Opuscula Mathematica

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.Krakówwersja wydawnicz