Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.