Research On Several Operator Inequalities And Its Applications

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. Since norm equalities and inequalities of operators contain lots elementary properties of operators, many scholars have studied many classes of norm equalities and inequalities of operators, among which some gave new inequalities, while some generalized and then improved classical inequalities. In addition, many applications of inequalities arise. Then all kinds of inequalities are united ownning to the common resources. An operator matrix is a matrix whose entries are bounded operators on the correponding Hilbert spaces. It is operator matrix, the convexity and concavity of function, the spectral decomposition of operator and functional calculus that the author is using to generalize some classical inequalities and then to give series of important operator inequalities and norm inequalities.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 inertia index, numerical range , the singular value of operator, etc. Subsequently we give some well-known theorems such as the spectral theorem, spectral mapping theorem and polar decomposition theorem .In chapter 2, we utilize the properties of convex function and seminorm, then generalize the Bohr inequality. The classical Bohr inequality , i.e., if a,b are all complex numbers , p,q > 1 and 1/p + 1/q = 1, then |a - b|2 ≤ p|a|² + q|b|2. It is shown that if X be a vector space with a seminorm u(·), if pi > 1(i=1, 2,..., n) withthen for all x₁,x₂∈X, which greatly generalizes the classical Bohr inequality.In chapter 3, considering the imitate relation of norm inequality and the singular value of operator, we begin the chapter with researching the property of singular value of compact operator. Thus some important norm inequalities are given, for example, Clarkson inequalities. In this chapter, we introduce the n roots of unity, i.e., ω₀,ω₁,...,ω_n-1, where ωj= e²Ï€ij/n , 0 ≤j ≤ n-1. Using the properties of n roots...