For a given generalized Nevanlinna function Q(z) 2 N� (H), we study decompo- sitions that satisfy: Q = Q1 +Q2; Qi2 Ni (H), 0 � �i, i = 1, 2, and �1 +�2 = �, which we call desirable decompositions. In this paper, necessary and sufficient con- ditions for such decompositions of Q are obtained. One of the main results is a new, operator representation of ˆ

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length l, we show that there exists a pole cancellation function of order at least l, and, moreover, the construction shows that it is of surprisingly simple form.