The Kreĭn Formula and its Weyl Families

Almost 50 years ago the well-known Kreĭn’s formula for the description of self-adjoint exit space extensions of a densely defined symmetric operator, originally established by M.G. Kreĭn in his famous 1946 paper, was presented by Langer and Textorius in the context of linear relations. The present paper offers a review of the parameters in the Kreĭn’s formula from the point of view of the general concept of boundary relations and their Weyl families. One of the main new results extends Kreĭn’s resolvent formula for an arbitrary boundary relation Γ Γ of a symmetric operator (or relation) S whose Weyl family is a Nevanlinna family Z . Analogous to the case of the standard Kreĭn’s resolvent formula, which describes self-adjoint extensions of a symmetric operator A with equal defect numbers in a Hilbert space H H , this Kreĭn type formula gives a one-to-one correspondence between (smooth) extensions S_Θ ⊂ dom Γ S Θ ⊂ dom Γ of S= Γ S = ker Γ and parameters Θ =Γ (S _Θ ) Θ = Γ ( S Θ ) . When such a Kreĭn type formula is orthogonally coupled with the standard Kreĭn’s resolvent formula for self-adjoint extensions of A this yields an elementary proof of the Kreĭn-Naĭmark resolvent formula for the self-adjoint exit space extension of A generated by the Nevanlinna family Z by projecting the coupled formula to the space H H .