Operator Positivstellens\"atze for noncommutative polynomials positive on matrix convex sets

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) $L(X)=A_0\otimes I+\sum_{j=1}^g A_{j}\otimes X_j\succeq 0,$ where $A_j$ are self-adjoint linear operators on a separable Hilbert space, $X_j$ matrices and $I$ is an identity matrix. If $A_j$ are matrices, then $L(X)\succeq 0$ is called a linear matrix inequality (LMI) and $D_L$ a free spectrahedron. For monic LMIs, i.e., $A_0=I$, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCollough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion $D_{L_1}\subseteq D_{L_2}$ from monic LMIs to monic LOIs. As a corollary one obtains the description of a polar dual of a free Hilbert spectrahedron $D_L$ and its projection, called a free Hilbert spectrahedrop. Using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials $F$ positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fej\'er-Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials $F$. Finally, focusing on the algebraic description of the equality $D_{L_1}=D_{L_2}$, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs.