Linear Maps Between Operator Algebras Preserving Centain Spectral Functions

In the21st Century there has been considerable interest in the linear preserver problems.The goal of studying linear preservers is to give structural characterizations of linear maps on algebras having some special properties.One of the most famous problem is Kaplansky s problem in this direction.Let φ be a surjective linear map between two semi-simple Banach algebra A and B.Suppose that σ(φ(x))=σ(x) for all x∈A,is it true that0is Jordan iso-morphism?In this article,we give the characterization of antomorphism on B(H) when the φ is a surjective linear maps preserving the set of upper semi-Weyl operators or the set of left invertible operators.And if φ preserves the upper semi-Browder spectrum or descent spectrum and the set of isolated points,the φ is an automorphism on B(H).This paper contains three chapters:In chapter1, given the historical background of this study, prepare knowledge and the characterization of the φ that preserves the set of semi-Fredholm operators and the set of Fredholm operators.In chapter2, given the characterization of the φ that preserves the set of left invertible operators or the set of upper semi-Weyl operators.And the relation be-tween the linear maps preserving the set of left invertible operatorss and the set of upper semi-Weyl operator is considered.In chapter3, Let B(H) be the algebra of all bounded linear operators on infinite dimensional complex Hilbert space H, and let φ:B(H)â†'B(H) be a surjective linear map. If φ preserves the upper semi-Browder spectrum or descent spectrum and the isolated points, then6is an automorphism on B(H). And if φpreserves the Drazin spectrum and the isolated points, we also give the φ two probable structure.