valued function spaces preserving the kernel of a linear operator

We prove Banach Stone type theorems for isometrics of continuously differentiable (continuously twice differentiable resp.) functions on the unit interval equipped with norms related to the first derivative (a linear differential operator of second-order resp.). They are consequences of a general theorem on isometrics of function spaces, on which a linear operator is defined, over compact Hausdorff spaces.