Ji Gen PENG,Zong Beng XU
In [1], a dual operator notion of a nonlinear operator, named Lipschitz dual operator, was introduced, and it was mainly shown that the Lipschitz dual operator of any a nonlinear Lipschitz operator in Banach space X is a bounded linear operator on the Lipschitz dual space X*L of X. In this paper, we further prove that, for a bounded linear operator U on X*L, if and only if U is a w*-continuous homomorphism, there is a Lipschitz operator T in X such that U is the Lipschitz dual operator of T. It is therefore deduced that the invertibility of any a nonlinear Lipschitz operator is equivalent to the invertibility of its Lipschitz dual operator. As an application example, by developing a new concept, named PX-dual operator, a generation theorem of nonlinear operator semigroup is established.
