The Properties Of The Least Normal Compact Operator

In operator theory,the problem can be solved more quickly by using approxi-mation method.In linear flag manifolds,the minimal matrix is used to approximate the minimal curve.Different operators have different properties and the correspond-ing minimal operators also have different properties and characteristics.For the best approximation of hermitian operators,positive operators and unitary operators has many definite results.In recent years,the necessary and sufficient conditions for the minimal hermitian matrix and the minimal compact hermitian operator have been given.In this paper,further research has been made on the existing conclusions:some properties of the minimal matrix have been extended to the compact normal operator,and some new conclusions have been obtained as follows.The content and structure of this paper are divided into three chapters:In Chapter 1,we mainly introduce the basic concepts and preliminary theorems used in this paper,which makes the article easier to read.In Chapter 2,we mainly give some properties and lemmas about minimal op-erators,and extend the spectrum of minimal compact hermitian operators.If a normal operator A?K(H)is minimal,then there exists 9 ?[0,2?)such that ?A?ei? ??(A)and-?A?ei??(A).The main conclusions of this paper are given as follows:Let A?K(H)be a non-zero normal operator.Then A is minimal if and only if there exists X?T(H)\{0}such that ?(X)=0 and AX=|X|?A?.In Chapter 3,we give the characterization of the concrete minimal matrix.Let M?M2h(C).M is trace-norm minimal if and only ifM=((?))where a,c ? R with a2?c2 and ??[0,2?).