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...

Abstract. It is known that the i-th order laminated microstructures can be resolved by the k-th order rank-one convex envelopes with k ≥ i. So the requirement of establishing an efficient numerical scheme for the computation of the finite order rank-one convex envelopes arises. In this paper, we develop an iterative scheme for such a purpose. The 1-st order rank-one convex envelope R1f is appro...

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form [ aI cT dT ∗ bI ] . It reduces to a quadratic operator if d = 0. In this paper, spectra, norms, and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a...

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.–D. Choi et al., Rep. Math. Phys. 58, 77 (2006); Lin. Alg. Appl. 418, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to o...