Numerical Range Of The Aluthge Transform And Not Attained Points Of Numerical Range

The study of numerical range started in 1918-1919 by Toeplitz and Hausdorff when they first proved that W(A) is always convex, since then, the study of numerical range theory has been one of the most active research areas. The subject is related and has applications to many different branches of pure and applied science such as operator theory, C*-algebras, Banach algebras, matrix norms, inequalities, numerical analysis, perturbation theory, matrix polynomials, systems theory, quantum physics, etc. In recent years, the Aluthge transform T and the *-Aluthge transform T^ were put forward by A. Aluthge and T. Yamazaki respectively. After that, many lecturers began to discuss the properties of T , T , T(*). In [26], Takeaki Yamazaki drew three conclusions about numerical ranges ? and numerical radius of T and T, that is, (1) w(T) > w(T); (2)if T is an n x n matrix, then W(T) D W(T); (3)if N(T) C N(T*), then W(T) W(T). But these results are not the best. In 2002, Wu Pei-yuan drew two conclusions about numerical range among T , T and f (*) on the basis of [26], such as W(T) = W(fW) and W(T) D W(T) (see [25]). At the same time, in [25], Wu put forward two open questions whether W(T) = W(T^} and W(T) D W(T) hold or not. The aim of this paper is to make a deep investigation on the result W(f) = W(T^).The main content as follows:Chapter 1 pays the emphasis on the result that T and T(*) have the same numerical ranges. First we introduce T , T(*) and discuss the spectrum and the norm among T , T and T(*). We show that W(T) = W(T(*). At last, we discuss the numerical range of the Duggal transform T =| T | V, draw that for a bounded and linear operator T,W(f) C W(T) holds.Chapter 2 deals with the numerical ranges between Tt and Tt(*) mainly. Firstly wegive the definitions and the basic properties of Tt and Tt(*) . We prove the fact that T* andTt(*) have the same numerical ranges. Furthermore some properties about closed range points and point spectrums of W(Tf) are discussed, which generalize the main results in[15], [22].Chapter 3 is mainly about not attained points and corner points of the closure of the numerical range. Firstly we introduce not attained points and the corner points and their basic properties. Secondly we prove two main results, such as (1) If A is a not attainedIIIpoint of W(T) and λ∈ex(W(T)), then there exists an orthonormal sequence {yn} such that (Tyn,yn) → A as n→∞; (2)If A is not attained for W(T) and not an extreme point of W(T), then there exists a sequence {xn} of linearly independent unit vectors such that (Txn, xn) → $ as n → ∞. Finally we discuss some properties of the corner points and the reduced point spectrums.