Deficiency indices and self-adjoint extensions of T0

1987 ◽  
pp. 52-71
Author(s):  
Joachim Weidmann
Keyword(s):  
Deficiency Indices

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 Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yan Liu ◽  
Meiru Xu
Keyword(s):  
Perturbation Theory ◽  
Hamiltonian Systems ◽  
Limit Point ◽  
Limit Circle ◽  
Linear Relations ◽  
Deficiency Indices ◽  
Bounded Perturbations

AbstractThis paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.

THE ASSESSMENT OF MARITIME SAFETY IN THE TURKISH STRAITS BASED ON THE PERFORMANCE OF FLAG STATES IN PORT STATE CONTROL REGIMES

2021 ◽  
Vol 160 (A3) ◽  
Author(s):  
E G Emecen Kara
Keyword(s):  
Risk Level ◽  
Annual Number ◽  
State Control ◽  
Maritime Safety ◽  
Deficiency Indices ◽  
Risk Levels ◽  
Maritime Shipping ◽  
Increased Risk ◽  
Pass Through ◽  
Port State Control

The Turkish Straits are well known for theirs intensive maritime traffic. The average annual number of transit ships passing through this waterway is approximately 50000 and more than 100 flag states pass through it. Moreover, this waterway presents a navigational challenge owing to its inherent geographic and oceanographic characteristics. Also, sub-standard ships navigating in this region lead to an increased risk levels and pose a threat to the marine environment. Over the years, serious maritime accidents occurring in the straits region had resulted in losses of life and constituted environmental disasters. The high risk arising from maritime shipping in these regions had always endangered public health in the vicinity of the Turkish Straits. In this study, maritime safety in the Turkish Straits region had been assessed based on the performance in the Port State Control inspections of flag states passing through this region. For the assessment of the performance of passing flag states, detention and deficiency indices of these flag states were generated for the MOUs. According to these values, the risk level of these flag states had been determined by the weighted risk point methods. Hereby, in addition to the determination of the risk level of flag states, the relationships between the inspections of MOUs had been also discussed on the basis of both the detention and the deficiency rates of flag states.

Finite-Dimensional Extensions of Certain Symmetric Operators(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 455-456
Author(s):  
I. M. Michael
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Inner Product ◽  
Selfadjoint Extension ◽  
Von Neumann ◽  
Deficiency Indices ◽  
Finite Dimensional ◽  
Symmetric Operators

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.

Odd-order differential expressions with positive supporting coefficients

1987 ◽  
Vol 105 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Bernd Schultze
Keyword(s):  
Essential Spectrum ◽  
Deficiency Index ◽  
Deficiency Indices ◽  
Compact Perturbations ◽  
Odd Order ◽  
Independent Variable

SynopsisThe deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

scholarly journals Deficiency Indices of Some Classes of Unbounded $$\mathbb H$$ H -Operators

2017 ◽  
Vol 13 (1) ◽  
pp. 85-113 ◽  
Author(s):  
B. Muraleetharan ◽  
K. Thirulogasanthar
Keyword(s):  
Deficiency Indices

scholarly journals Generalized resolvents and spectrum for a certain class of perturbed symmetric operators

2005 ◽  
Vol 2005 (1) ◽  
pp. 81-92 ◽  
Author(s):  
A. Hebbeche
Keyword(s):  
Deficiency Indices ◽  
Symmetric Operators

The generalized resolvents for a certain class of perturbed symmetric operators with equal and finite deficiency indices are investigated. Using the Weinstein-Aronszajn formula, we give a classification of the spectrum.

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