Additive Maps Preserving Normal Eigenvalues

As an important role in operator algebra,spectrum can deeply reflect the essential attributes of operators and naturally concerned by many experts and scholars at home and abroad.There are profound and comprehensive research results of spectral properties of various operators.The spectral preserving problems is also deeply concerned and studied as an important means to study the operator structure in operator algebras.In the infinite-dimensional Banach space,the normal eigenvalues which is closely related to the spectrum of operators are also in the light of the discovery.In this paper,the normal eigenvalues of operators are studied.We discuss the preserving problems related to normal eigenvalues.Let X be an infinite-dimensional complex Banach space and Β(χ)the algebra of all bounded linear operators on χ.For T∈Β(χ)and a given integer n≥ 1,σ0(T)denotes the set of all normal eigenvalues of T and Nn(T)denotes the set of normal eigenvaluesλ of T satisfying dimker(T-λI)≤n.The main results are as follows:In the first part,we characterized the additive surjection φ which preservingσ0(T)in both directions.That is,φ satisfies σ0(T)=σ0(T)),(?)T∈Β(χ).We firstly considered the relavent properties of one rank perturbations of σ0(T).By using the space decomposition,we get the necessary condition of dimran(T)≥ 2 and T≠0.Meanwhile,we get the equal characterization of operators equal inΒ(χ).Eventually we proved that if φ preserves σ0(T)in both directions,then φpreserves one rank idempotents and their linear spans in both directions.So we can characterize φ.We show that such a map is an isomorphism or anti-isomorphism.In the second part,we discussed the characterization of the additive surjectionφ preserving Nn(T)in both directions.By studying the properties of Nn(T)we proved φ preserves one rank idempotents and their linear spans in both directions.Similarly,φ is either an automorphism or an anti-automorphism.In this part,we get the construction of φ in the whole space while φ preserving only countable elements of Β(χ).And these results enlighten us about the learning of preserving problems in operator algebras.