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

We prove Banach—Stone type theorems for isometries 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 isometries of function spaces, on which a linear operator is defined, over compact Hausdorff spaces.