On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators

If the Q-function Q corresponding to a closed symmetric operator S with defect numbers (1, 1) and one of its selfadjoint extensions belongs to the Kac class N1 then it is known that all except one of the Q-functions of S belong to N1, too. In this note the situation that the given Q-function does not belong to the class N1 is considered. If Q ∈ Np, i.e., if the restriction of the spectral measure of Q on the positive or the negative axis corresponds to an N1-function, then Q itself is the Q-function of the exceptional extension, and, hence, it is associated with the generalized Friedrichs extension of S. If Q or, equivalently, the spectral measure of Q is symmetric, or if the difference of Q and a symmetric Nevanlinna function belongs to the class N1 or Np, then Q is still exceptional in a wider sense. Similar results hold for the generalized Kreĭn-von Neumann extension of the symmetric operator