The higher rank numerical range is useful for constructing quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a1, . . . , an, then its rank-k numerical range Λk(A) is the intersection of convex polygons with vertices aj1 , . . . , ajn−k+1 , where 1 ≤ j1 < · · · < jn−k+1 ≤ n. In this paper, it is shown that the higher rank numeri...

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...

Here we give a closure free description of the higher rank numerical range normal operator acting on separable Hilbert space. This generalizes result Avendaño for self-adjoint operators. It has several interesting applications. We show using Durszt's example that there exists contraction T which intersection ranges all unitary dilations contains as proper subset. strengthen and generalize Wu by...