On The Range Of The Aluthge Transform

The study of operator theory began in 20th century. Since it is used widely in mathematics and other scientific branches, it got great development at the beginning of the 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in morden mathematics. It has close relations and interinfiltrations with quantum mechanics, linear system and control theory, as well as some other important branches of mathematics.In this article, we study the algebraic operators and idempotent operators in the range of the Aluthge transform, and the translation property of the Aluthge transform.At the same time,we dicuss the closed and dense properties about the range of the Aluthge transform. The details as following:In chapter 1, some notations, definitions are introduced and some well-known theorems are given. In section 2, we introduce the definitions of spectrum, point spectrum,approximate point spectrum, idempotent operator,Aluthge transform,*-Aluthge transform and so on.In section 3, we give some well-known theorems and the relationship of some spectrum between T and T(*).In chapter 2, we first discuss the algebraic operators and idempotent operators in the range of the Aluthge transform,and the translation property of the Aluthge transform.We show that T is an algebraic operator if and only if T is. Subsequently, we give a necessary and sufficient condition for T to be an idempotent. When H is a finite dimensional Hilbert space, we prove that T is a normal operator if T satisfied translation property T + λ = T + λ, for all A in the complex plane.In chapter 3,we discuss the closed and dense properties of the range R(△) — {T : T ∈ B(H)} of △. We prove that R(△) is neither closed nor dense in B(H). However R(△) is strongly dense if H is infinite dimensional.