Numerical Range Of Operator

The concepts of numerical range of operators on a Banach and Hilbert space were introduced by Bauen and Toeplitz in 1962 and 1918. From then on,there is a great deal of reseach between the numerical range,the numerical ra-dius.In particular,the subject on them has many connections and applications to lots of areas.such as iterations methods,dilation theory,C*-algebras,several oper-ator theory, factorizations of matrix polynomials, Krein space operators,unitary similarity and so on.After Toeplitz and Hausdouff who Proof of the above two theorems in the twenties of the last century,the studying between the numerical range and numerical radius is becoming more and more actively. In real life, they also played an increasingly important role. Here this artical only sum up three category operators which are all in Hilbert space.That is, the numerical range of Aluthge transform of operators, the numerical range of nipotent operators and the numerical range of normal operators.This paper consist of four parts.In the first part reference language.In this chapter, we briefly introduced the concept of the numerical of op-erator and basic theorems.In this article,a operator T is always a linear bounded operator for the complex Hilbert space H.W(T) denotes numerical range, w(T) denotes numerical radius.They are defined as follows Definition1.1. x is a unit vector of complex Hilbert spaces HThe numerical range of operator issues,in the Hilbert space,there are many classic conclusion.After the definition of the numerical range of operator were intrductd by Toeplitz in 1918,the following two are more representative.Theoreml.2. (Toeplitz-Hausdouff Theorem [1],[2]) An operator T∈B(H),its numerical range is always convex.Theoreml.3. W(T) (?) conσ(T)In the second part the numerical range of Aluthge transform of operators.In the second part,we conside the numerical range of Aluthge transform of operators. Summed up the results among the Aluthge operators,transform of Aluthge operators and transform of*-Aluthge operators in the past few years,In document [3] by Tung,there is T* (?) T,but it is only effective on Matrix of 2x2. so,in 2001,in document [4] by Yamazaki,he proofed that the above theorem still holds in n x n.And in 2002,Taiwan scholar Wu Pei Yuan who promote the above conclusion proofed that it was established for any operators.In 2005,Liu Xiumei,Yang Xinbin promote the results of the WU Pei-yuan which make their relationship more perfect.In 2008,Chen Dongjun,Zhang Yun prooded that there are similar results between the generalized Aluthge formation and the generalized * - Aluthge formation.In last,for there essencial numerical range,there are also similar results.The main conclusions are as follow:If T∈B(H),T is an Aluthge operators in Hilbert space, (1) If T is Aluthge transformation of T, (?);(2) If Tt is*-Aluthge transformation of (?) denotes*-Aluthge of T,then if t∈(0,1), (?);(3)If We(T) is the ess.numerical range,if t∈(0, 1),then (?).In the third part the numerical range of nipotent operators.In the third part, We have summarized the development process of the numerical range of nilpotent operator, make a special study of make a go of following result.(?),and radius is less than (?)The main corollary is as follow:If T∈B(H) is a compact operator of Toeplitz, then W(T) is a closed circle which is a zero as the center, and the radius is less then (?).Be extended to the general operator, will beIf T is an n-dimensional space operator, and its numerical range is zero as the center of the disc,then (?).In the last part the numerical range of normal operators.In the last part,Summed up the value of its numerical range on the con-centration of expression. In the beginning,using spectral measure denoted the numerical range,that is[z1, z2]is the longest internal in (?) After that,we have W(T)=conσ(T)Now there is a more precise expression of positive,that is If T is a normal operator on H and f is the function,then W(T)=(Intcona(T))∪{z∈(?)(conσ(T))}[z1,z2]is the longest internal in (?)和(?)are positive.