Projections in the convex hull of isometries on $$C^2[0,1]$$

Let $$C^2[0, 1]$$ be the Banach space of all functions that have continuous derivatives $$f'$$ and $$f''$$ on the closed interval [0, 1], equipped with norm $$\Vert f\Vert = |f(0)| + |f'(0)| + \Vert f''\Vert _{\infty }$$ , where $$\Vert \cdot \Vert _{\infty }$$ is the usual supremum norm. In this paper, we characterize projections on $$C^2[0, 1]$$ that can be written as convex combination of two surjective linear isometries. We also find out the structure of Hermitian projections and generalized bi-circular projections on $$C^2[0, 1]$$ . Finally, we discuss the relationship of these two types of projections (Hermitian and generalized bi-circular projections) with the convex combination of two isometries.