Extensions of positive symmetric operators and Krein's uniqueness criteria

Sebestyén, Zoltán and Tarcsay, Zsigmond (2022) Extensions of positive symmetric operators and Krein's uniqueness criteria. LINEAR AND MULTILINEAR ALGEBRA. ISSN 0308-1087 (Submitted)

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Abstract

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $T^*T$ of a densely defined linear transformation $T$ and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised.

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