A Study of Preserver Problems

This dissertation, "A Study of Preserver Problems" by Nung-sing, Sze, 施能聖, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled A STUDY OF PRESERVER PROBLEMS Submitted by Nung-Sing Sze for the degree of Doctor of Philosophy at The University of Hong Kong in April 2005 Preserver problems concern the characterization of maps between matrix spacesleavinginvariantcertainfunctions, subsets, andrelations. Asolutionof a preserver problem is a structural description of the preserver. Traditionally, linearity was assumed for preservers acting on linear spaces. A classical exam- ple, due to Frobenius, was that linear maps φ on complex matrices satisfying det(φ(A)) = det(A) for all complex matrices A have the form A7→MAN or A7→MAN for nonsingular matrices M and N with det(MN) = 1. Recently, the more general and challenging problem of characterizing preservers not necessarily linear was studied. In this thesis, nine preserver problems were solved including some of the challenging types. New techniques were developed in the course. LetS andT betwosetsofmatriceswhosenumericalrangebelongtocertain prescribed subsets. Linear maps φ satisfying φ(S) = T were characterized. The maps usually have the form ∗ ∗ A7→μT AT or A7→μT ATfor some complex numberμ and invertible / unitary matrixT. By considering differentS andT, earlier results of Schneider and Marcus were obtained. A difficulty in characterizing non-surjective linear preservers from one ma- trix space to another is that such maps may not exist. Three problems along thisdirectionwerediscussedinthisthesis. First, linearmapsfromthespaceof 0 0 nn matrices to that ofn n matrices transforming thek-numerical ranges tok-numerical ranges were studied. It was shown that such maps exist if and 0 0 only if n is a multiple of n, and k is a multiple of k or n-k. If such a map exists, it has the form ∗ t t A7→U h(A)(+)---(+)h(A)(+)h(A) (+)---(+)h(A) U, whereU isunitary, andh(A) =Aorh(A) = [(trA)I -(n-k)A]/k. Complete 0 0 descriptions were also given for linear maps from nn matrices to n n matrices transforming theH -unitary group to theH -unitary group, and also 1 2 0 0 linear maps frommn matrices tom n matrices transforming the Ky Fan k-norm to Ky Fan k-norm. Multiplicative preservers of (semi)groups of matrices are just (semi)group homomorphisms. There were not much results in the literature. In this the- sis, automorphisms of various (semi)groups of nonnegative matrices including stochastic matrices, monomial matrices, positive matrices, and etc, were char- acterized. The rank one preserver is an important tool in the study of preserver prob- lems. Many preserver problems can be solved using rank one preservers. Ad- ditive rank one preservers from the space of mn matrices to that of rs matrices were characterized. This result is a key steps in the subsequent char- acterization of distance preserving maps on rectangular matrices with respect to general unitarily invariant norms. Intheabsenceofanadditivity/linearity/multiplicativityassumption, apre-server can be quite arbitrary. In this thesis, maps φ satisfying only F(φ(A)φ(B)) =F(AB) for all matrices A and B were studied. The maps have the form -1 -1 t A7→S AS or A7→S AS fornonsingularS whenF isthespectrum. Acompletestructuraltheoremwas also given whenF is a unitary similarity invariant norm. Different approaches were required as for the spectrum, F(A) = 0 does not imply A = 0. DOI: 10.5353/th_b3160799 Subjects: Linear operators Matrices