Research On Inequlities Of Unitarily Invariant Norms Of Operators And Idempotency Of Linear Combination Of Idempotents

Abstract The study of operator theory began in 20th century. Since it is used widely in mathematics and other sinentific branches, it got great development at the beginning of the 20th century. In this article, we study inequalities of unitarily invariant norms of operators, idempotency of linear combination of idempotents and the problem of operator pairs in Hilbert space. These problems are all red points on operator algebra and operator theory.This paper contains four chapters. Chapter 1 mainly introduces some notations, definitions and some well-known theorems. Firstly, we give some technologies and notations, and introduce the definitions of approximate point spectrum, positive operator, Moore-penrose inverse etc. Subsequently we give some well-known theorems such as the Lyapunov theorem, polar decomposition theorem and spectral theorem.In chapter 2 we generalize the results of [1]. That is, if Ai(i = 1,...,n) are positive operators on a separable complex Hilbert space and |||.||| is any unitarily invariant norm, thenwhere A_n+1 = A1.AndwhereIn chapter 3 By using the techniques of block operator matrices, we have characterise the idempotency of two and three idempotents. Especially, using the techniques of block operator matrices, we have obtained a complete characterizations of...