Extensions of positive symmetric operators and Krein's uniqueness criteria

TL;DR

This paper revises Krein's extension theory for positive symmetric operators, introducing a new approach via factorization that applies to non-dense transformations and both real and complex spaces, with applications to self-adjoint extensions.

Contribution

It presents a novel factorization method for positive symmetric operators, extending Krein's theory to non-densely defined transformations and broadening its applicability.

Findings

Applicable to non-densely defined transformations

Works in both real and complex spaces

Revises Krein's criteria for uniqueness of extensions

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.