Research On Spectral Assignment And Nth Roots Of Operator Matrix As Well As Norm Inequalities Of Operators

The study of operator theory begin 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. A operator matrix is a matrix whose entries are bound operators on the correponding Hilbert spaces. The parital operator matrix is a operator matrix whose some entries fixed and others are unknown. Operator completion problem concentate on the impact of the unknown entries on the whole operator matrix. In this article we discuss the spectral assignment problem based on the controllable operator pairs and admissible operator pairs which are origined from systems theory. Meanwhile we study the uniqueness of operator nth roots by means of operator matrix. 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. Two spectial classes of norm inequalities are studied In the artical.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 numerical range , convex set, extreme point, maximal partial isometry etc. Subsequently we give some well-known theorems such as the Krein-Milman theorem, polar decomposition theorem and spectral theorem.Chapter 2 we discuss spectral assigiment problem based on the controllable operator pairs and admissible operator pairs. We get that if and only if operator pair (A, B) is controllable, if and only if (A, B) is controllable . If operator pair (A, B) is admissible and dimR(B) = ∞, denote θ(A, B) := { λ ∈ C : (A - A, B) not right invertible } then we haveand , respectively.Chapter 3 we analysis the uniqueness and existence of operator square roots in finite dimensional space, futhermore we study the operator nth roots in infinite dimensional space and draw the conclusions by restrict spectra and numerical range of operators in a subset of complex plane. The results cover the conclusions of Charles R. Johnson , Kazuyoshi Okubo.Chapter 4 we characterise the set {B ∈ A : there is a a > 0 such that and give a necessary and sufficient condition for operator pair (A, B) satisfied ||A + B|| + ||A - B|| = 2||A||. In the second section of this chapter we give the definition of orthogonality of operators, which is a extention of the concept in Hilbert space. A is orthogonal to B if for every scalar . K. Li, R. Bhatia and P. Semrl concertrate on finte dimension space , in this article, we characterise the orthogonality of two operators in infinite dimension Hilbert space, the proof is different with R.Bhatia.