Reduction of positive self-adjoint extensions

Abstract

We revise Krein’s extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the “resolvent operator” (I + T )−1 of T . Our treatment is somewhat simpler and more natural than Krein’s original method which was based on the Krein transform (I−T )(I+T )−1. Apart from being positive and symmetric, we do not impose any further constraints on the operator T : neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces.