Research On Several Kinds Of Operator Inequalities

Matrix inequalities have a pivotal position in matrix theory,and they are ex-tended to infinite-dimensional cases to generate operator inequalities in Hilbert s-pace,which has become an important branch of operator theory.In this thesis,we mainly study the related problems of operator partial order,sector matrix in-equality,Young type operator inequality,Rotfel'd type matrix inequality,functional matrix inequality,Harnack type matrix inequality and other related issues.This thesis mainly contains four chapters.In chapter 1,we mainly introduce the research background and some basic definitions of operator inequalities.Moreover,we briefly introduce some of the main conclusions of this thesis.In chapter 2,in terms of operator partial order,first,we give the Kantorovich inequality of the positive operator with more optimized coefficients and the reverse weighted arithmetic geometric mean inequality of the positive operator,extending these inequalities to accretive operators which are more general than positive op-erators.Second,we use two different methods to obtain a Young type operator inequality,and then we give two related p(p>0)-th power inequality.Moreover,we give some inequalities involving the Hilbert-Schmidt norms.Finally,we give a new function on the unitary invariant norm of matrices.This function is log-convex in each variable.Some related inequalities for unitarily invariant norms are also given.In chapter 3,based on the theory of majorization,we prove some singular value inequalities,unitary invariant norm inequalities,and determinant inequalities of ma-trices whose numerical ranges are contained in a sector,generalizing the Garg-Aujla type inequalities for positive semi-definite matrices.Next we give some generaliza-tions of Rotfel'd type inequality for sector matrices.In the end we present some sector matrix inequalities related to Heinz mean.In chapter 4,we give new upper and lower bounds on Harnack-type inequal-ities for matrices in terms of eigenvalues and singular values,refining the existing results.We also discuss and compare the upper and lower bounds obtained by d-ifferent arguments.In addition,we propose some questions about spectral norms and eigenvalues.