The Theory Of Lipschitz-α Operators On Non-Compact Metric Spaces

In this article, we study the theory of Lipschitz-α operators on non-compact metric spaces. At first, we introduce and discuss various nonlinear Lipschitz-α operators on non-compact metric space D and obtain some properties of these operators. Secondly, we discuss relative theory of operator spaces and operator algebras. We divide this article into three chapters, the following is the topic of this paper.In Chapter 1, we define nonlinear Lipschitz-a operators among metric spaces and metric space into Banach space and prove some properties of them. Secondly, the invertibility of these operators is discussed. Moreover, it is proved that all of these operators among metric spaces constitute a linear space. Finally, we give some examples of Lipschitz-a operators.In Chapter 2, operator spaces L^α(D,Y);L_e^α(D,Y);l^α{D,Y);L_B^α(D,Y) and l_B^α(D, Y), consisting of big Lipschitz-a operators and of little Lipschitz-a operators from D into Banach spaces Y, respectively, are introduced and discussed. Firstly, It is proved that L^α(D, Y) and l^α(D, Y) ∩ L^α(D, Y) are Banach spaces with respect to the norm ||T||_α,e = ||Te|| + L_α(T). Secondly, the inclusion relationships between these spaces are discussed. Thirdly, bounded Lipschitz-a operators spaces are introduced and discussed. Moreover, it is proved that operator space L_B^α(M,R) is Riesz space and (B₁(L_B^α(M,R)),∨,∧) is completely distributive lattice. Finally, we discuss Lipschitz-a dual operators and dual spaces.In Chapter 3, we investigate the algebra theory of Lipschitz-a operators. At first, the Lipschitz-a operator from (D, d) to Banach algebra A and relative operator algebras are introduced and discussed. Secondly, it is proved that these algebras are Banach algebras with respect to some norms, respectively. And, their relations are disscussed. Thirdly, the *-algebra of Lipschitz-a is introduced. Finally, we discuss the closed ideals of these algebras.