Opuscula Math. 44, no. 3 (2024), 425-438

Opuscula Mathematica

Reduction of positive self-adjoint extensions

Zsigmond Tarcsay
Zoltán Sebestyén

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.

Keywords: positive selfadjoint contractive extension, nonnegative selfadjoint extension, Friedrichs and Krein-von Neumann extension.

Mathematics Subject Classification: 47A57, 47A20, 47B25.

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  • Zsigmond Tarcsay (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-8102-5055
  • Corvinus University of Budapest, Department of Mathematics, IX. Fővám tér 13-15., Budapest H-1093, Hungary
  • Eötvös Loránd University, Department of Applied Analysis and Computational Mathematics, Pázmány Péter sétány 1/c., Budapest H-1117, Hungary
  • Zoltán Sebestyén
  • Eötvös Loránd University, Department of Applied Analysis and Computational Mathematics, Pázmány Péter sétány 1/c., Budapest H-1117, Hungary
  • Communicated by Aurelian Gheondea.
  • Received: 2023-01-10.
  • Revised: 2023-08-10.
  • Accepted: 2023-08-16.
  • Published online: 2024-02-15.
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Cite this article as:
Zsigmond Tarcsay, Zoltán Sebestyén, Reduction of positive self-adjoint extensions, Opuscula Math. 44, no. 3 (2024), 425-438, https://doi.org/10.7494/OpMath.2024.44.3.425

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