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.
