Computation of realizations of 2D cone-systems

A new notion of a 2D (P,Q,V)-cone system is introduced. Necessary and sufficient conditions for the existence of a realization of a given proper transfer matrix are established. A procedure is proposed for computation of a realization of a given transfer matrix of the 2D (P,Q,V)-cone system. It is shown that there exists a realization of the transfer matrix T(z
1
,z
2
) of a 2D (P,Q,V)-cone system if and only if there exists a positive realization of T̄ (z
1
,z
2
)=VT(z
1
,z
2
)Q
⁻¹
, where V, Q and P are nonnegative matrices generating the cones V, Q, and P.