Abstract. In the space ${\msbm M}_m({\msbm M}_n)$ of $m\times m$ block-matrices with entries in ${\msbm M}_n$ 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.

AMS Subject Classification (1991): 47L07, 47A20, 47A68, 47B35, 81P16

Keyword(s): cones of matrices, duality of cones, separability and decomposability, dilation theorem, matrix truncated moment theorems, matrix Fejér--Riesz theorem, matrix Schwarz inequality

Received November 23, 2012, and in revised form March 15, 2013. (Registered under 9/2013.)