On Essential Numerical Range And Maximal Numerical Range Of The Aluthge Transform

Numerical range is an importent part of functional analysis, this subject is realated and has applications to many different branches of pure and applied science such as functional analysis, operator theorem, C*-algebras, inequalities, numerical analysis, perturbation theorem, martix polynomials, systems theorem, quantum physics and so on. In 1919, the famous Toeplitz-Hausdorff Theorem was proved, then the researches on the properties of numerical range and numerical radius became active. As a result of the development of numerical range, various generalized numerical range were studied, such as maximal numerical range, essential numerical range, essential maximal numerical range, joint numerical range, c-numerical range and joint essential maximal numerical range.For any bounded linear operator T on Hilbert space H, in 1990 A.Aluthge gave the definition of the Aluthge transform (T|⁾ = |T|^1/2∪|T|^1/2 when studying p- hy-ponormal operator([1]). In 2001 when discussing the relationship between T and (T|⁾, T.Yamazaki introduced the *-Aluthge transform (T|⁾(*) = |T*|^1/2∪|T*|^1/2 ([2]). After that, many lectures began to discuss the properties of T, (T|⁾ and (T|⁾(*) such as p-hyponormal, log-hyponormal, spectrum, numerical range ect. In [6] Pei Yuan Wu drew two conclusions about the numerical range of T, (T|⁾ and (T|⁾(*), that is for any bounded linear operator T, we have (1)(W((T|⁾)|——) (?) (W(T)|——), (2)(W((T|⁾)|——) = (W((T|⁾(*))|——). Recently, in [3] the author Xiumei Liu proved that W((T|⁾) = W((T|⁾(*)) was also true. The aim of this paper is to make an investigation on the essential numerical range, maximal numerical range and essential maximal numerical numerical range about T, (T|⁾ and (T|⁾(*).The main content as follows: Chapter 1 pays the emphasis on the result that W_e(T|⁾ (?) W_e(T), which generalized the main result in [6], also we prove that T and (T|⁾(*) have the same essential numerical range. At the same time, the Weyl spectrum, Kato spectrum and reduce spectrum of the three operators are discussed.Chapter 2 deals with the maximal numerical range, essential maximal numerical range and some generalized inverse of (T|⁾ and (T|⁾(*). In this chapter we prove three main results:(1) W₀(T) (?) (W((T|⁾)|——);If ||T|| = ||(T|⁾||, then W₀((T|⁾) (?) W₀(T). (2)For anyA e C, we have WQ(T - A) = W0(T^ - A). (3) essW0(f - A) = essW0(f W - A) holds for any A G C.