Relevant Conclusions About Aluthge Transformation

Numerical range is one of the topics in mathematics comparison,since the advent of the Toeplize-Hausdorff theorem,the study of the numerical range become very active.The research on numerical range is also related to many branches of basic mathematics and applied mathematics and it has been widely used in these branches.Since 1990,Ariyadasa Aluthge introduced the Aluthge transformation and in 2001,Takeaki Yamazaki introduced*-Aluthge transformation,then research on the nature of T,(?),(?)(*)attracts the attention of most scholars.This paper mainly deals with the previous results.The following is the main content of this article:Chapter 1 is introduction and related preparatory knowledge.Chapter 2 is relevant knowledge about Aluthge transformation and generalized Aluthge transformation.Firstly,we introduce the definition of (?),(?)(*),(?) ?,(?) ?(*).Secondly,we introduce some basic properties of them.Lastly,we introduce W((?))=W((?)(*))and W((?)?)=W((?)?(*).Chapter 3 summarizes some conclusions about Aluthge transform spectral picture mainly.First,we introduce the definition of the spectral picture,then through some lemmas and theorems,we can find that in most cases,the spectral picture of T is coincide with that of (?).Chapter 4 summarizes some conclusions about the Aluthge transform of complex symmetric operator.First,we give the definition of conjugation and complex symmetric,then through some lemmas and theorems,we can summerize five main results:(1)The Aluthge transform of a complex symmetric operator is remain complex symmetric.(2)If T is a complex symmetric operator,then((?))* and((?) *)are unitary equivalent.(3)If T is a complex symmetric operator,then (?) = T if and only if T is normal.(4)(?) = 0 if and only if T 2 = 0.(5)Every operator which satisfies T2= 0 is necessarily a complex symmetric operator.Chapter 5 summarizes some conclusions about the polar decomposition of Aluthge transformation,and gives some conclusions about the polar decomposition of Aluthge transformation and the binormal operator.