Cones in the space of block-matrices

Abstract

In the space $$\mathcal{M}$$ $$\mathcal{m}$$ ( $$\mathcal{M}$$ $$\mathcal{n}$$ ) of $$\mathcal{m}$$ X $$\mathcal{m}$$ block-matrices with entries in Mn three natural cones related to positivity are introduced. In this paper several known results related to positivity: dilation theorem, truncated moment theorems and the Fejér–Riesz theorem on factorization of positive polynomials, are applied to derive interrelationships among those three cones. At the end of the paper, some form of matrix Schwarz inequalities is presented. The paper is largely of expository character.