2021/10/20 by Klobouk, Abel, Varela, Alejandro
#15A60 #15B57 #52A20 #65F35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2110.10584
For a given complex finite dimensional subspace S of ℂn and a fixed basis, we study the compact and convex subset of (ℝ≥ 0)n that we call the moment of S mS= convex hull (\|s|2∈ℝn≥ 0: s∈ S \wedge ‖s‖=1\ ) ≃ \ Diag(Y) ∈ Mnh(ℂ):Y≥ 0, tr(Y)=1, PS Y PS=Y\ where |s|2=(|s1|2,|s2|2,…,|sn|2). This set is relevant in the determination of minimal hermitian matrices (M∈ Mhn such that ‖M+D‖≤ D for every diagonal D and ‖ ‖ the spectral norm). We describe extremal points and curves of mS in terms of principal vectors that minimize the angle between S and the coordinate axes. We also relate mS to the joint numerical range W of n rank one n× n matrices constructed with the orthogonal projection PS and the fixed basis used. This connection provides a new approach to the description of mS and to minimal matrices. As a consequence the intersection of two of these joint numerical ranges allow the construction or detection of a minimal matrix, a fact that is easier to corroborate than the equivalent condition for moments. It is also proved that mS is a semi-algebraic set equal to the intersection of the mentioned W with a hyperplane and whose generated positive cone coincides with that of W.