Let ‎X be an ‎‎n-‎‎‎‎‎‎square complex matrix with the ‎Cartesian decomposition ‎‎X = A + i ‎B‎‎‎‎‎, ‎where ‎‎A ‎and ‎‎B ‎are ‎‎‎n ‎‎times n‎ ‎Hermitian ‎matrices. ‎It ‎is ‎known ‎that ‎‎$Vert X Vert_p^2 ‎leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)‎‎‎$, ‎where ‎‎$‎p ‎‎geq 2‎$‎ ‎and ‎‎$‎‎Vert . Vert_p$ ‎is ‎the ‎Schatten ‎‎‎‎p-norm.‎ ‎‎ ‎‎In this paper‎, this inequality and some of its improvements ...

It is shown that for $n \le 3$ the joint numerical range of a family commuting $n\times n$ complex matrices always convex; \ge 4$ there are two whose not convex.

We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so thatWk(A) is convex. It is shown that m can reach the upper bound 2k(n− k) + 1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized...

In this note we characterize polynomial numerical hulls of matrices $A in M_n$ such that$A^2$ is Hermitian. Also, we consider normal matrices $A in M_n$ whose $k^{th}$ power are semidefinite. For such matriceswe show that $V^k(A)=sigma(A)$.

For a subspace S of Cn and fixed basis, we study the compact convex set mS=convexhull {|s|2∈R≥0n:s∈S ‖s‖=1}≃{Diag(Y)∈Mnh(C):Y≥0,tr(Y)=1,PSYPS=Y}that call moment S, where |s|2=(|s1|2,|s2|2,…,|sn|2). This is relevant in determination minimal hermitian matrices (M∈Mnh such that ‖M+D‖≤D for every diagonal D spectral norm ‖⋅‖). We describe extremal points certain curves mS terms principal vectors mi...

In this talk I will discuss some instances in quantum computing where numerical range techniques arise. I will also try to formulate some open problems. Elliptical range theorems for generalized numerical ranges of quadratic operators Speaker Chi-Kwong Li, William and Mary, [email protected] Co-authors Yiu-Tung Poon, Iowa State University, [email protected]; Nung-Sing Sze, University of Connect...