Matricial inner products and pointed cones of hermitian-preserving linear transformations Original Research Article

Let σ be a conjugate-homogeneous reflector on a vector space V (over R or C) with image a pointed cone contained in specσ (V). A mapping on V × V whose range is contained in image which generalizes the usual inner product properties is called a vectorial inner product. A certain family of these vectorial inner products on matrices (which we call matricial inner products) is used to generate a set of pointed cones in the ambient space of hermitian-preserving linear transformations. Some basic results on these cones [including Π(PSD), Π(PSD)*, and image] and on the partial orders that they induce are given.