The Approximation Of The Almost Surjective ε-Isometric Operator And The Stability Of The Operator Equation

The reasearch of the isometric operator theory on metric spaces is a very wellknown and active fled. And we can combine the operator theory and the property ofthe space to study the stability of the isometric operator. The study of the isomet-ric approximation of the almost surjectiveε-isometric operator on metric space isimportant in the theory of the Banach space. Recently, as the subject of the Hyers-Ulam-Rassias stability of operator equations on metric linear space has infiltratedto more areas both within mathematics and outside, the study experiences growthand vitality in international.Master thesis "The Approximation of the Almost Surjective G-Isometric Oper-ator and the Stability of the Operator Equation" applies the theory of Banach spaceand the operator theory comprehensively. There are three chapters in the thesis.Chapter 1 mainly introduces some study backgrounds and corresponding preliminar-ies. In order to build a good foundation for later parts, introduction inculdes someconceptions and notations needed in the following chapters. In chapter 2, on thebasis of the study of the scholars, we investigate the isometric approximation of thealmost surjectiveε-isometric operator between Banach spaces, weakening uniformlyconvex condition and strengthening the condition of theε-isometric operator. Inaddition, we also consider the problem on completeβ-normed (0＜β＜1) spacesand derive related results. In chapter 3, we focus our attention mainly on the ad-ditive approximation of the six-variate Jensen type operator equation. Taking intoaccount the property of the approximate remainderφ, we derive some importantconclusions and the corresponding error formulas.In particular, we make affirmativeor negative answer for the Hyers-Ulam stability on some area.