Matrix inequalities

Abstract

The theory related to the generalization of Furuta s inequality and inequalities related to indefinite forms of matrices has been developed in the thesis. The first chapter discusses a literature review of research work done on desirable problems in previous and recent years. Basic notations along with the basic concepts related to positive definite matrices, positive linear maps, operator monotone functions, operator convex functions and operator means are covered in Chapter 2. In Chapter 3, we characterise chaotic order using an arbitrary operator mean and#963;. The mean theoretic generalisation of Furuta s inequality is constructed as a generalisation of Furuta s inequality. In addition to generalised Furuta s inequality, generalized Kadison-type inequality for operator means is investigated which yields asymmetric Kadison inequality and Choi-Davis type inequalities induced by power mean. Basic results and definitions related to the indefinite form of matrices are covered in Chapter 4. In Chapter 5, we introduced the concept of J-means and#963;and#119891;, which is denoted by Aand#963;and#119891;B of J-selfadjoint matrices with spectra in (0, and#8734;) having properties homogeneity, monotonicity, transpose, duality and adjoint. The theory of power monotonicity of J-means is established and as an application indefinite version of Ando-Hiai inequality is proved. In addition, an indefinite version of arithmetic-harmonic mean inequality is also obtained. newline