A general realization theorem for matrix-valued Herglotz–Nevanlinna functions

New special types of stationary conservative impedance and scattering systems, the so-called non-canonical systems, involving triplets of Hilbert spaces and projection operators, are considered. It is established that every matrix-valued Herglotz–Nevanlinna function of the formV(z)=Q+Lz+∫R1t-z-t1+t2dΣ(t)can be realized as a transfer function of such a new type of conservative impedance system. In this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if L = 0. It is also shown that V(z) with an additional condition (namely, L is invertible or L = 0), can be realized as a linear fractional transformation of the transfer function of a non-canonical scattering F+-system. In particular, this means that every scalar Herglotz–Nevanlinna function can be realized in the above sense.Moreover, the classical Livšic systems (Brodskii˘–Livšic operator colligations) can be derived from F+-systems as a special case when F+ = I and the spectral measure dΣ(t) is compactly supported. The realization theorems proved in this paper are strongly connected with, and complement the recent results by Ball and Staffans.