非线性Lipschitz算子的Lipschitz对偶算子及其应用

A Novel Dual Notion of a Nonlinear Lipschitz Operator: Lipschitz Dual Operator

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.