Abstract 2

We consider the Riccati equation

A^*X + XA - C + XBX = 0,

which is a quadratic equation for possibly unbounded operators acting on a Hilbert space. The question, whether there is a solution X, is non-trivial since the operators above need not commute.

There are two examples where the existence of a solution is known. One is from control theory, the other involves the so-called Neumann-to-Dirichlet map.

By using operator theoretic methods, the existence of a solution in a general setting can be shown. We will in fact construct all possible solutions of the Riccati equation and derive some of their properties.

However, it turns out that the needed assumptions are not suitable for the case of the Neumann-to-Dirichlet map. In this situation, the original proof of existence can be generalised to a more abstract setting.

	Christian Wyss
	An operator Riccati equation

We consider the Riccati equation A^*X + XA - C + XBX = 0, which is a quadratic equation for possibly unbounded operators acting on a Hilbert space. The question, whether there is a solution X, is non-trivial since the operators above need not commute. There are two examples where the existence of a solution is known. One is from control theory, the other involves the so-called Neumann-to-Dirichlet map. By using operator theoretic methods, the existence of a solution in a general setting can be shown. We will in fact construct all possible solutions of the Riccati equation and derive some of their properties. However, it turns out that the needed assumptions are not suitable for the case of the Neumann-to-Dirichlet map. In this situation, the original proof of existence can be generalised to a more abstract setting.