A note on the existence, uniqueness and symmetry of par-balanced realizations

We give a proof of the realization theorem of N.J. Young which states that analytic functions which are symbols of bounded Hankel operators admit par- balanced realizations. The main tool used in this proof is the induced Hilb ert spaces and a lifting lemma of Krein-Reid-Lax-Dieudonne. Alternatively o ne can use the Loewner inequality. A short proof of the uniqueness of par-b alanced realizations is included. As an application, it is proved that par- balanced realizations of real symmetric transfer functions are J-self-adjoi nt.