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Reduction of positive self-adjoint extensions

Tarcsay Zsigmond, Sebestyén Zoltán

Opuscula Mathematica

2024

Vol. 44, no. 3

425--438

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.