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 noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...

In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural genera...

in this paper, we introduce the notions of c-numerical range and c-spectrum of matrix polynomials. some algebraic and geometrical properties are investigated. we also study the relationship between the c-numerical range of a matrix polynomial and the joint c-numerical range of its coefficients.

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...

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...