### On an approach to the construction of the Friedrichs and Neumann-Krein extensions of nonnegative linear relations

#### Abstract

Let $L_{0}$ be a closed linear nonnegative (probably, positively defined) relation ("multivalued operator") in a complex Hilbert space $H$. In terms of the so called boundary value spaces (boundary triples) and corresponding Weyl functions and Kochubei - Strauss characteristic ones, the Friedrichs (hard) and Neumann -- Krein (soft) extensions of $L_{0}$ are constructed.

It should be noted that every nonnegative linear relation $L_{0}$ in a Hilbert space $H$ has two extremal nonnegative selfadjoint extensions: the Friedrichs extension $L_{F}$ and the Neumann -- Krein extension $L_{K},$ satisfying the following property: $$(\forall \varepsilon > 0) (L_{F} + \varepsilon 1)^{-1} \leq (\widetilde{L} + \varepsilon 1)^{-1} \leq (L_{K} + \varepsilon 1)^{-1}$$

in the set of all nonnegative selfadjoint subspace extensions $\widetilde{L}$ of $L_{0}.$

The boundary triple approach to the extension theory was initiated by F. S. Rofe-Beketov, M.~L. and V. I. Gorbachuk, A. N. Kochubei, V. A. Mikhailets, V. O. Dercach, M. N. Malamud, Yu. M. Arlinskii and other mathematicians.

In addition, it is shown that the construction of the mentioned extensions may be realized in a more simple way under the assumption that initial relation is a positively defined one.

It should be noted that every nonnegative linear relation $L_{0}$ in a Hilbert space $H$ has two extremal nonnegative selfadjoint extensions: the Friedrichs extension $L_{F}$ and the Neumann -- Krein extension $L_{K},$ satisfying the following property: $$(\forall \varepsilon > 0) (L_{F} + \varepsilon 1)^{-1} \leq (\widetilde{L} + \varepsilon 1)^{-1} \leq (L_{K} + \varepsilon 1)^{-1}$$

in the set of all nonnegative selfadjoint subspace extensions $\widetilde{L}$ of $L_{0}.$

The boundary triple approach to the extension theory was initiated by F. S. Rofe-Beketov, M.~L. and V. I. Gorbachuk, A. N. Kochubei, V. A. Mikhailets, V. O. Dercach, M. N. Malamud, Yu. M. Arlinskii and other mathematicians.

In addition, it is shown that the construction of the mentioned extensions may be realized in a more simple way under the assumption that initial relation is a positively defined one.

#### Keywords

Hilbert space, relation, operator, extension, boundary value space

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