arXiv subjects
Publications and source records attributed to John Maroulas.
The $k$-rank numerical range $Λ_{k}(A)$ is expressed via an intersection of a countable family of numerical ranges $\{F(M^{*}_νAM_ν)\}_{ν\in\mathbb{N}}$ with respect to $n\times (n-k+1)$ isometries $M_ν$. This implication for $Λ_{k}(A)$ provides further elaboration of the $k$-rank numerical radii of $A$.
In this article the well known "Perron-Frobenius theory" is investigated involving the higher rank numerical range $Λ_{k}(A)$ of an irreducible and entrywise nonnegative matrix $A$ and extending the notion of elements of maximum modulus in $Λ_{k}(A)$. Further, an application of this theory to the $Λ_{k}(L(λ))$ of a Perron polynomial $L(λ)$ is elaborated via its companion matrix $C_{L}$.
The notion of the higher rank numerical range $Λ_{k}(L(λ))$ for matrix polynomials $L(λ)=A_{m}λ^{m}+...+A_{1}λ+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further, the sharp points of $Λ_{k}(L(λ))$ are defined and their relation to the numerical range $w(L(λ))$ is presented. A connection of $Λ_{k}(L(λ))$ with the vector-valued higher rank numerical range $Λ_{k}(A_{0},..., A_{m})$ is also discussed.
A presentation of numerical range for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Then we are extended to the treatment of rank-k numerical range.
In this paper necessary and sufficient conditions are stated for the Craig-Sakamoto equation det(I-sA-tB) = det(I-sA)det(I-tB), for all scalars s, t. Moreover, spectral properties for A and B are investigated.