On three Krein extension problems and some generalizations

1998 ◽  
Vol 31 (1) ◽  
pp. 1-91 ◽  
Author(s):  
Damir Z. Arov ◽  
Harry Dym
Keyword(s):  
Krein Extension

Generalized Krein extension problems

2010 ◽  
pp. 420-469
Author(s):  
Damir Z. Arov ◽  
Harry Dym
Keyword(s):  
Krein Extension

Bitangential Krein extension problems

2012 ◽  
pp. 310-354
Author(s):  
Damir Z. Arov ◽  
Harry Dym
Keyword(s):  
Krein Extension

Krein Extension of an Even-Order Differential Operator

2018 ◽  
Vol 54 (4) ◽  
pp. 551-556
Author(s):  
Ya. I. Granovskyi ◽  
L. L. Oridoroga
Keyword(s):  
Differential Operator ◽  
Order Differential Operator ◽  
Even Order ◽  
Krein Extension

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

2018 ◽  
Vol 10 (2) ◽  
pp. 387-394
Author(s):  
O.G. Storozh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Boundary Value ◽  
Complex Hilbert Space ◽  
Multivalued Operator ◽  
Friedrichs Extension ◽  
Boundary Triple ◽  
Krein Extension ◽  
Selfadjoint Extensions ◽  
Weyl Functions

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 showed 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.

Sign in / Sign up

Export Citation Format

Share Document