机构地区: 西安交通大学理学院应用数学研究中心
出 处: 《数学学报(中文版)》 2002年第3期469-480,共12页
摘 要: 在文山中我们对非线性Lipschitz算子定义了其Lipschitz对偶算子,并证明了任意非线性Lipschitz算子的Lipschitz对偶算子是一个定义在Lipschitz对偶空间上的有界线性算子.本文还进一步证明:设C为 Banach空间 X的闭子集,C*L为C的 Lipschitz对偶空间,U为 C*L上的有界线性算子,则当且仅当 U为 w*-w*连续的同态变换时,存在Lipschitz连续算子T,使U为T的Lipschitz对偶算子.这一结论的理论意义在于:它表明一个非线性Lipschitz算子的可逆性问题可转化为有界线性算子的可逆性问题.作为应用,通过引入一个新概念──PX-对偶算子,在一般框架下给出了非线性算子半群的生成定理. 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.