THE OPERATOR A(X)D DX ON BANACH-SPACES

The operator a(x)d/dx on C(I), where I is an interval contained in the real line, is considered in many places. In this paper, we attempt to reconsider it in the subspace of C-0(-infinity, infinity,) containing all even functions, and show that it generates a strongly continuous semigroup. It is interesting that our main conditions seem contradicto ry to previous ones. It is due to the symmetry of the functions and th e different domain of the operator than usual.