The Absolute Numerical Index Of The Real L_p² Space And The Isometries Extension Of L₁-sum Spaces

The numerical indices and the isometric extension problem are two very important research in functional analysis, and various characteristics of our study of space (in particular the norm properties and spatial geometric characteristics) has a significance.In chapter 1, we introduce the numerical index of a Banach space and the isometric extension problem. Some important results in the development of the related problems are outlined in this paper and some recent advancement are repointed.The numerical index of a Banach space is a constant relating the behavior of the numerical range with that of the usual norm on the Banach algebra of all bounded linear operators on the space. The theory of numerical index, different from the spectral theory, has played a crucial role in contacting the operator norm structure and the algebraic structure, and recent years have seen many scholars’ devotions to study the classic space and its conjugate space and meaningful findings have been sprouting.Isometric extension problem, also known as the Tingley’s problem. Expressed as follows:Tingley’s Problem:Let E and F be two real Banach spaces. If V0 is a surjective isometry between the two unit spheres S(e) and S(F), does V0 have an isometric affine extension, i.e. does there exist an affine isometric mapping V:E→F such that V|S(E)=V0?The study of this issue needs a good understanding of the geometric properties and algebraic properties of the normed space. In chapter 2, we study the absolute numerical index of the real lp2 space. In 2011, M.Martin, J.Meri, and M.Popov first proposed the concept of absolute numerical index of a Banach space. In this chapter, we study the absolute numerical radius of the operator on the real lp2 space, and obtain a estimate of the absolute numerical index of real lp2. Let 1< p<∞and is the absolute numerical index of the real lp2. In chapter 3, we study the extension of isometries between the unit spheres of the l2-sumof a strictly convex normed space and a general normed space which satisfys some conditions. We obtain that the Tingley’s problem is affirmative under this condition. Let E1,E2 be two normed spaces and E1 is strictly convex, E2 satisfying:if x, y∈S(E2) and‖x+y‖= 2, then‖x-y‖< 2. Suppose V0:S(E1 (?)l1 E2)→S(E1 (?)l1 E2) is an onto isometry. Then for any Vo, there exists real linear isometry U:E1 (?)l1 E2→E1 (?)l1 E2 such that U|S(E1 (?)l1 E2)= V0.