# Linear preservers of operators with non-negative generalized numerical ranges

Abstract (Summary)

(Uncorrected OCR) Abstract of thesis entitled 'LINEAR PRESERVERS OF OPERATORS WITH NON-NEGATIVE GENERALIZED NUMERICAL RANGES' Submitted by Kong Chan for the degree of Master of Philosophy at the University of Hong Kong in August, 1999 Let Mn be the set of all n x n complex matrices, Hn the set of all n x n hermitian matrices and Un the group of all unitary matrices in Mn. Given c = (ci, ...,cn)* G M71, denote by [c] the diagonal matrix with diagonal entries Ci,..., cn. The c-numerical range of A in Mn is defined by WC(A) = {ti([c]UAU*) : U G Un}. Let Pc be the set of all matrices in Hn with non-negative c-numerical range, in symbol, Pc = {AeHn:Wc(A)c[0,oo)}. The set Pc is a cone in Hn. In this thesis, we study linear operators T on Hn which leave Pc invariant. Such T will be called linear preservers of Pc. They are closely related to linear preservers of the c-numerical range (or the c-numerical radius). Altogether, there are 6 chapters. In Chapter 0, we give a survey on linear preservers of different generalized numerical ranges or their numerical radii. In Chapter 1, extreme directions of Pc are completely characterized when Ym=i ci 7^0 and not all c^'s are equal. A matrix P G Pc is called an extreme direction of Pc if whenever P = Pi + P2 for some nonzero P\,P<2, G Pc, then P\ and P2 are positive multiples of P. Extreme directions are instrumental in understanding the cone Pc. Using the knowledge of extreme directions obtained in Chapter 1, we obtain a description of linear preservers of Pc. This is done in Chapter 2. Let (?, ?) denote the inner product on Hn defined by (A, B) = tr(AB*) for all A,B e H, Suppose T : Hn ??Hn is a linear operator. The dual operator of T is the unique linear map T* : Hn ??Hn that satisfies (T(A),B) = {A,T*(B)) for all A,B G Hn. When rc is a norm, it is shown in Chapter 3 that T(PC) = Pc if and only if T*(?C)) = ?C), where ?C) = {XUCU* : X > 0, U G Un} and C is any hermitian matrix with eigenvalues ci,... ,cn. Consequently, linear preservers of ?C) are completely characterized. In Chapter 4, we study the infinite dimensional version of the linear preserver problem. Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. The cone Pc can be defined analogously as in the finite dimensional case. We study linear preservers of Pc when all ci,... ,cn are nonnegative. If c, d are different vectors in R", we have two different cones Pc and Pa- In Chapter 5, we study linear mappings T on Hn such that T(PC) ?Pj. Nontrivial results are obtained when there exist v,v' G K. such that c ?vd + v'e, where e G M.n has all entries equal to one.
Bibliographical Information:

Advisor:

School:The University of Hong Kong

School Location:China - Hong Kong SAR

Source Type:Master's Thesis

Keywords:linear operators matrices

ISBN:

Date of Publication:01/01/2000