BOUNDARY TRIPLETS AND KREIN TYPE RESOLVENT FORMULA FOR SYMMETRIC OPERATORS WITH UNEQUAL DEFECT NUMBERS

Abstract. Let H be a Hilbert space and let A be a symmetric operator in H with ar-bitrary (not necessarily equal) deficiency indices n±(A). We introduce a new concept of a D-boundary triplet for A∗, which may be considered as a natural generalization of the known concept of a boundary triplet (boundary value space) for an operator with equal deficiency indices. With a D-triplet for A ∗ we associate two Weyl func-tions M+(·) and M−(·). It is proved that the functions M±(·) posses a number of properties similar to those of the known Weyl functions (Q-functions) for the case n+(A) = n−(A). We show that every D-triplet for A ∗ gives rise to Krein type for-mulas for generalized resolvents of the operator A with arbitrary deficiency indices. The resolvent formulas describe the set of all generalized resolvents by means of two pairs of operator functions which belongs to the Nevanlinna type class eR (H0,H1). This class has been earlier introduced by the author. 1