Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Matrices

This dissertation, "Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Matrices" by 施能聖, Nung-sing, Sze, 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 "LINEAR OPERATORS PRESERVING GENERALIZED NUMERICAL RANGES AND RADII ON CERTAIN TRIANGULAR MATRICES" Submitted by Nung-Sing Sze for the degree of Master of Philosophy at The University of Hong Kong in July 2002 LetC be annn matrix. TheC-numerical range ofA∈M is defined by W (A) ={tr(CUAU ): U ∈U } C n where U is the set of all nn unitary matrices, and the C-numerical radius of A is the quantity r (A) = max{z: z ∈W (A)}. C C Many researchers have studied the structure of linear operators φ on M that preserve the C-numerical range and radius. These are linear operators φ sat- isfying W (φ(A)) =W (A) for every A∈M C C n or r (φ(A)) =r (A) for every A∈M C C n respectively. Some previous results of C-numerical range and C-numerical radius preservers on M and on certain triangular matrices, T(n, ..., n ), n 1 k which is the subalgebra of M of all k k block triangular matrices were described. A matrix A ∈ T(n, ..., n ) if and only if A = (A ) where A ∈ 1 k pq pp M for 1q. n pq pCheung and Li (2001) studied C-numerical range and radius preservers, when C is hermitian, on T(n, ..., n ) by using the duality technique. An 1 k alternative proof to this result was given without using dual operators. A C-numerical range preserver on T(n, ..., n ) was extended to a C-numerical 1 k range preserver on M and a result of Li and Tsing (1988) was employed to classify such a preserver. Using the result on M, they determined the structure of preservers on T(n, ..., n ). An important lemma was proven 1 k which yields that anyC-numerical radius preserver must map identity matrix to a unit scalar matrix when trC is nonzero. This lemma also holds when C is normal. The structure of C-numerical range and radius preservers was determined when C is normal on T(n, ..., n ). For C-numerical radius preservers, the 1 k same technique as for hermitian C was used and again it was found that, whenthetraceofC isnonzero, itisjustaunitmultipleofaC-numericalrange preserver. For C-numerical range preserver, the image of D(n, ..., n ) was 1 k first classified, where D(n, ..., n ) denotes the subalgebra of M consisting 1 k n of k k block matrices such that all A = 0 whenever p 6= q. The dual pq operator which preserves the unitary orbit was then studied, and a special relation between the diagonal entries of a matrix and those of its image was found. Usingthisrelation, thestructureofthedualoperatorwascharacterized, hence the C-numerical range preserver problem was settled. DOI: 10.5353/th_b2979773 Subjects: Linear operators Triangularization (Mathematics) Matrices