Searcharxiv⌕ Search

arXiv subjects

Kuo-Zhong Wang

Publications and source records attributed to Kuo-Zhong Wang.

10 recordsLinked to original sources
arXiv2026

Numerical radius of certain two-by-two block matrices

We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T = \begin{pmatrix} A & B \\ 0 & 0 \end{pmatrix}$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U = \begin{pmatrix} U^*AU & B \\ 0 & 0 \end{pmatrix}$ over all unitary operators $U$. Finally, we study matrices $T \in M_n$ satisfying $\|T^m x\| = \|T^m\| = \|T\|$ for a unit vector $x$ and all positive integers $m$. This analysis connects these matrices to the aforementioned block form and provides a counterexample to the conjecture that if $\|T^k\| = \|T\|$ for all $k \ge 1$, then some power of the matrix has a direct summand that is a scalar multiple of an idempotent.

math.FA↗
arXiv2024

Norm of an operator with numerical range in a sector

We refine a recent result of Drury concerning the optimal ratio between the norm and numerical radius of a bounded linear operator $T$ with numerical range lying in a sector of a circular disk. In particular, characterization is given to the operators attaining the optimal ratio, and properties of such operators are explored.

math.FA↗
arXiv2016

Numerical Ranges of the product of Operators

We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of $AB$, and conditions on $A$ and $B$ for the set equality holding are obtained. The results cover the case when $A$ and $B$ are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections.

math.FA↗
arXiv2016

Optimal Bounds on Functions of Quantum States under Quantum Channels

Let $ρ_1, ρ_2$ be quantum states and $(ρ_1,ρ_2) \mapsto D(ρ_1, ρ_2)$ be a scalar function such as the trace norm, the fidelity, and the relative entropy, etc. We determine optimal bounds for $D(ρ_1, Φ(ρ_2))$ for $Φ\in \mathcal{S}$ for different class of functions $D(\cdot, \cdot)$, where $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.

quant-ph↗
arXiv2016

Product of two positive contractions

Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) projections. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is written to check the condition and construct the two positive projections whose product equal to the given matrix, if they exist.

math.RA↗
arXiv2015

Minkowski product of convex sets and product numerical range

Let $K_1, K_2$ be two compact convex sets in $\mathit{C}$. Their Minkowski product is the set $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$. We show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk. Examples for $K_1$ and $K_2$ are given so that $K_1$ and $K_2$ are triangles (including interior) and $K_1K_2$ is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.

math.MG↗
arXiv2014

The spectrum of the product of operators, and the product of their numerical ranges

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ σ(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators $A$ so that equivalence conditions hold.

math.FA↗
arXiv2013

Numerical Radii for Tensor Products of Matrices

For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\|A\|=\|A^k\|=1$ for some $k$, $1\le k<\infty$, then $w(A)\ge\cos(π/(k+2))$, and, moreover, the equality holds if and only if $A$ is unitarily similar to the direct sum of the $(k+1)$-by-$(k+1)$ Jordan block $J_{k+1}$ and a matrix $B$ with $w(B)\le\cos(π/(k+2))$, and (3) if $B$ is a nonnegative matrix with its real part (permutationally) irreducible, then $w(A\otimes B)=\|A\|w(B)$ if and only if either $p_A=\infty$ or $n_B\le p_A<\infty$ and $B$ is permutationally similar to a block-shift matrix \[[ {array}{cccc} 0 & B_1 & & & 0 & \ddots & & & \ddots & B_k & & & 0 {array} ]\] with $k=n_B$, where $p_A=\sup\{\ell\ge 1: \|A^{\ell}\|=\|A\|^{\ell}\}$ and $n_B=\sup\{\ell\ge 1 : B^{\ell}\neq 0\}$.

math.FA↗
arXiv2013

Zero-dilation Index of a Finite Matrix

For an $n$-by-$n$ complex matrix $A$, we define its zero-dilation index $d(A)$ as the largest size of a zero matrix which can be dilated to $A$. This is the same as the maximum $k$ ($\ge 1$) for which 0 is in the rank-$k$ numerical range of $A$. Using a result of Li and Sze, we show that if $d(A) > \lfloor 2n/3\rfloor$, then, under unitary similarity, $A$ has the zero matrix of size $3d(A)-2n$ as a direct summand. It complements the known fact that if $d(A)>\lfloor n/2\rfloor$, then 0 is an eigenvalue of $A$. We then use it to give a complete characterization of $n$-by-$n$ matrices $A$ with $d(A)=n-1$, namely, $A$ satisfies this condition if and only if it is unitarily similar to $B\oplus 0_{n-3}$, where $B$ is a 3-by-3 matrix whose numerical range $W(B)$ is an elliptic disc and whose eigenvalue other than the two foci of $\partial W(B)$ is 0. We also determine the value of $d(A)$ for any normal matrix and any weighted permutation matrix $A$.

math.FA↗