The Generalization Of Matrix Inequality Under Unitary Transformation

Recently more and more researchers have begun to focus on inequality in almost all branches of science and mathematics, in which the inequality plays a very important role. Researching on it has been very active and attractive. As an important part of the inequality, the content of the matrix inequality is also very rich. This paper considers the generlization of matrix inequality by unitary transformation.This paper is divided into three chapters:In the first chapter, we consider norm inequalities for an accretive-dissipative op-erator matrix. An operator on a Hilbert space is accretive-dissipative if its real and imaginary part are both positive. The inequalities obtained extend some known results.In the second chapter, we consider eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks. be a positive semidefinite block matrix with square matrices M and N of the same order and denotebe i=(?).The main results are the following eigenvalue majorization inequalities: for any complex number z, λ(H)<1/2(λ[M+N+(z2/|z|2K*+z2|z|2K)+1/2(λ[M+N-(z2/|z|2K*+z2/|z|2K)](?)0). If, in addition, K is Hermitian, then for any real number r ∈[-2, 2], λ(H)<1/2(λ(M+N+rK)(?)0)+1/2(λ(M+N-rK)(?)0), while if K is skew-Hermitian,then for any real numberr ∈[-2,2], λ(H)<1/2(λ(M+N+riK)(?)0)+1/2(λ(M+N-riK)(?)0), where 0 is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz.In the third chapter, we present a Jensen operator inequality for superquadratic functions. In particular we extend the inequality f(<Ax,x>)≤<f(A)x,x>. for superquadratic functions and give some applications for our result.