Singular Operator as a Parameter of Self-adjoint Extensions

Abstract

Let Å be a symmetric restriction of a self-adjoint bounded from below operator A in a Hilbert space H and let A ∞ denote the Friedrichs extension of Å. We prove that in the case, where A ∞ ≠ A, under natural conditions, each self-adjoint extension A of Å has a unique representation in the form of a generalized sum, Ã = A + V, where V is a singular operator acting in the A-scale of Hilbert spaces, from H 1(A) to H -1(A). In the particular case, where Å has deficiency indices (1, 1), this result has been proven by Krein and Yavrian.