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.

scholarly journals Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

2020 ◽  
Vol 12 (2) ◽  
pp. 289-296
Author(s):  
O.G. Storozh
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Differential Operators ◽  
Boundary Value ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Multivalued Operator ◽  
Linear Transformations ◽  
Accretive Extension

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

scholarly journals On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

1997 ◽  
Vol 20 (3) ◽  
pp. 457-464
Author(s):  
A. A. El-Sabbagh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Matrix ◽  
Linear Relations ◽  
Symmetric Linear Relation ◽  
Necessary And Sufficient ◽  
Selfadjoint Extensions ◽  
Pick Problem

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

scholarly journals Traces of localisation operators with two admissible wavelets

2003 ◽  
Vol 45 (1) ◽  
pp. 17-25 ◽  
Author(s):  
M. W. Wong ◽  
Zhaohui Zhang
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Resolution Of The Identity

AbstractThe resolution of the identity formula for a localisation operator with two admissible wavelets on a separable and complex Hilbert space is given and the traces of these operators are computed.

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

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