On generalized Friedrichs and Krein-von Neumann extensions and canonical systems

Let S be a densely defined and closed symmetric relation in a Hilbert space H with defect numbers (1, 1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so-called Q-function from the Nevanlinna class N. In this note we show to which subclasses N-gamma of N the Q-functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q-functions of the generalized Friedrichs and Krein-von Neumann extensions. A result of L. DE BRANGES implies that to each function Q is an element of N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh-Weyl coefficient of the two-dimensional canonical system Jy' = -zHy on [0,infinity) where Weyl's limit point case prevails at infinity. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T-min with defect numbers (1, 1) in the Hilbert space L-H(2), and Q is equal to the Q-function,with respect to the extension corresponding to the boundary condition y(1)(0) = 0. If H satisfies some growth conditions at 0 or infinity, we present results on the corresponding Q-functions and show under which conditions the generalized, Friedrichs or Krein-von Neumann extension exists.