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On the Range of the Aluthge Transform

Author: LiZe
Tutor: JiGuoXing
School: Shaanxi Normal University
Course: Basic mathematics
Keywords: Aluthge transform Polar decomposition Algebraic operator Idempotent operator Translational nature Range
CLC: O177
Type: Master's thesis
Year: 2006
Downloads: 15
Quote: 0
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Abstract


Operator theory in the 20th century , due to its wide application in mathematics and other sciences , in the first three decades of the 20th century has been a great development . With the continuous development of this theory , this theory has now become a popular branch of modern mathematics . And quantum mechanics , linear systems and control theory , as well as some other important mathematical branch are closely linked and mutual penetration . This paper mainly of algebra in the range of Aluthge transform operator idempotent operator , and Aluthge transform translational properties . Also consider the Aluthge transform the range closed and thick . The details are as follows : The first chapter introduces some symbols to be used in this article , definitions, and after two chapters to use theorems . Section II , we introduce the spectrum , point spectrum , the approximate point spectrum idempotent operator , Aluthge transform * - Aluthge transform concepts such . Section III are given for some of the well - known theorem also given ( ? ) And ( ? ) ( * ) < / Sup> of several relations between the spectrum . Chapter II First transform the range of B (H) on Aluthge the algebraic operators son idempotent operator and Aluthge transform the translational nature . Prove the the operator T Aluthge transform ( ? ) Is a necessary and sufficient condition for algebraic operator T is algebraic operators , given ( ? ) Is a necessary and sufficient condition for idempotent operator . When H is a finite -dimensional Hilbert space , a to operator T Aluthge transform translational nature of ( ? ) = ( ? ) Λ, ( ? ) Λ ∈ C , then T is a normal operator . The third chapter we consider the Aluthge transform the range closed and thick . R (Δ) in the B (H) is neither closed nor fused However , if H is infinite dimensional , R (Δ) in the B (H) is strong thick .

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