Wai-Shun Cheung

The C-numerical range of matrices is star-shaped

Let A C be n× ncomplex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by is always star-shaped with respect to star-center (tr A)(tr C)/ n. This result is equivalent to that the image of the unitary orbit {U ∗ AU:U} of A under any complex linear functional is always star-shaped.

Multiplicative preservers on semigroups of matrices

We characterize multiplicative maps φ on semigroups of square matrices satisfying φ(P)⊆P for matrix sets P, such as rank k (idempotent) matrices, totally nonnegative matrices, P0 matrices, M0 matrices, positive semidefinite matrices, Hermitian matrices, normal matrices, and contractions. We also characterize multiplicative maps φ satisfying φ(g(X))=φ(X) for various functions g on square matrices, such as the spectrum, spectral radius, numerical range, numerical radius, and matrix norms.

Isometries between matrix algebras

As an attempt to understand linear isometries between C-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m m complex matrices. If k n and V Mn ! Mk has the form X 7! UTX f.X/UV or X 7! UTX t f.X/UV for some unitary U; V 2 Mk and contractive linear map f V Mn ! Mk, then k.X/ kD kXk for all X 2 Mn . We prove that the converse is true if k 2n 1, and the converse may fail if k 2n. Related results and questions involving positive linear maps and the numerical range are discussed.

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

Let c = (c1, . . . , cn) be such that c1 ≥ · · · ≥ cn. The c-numerical range of an n×n matrix A is defined by

Elementary proofs for some results on the circular symmetry of the numerical range

Let W(A) be the numerical range of an n × n complex matrix A. Using algebraic geometry technique and a result of Kippenhahn, Anderson showed that if W(A) is contained in a circular disk, and W(A) contains more than n boundary points of the circular disk, then W(A) equals to the circular disk, and the center of the circular disk will be an eigenvalue of the matrix A. Many researchers have reproved and refined the result of Anderson. Very recently, Wu unified these results and proved the following statements for a matrix A of the form . 1. If W(A) is contained in a circular disk 𝒟 centered at α, and W(A) contains at least m + 1 boundary points of 𝒟, then W(A) = 𝒟. 2. If W(A) contains a circular disk 𝒟 centered at α, and the boundary of W(A) contains at least m + 1 boundary points of 𝒟, then the boundary of W(A) contains a circular arc which is the boundary of 𝒟. 3. If the boundary of W(A) contains at least 2m + 1 boundary points of a circular disk 𝒟 centered at α, then 𝒟 ⊆ W(A). Moreover, under any one of the three conditions, α is an eigenvalue of A with algebraic multiplicity larger than its geometric multiplicity. The proofs of Wu utilized the Bézouts theorem, Riesz-Fejér theorem, etc. In this note, short and elementary proofs using only simple properties of polynomials and continuous functions are given to Wus results. Furthermore, the results are extended to the higher numerical ranges of matrices.

On the boundary of rank-k numerical ranges

Using the properties of the polynomial , Cheung and Li provided an alternative proof of Wu’s result on the circular symmetry of the classical numerical range. In this article, we further extend the idea and generalize Wu’s result to higher rank numerical ranges.

Optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems

Abstract Let A be an adjacency matrix of a tree T with n vertices. Conditions are determined for the existence of a fixed permutation matrix P that maximizes the quadratic form x t P t APx over all nonnegative vectors x with entries arranged in nondecreasing order. This quadratic form problem is completely solved, and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix P that maximizes the largest eigenvalue of matrices of the form PDP t +A , over all real diagonal matrices D with nondecreasing diagonal entries. It is shown that there is a tree with six vertices for which neither of the problems has a solution, and all other trees with six or fewer vertices have solutions for both problems. By duality, the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form PDP t +A .

New pairs of matrices with convex generalized numerical ranges

In this article, we are going to search for matrices A and B such that their generalized numerical range is convex. More specifically, we consider and where and are . If then it is a convex set.

On the boundary of weighted numerical ranges

In this article, we introduce the weighted numerical range which is a unified approach to study the -numerical range and the rank numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of different sizes) overlap at sufficiently many points, then the two matrices share common generalized eigenvalues.

Maximizing the numerical radii of matrices by permuting their entries

Let be an complex matrix such that every row and every column has at most one non-zero entry. We determine permutations of the non-zero entries of so that the resulting matrix has maximum numerical radius. Extension of the results to operators acting on separable Hilbert spaces are also obtained. Related results and additional problems are also mentioned.