We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior.

Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are multiplicative sections of a line bundle on a real compact Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface. 1991 Mathematics Subject Classification. 47A48,47B32,46E22. Key words and phrases. compact Riemann surface, Beurling-Lax theorem, de Branges Rovnyak spaces, operator vessels, joint transfer function. The first author thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research.

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