Hill representations for *-linear matrix maps

2021/03/26 by ter Horst, Sanne, Naude, Alma
#15A69 #15B48 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2103.14500

Abstract

In a paper from 1973 R.D. Hill studied linear matrix maps L:ℂ^{q × q}→ℂ^{n × n} which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L(V^*)=L(V)^*, via representations of the form L(V)=∑_k,l=1^m ℍ_kl A_l V A_k^*, V∈ℂ^{q × q}, for matrices A₁,…,A_m ∈ℂ^{n × q} and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A₁,…, A_m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix ℍ=[ℍ_kl] explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of *-linearity on the linear map L.

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