Characterizations of majorization on certain spaces and their linear preservers

Abstract

In this thesis, we study characterizations of majorization on certain spaces and formulate the structure of linear preservers and strong linear preservers of the same. In this direction, we first introduce a novel notion, cone type majorization on Rn and establish its relation with classical majorization and weak majorization. We give a characterization of this majorization as well as produce a Hardy-Littlewood-P olya type theorem for cone type majorization. We also obtain the structure of linear preservers and strong linear preservers of cone type majorization. Using the notions of Hadamard product and circulant matrices, we introduce and study Hadamard circulant majorization of Mn. Further, we obtain the structure of linear preservers of Hadamard circulant majorization. We also prove that a linear operator T on Mn is invertible and preserves Hadamard circulant majorization if and only if T strongly preserves Hadamard circulant majorization. Ordering an absolutely summable sequence with infinitely many positive and infinitely many negative terms, and comparing the inequalities of partial sums of two such sequences is a daunting task. To overcome this, we propose a reformulation of the notion of majorization on l1 and therein investigate properties of this majorization. We prove Schur-Horn type and Hardy-Littlewood-P olya type theorems for sequences in l1. We also give a characterization of this majorization using convex functions. Finally, we propose a notion called p-weighted majorization on Rn. We adopt the Lorenz technique as a tool to introduce a type of stable vector for a set of vectors in Rn. We prove the existence of such a stable vector for any given subset of R2 and provide sufficient conditions for the existence of such a stable vector for a subset of Rn with n 3. Keywords: Majorization, cone type majorization, Hadamard product, circulant matrix, linear preserver, doubly stochastic matrix, Hadamard circulant majorization, circulant doubly stochastic matrix, self-adjoint operator, convex functions, Lorenz curve, weighted majorization, stable vector.