Hill representations for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e34" altimg="si12.svg"><mml:mo>∗</mml:mo></mml:math>-linear matrix maps

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