  In this work we study the problem of writing a Hermitian polynomial as a
Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea
of Putinar-Scheiderer to obtain necessary matrix positivity conditions for
Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals.
We show that the conditions are sufficient for a class of examples making a
connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms.
The work fits into the larger themes of Hermitian versions of Hilbert's 17-th
problem and characterizations of positivity.
