scholarly journals Deficiency indices and discreteness property of block Jacobi matrices and Dirac operators with point interactions

2021 ◽  
pp. 125582
Author(s):  
Viktoriya S. Budyka ◽  
Mark M. Malamud
Keyword(s):  
Dirac Operators ◽  
Jacobi Matrices ◽  
Point Interactions ◽  
Deficiency Indices

scholarly journals On the completely indeterminate case for block Jacobi matrices

2017 ◽  
Vol 4 (1) ◽  
pp. 48-57
Author(s):  
Andrey Osipov
Keyword(s):  
Orthogonal Polynomials ◽  
Moment Problem ◽  
Jacobi Matrices ◽  
Hamburger Moment Problem ◽  
Block Matrices ◽  
Deficiency Indices ◽  
Jacobi Operators ◽  
Indeterminate Case

Abstract We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.

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 $n_+(A)=n_-(A)$.

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