Isometric operators with equal deficiency indices, quasi-unitary operators

1960 ◽  
pp. 85-103 ◽  
Author(s):  
M. S. Livšic
Keyword(s):  
Unitary Operators ◽  
Deficiency Indices

scholarly journals STABILITY OF THE DEFICIENCY INDICES OF SYMMETRIC OPERATORS UNDER SELF-ADJOINT PERTURBATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 383-394 ◽  
Author(s):  
Edward Kissin
Keyword(s):  
Symmetric Operator ◽  
Sufficient Conditions ◽  
Strong Operator Topology ◽  
Stationary Points ◽  
Unitary Operators ◽  
Unbounded Operators ◽  
Deficiency Indices ◽  
Adjoint Extension ◽  
The Stability ◽  
Weyl Relation

AbstractLet $S$ and $T$ be symmetric unbounded operators. Denote by $\overline{S+T}$ the closure of the symmetric operator $S+T$. In general, the deficiency indices of $\overline{S+T}$ are not determined by the deficiency indices of $S$ and $T$. The paper studies some sufficient conditions for the stability of the deficiency indices of a symmetric operator $S$ under self-adjoint perturbations $T$. One can associate with $S$ the largest closed $^*$-derivation $\delta_{S}$ implemented by $S$. We prove that if the unitary operators $\exp(\ri tT)$, for $t\in\mathbb{R}$, belong to the domain of $\delta_{S}$ and $\delta_{S}(\exp(\ri tT))\rightarrow0$ in the strong operator topology as $t\rightarrow0$, then the deficiency indices of $S$ and $\overline{S+T}$ coincide. In particular, this holds if $S$ and $\exp(\ri tT)$ commute or satisfy the infinitesimal Weyl relation.We also study the case when $S$ and $T$ anticommute: $\exp(-\ri tT)S\subseteq S\exp(\ri tT)$, for $t\in\mathbb{R}$. We show that if the deficiency indices of $S$ are equal, or if the group $\{\exp(\ri tT):t\in\mathbb{R}\}$ of unitary operators has no stationary points in the deficiency space of $S$, then $S$ has a self-adjoint extension which anticommutes with $T$, the operator $S+T$ is closed and the deficiency indices of $S$ and $S+T$ coincide.AMS 2000 Mathematics subject classification: Primary 47B25

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

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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.

scholarly journals Understanding mathematics of Grover’s algorithm

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Phase Change ◽  
Unitary Operator ◽  
Mathematical Structure ◽  
Complex Numbers ◽  
Unitary Operators ◽  
Grover’S Algorithm ◽  
Great Probability ◽  
Grover's Algorithm ◽  
One Step ◽  
The One

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

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.

scholarly journals Spectral asymptotics of semiclassical unitary operators

2019 ◽  
Vol 473 (2) ◽  
pp. 1174-1202 ◽  
Author(s):  
Yohann Le Floch ◽  
Álvaro Pelayo
Keyword(s):  
Spectral Asymptotics ◽  
Unitary Operators

MONODROMY APPROACH TO QUANTUM COMPUTING

2002 ◽  
Vol 16 (30) ◽  
pp. 4593-4605 ◽  
Author(s):  
G. GIORGADZE
Keyword(s):  
Quantum Computing ◽  
Hilbert Problem ◽  
Holomorphic Vector Bundle ◽  
Classical System ◽  
Unitary Operators ◽  
Unitary Matrices ◽  
Classical Procedure ◽  
Logarithmic Singularities ◽  
The Given ◽  
Riemann Hilbert Problem

In this work, a gauge approach to quantum computing is considered. It is assumed that there exists a classical procedure for placing certain classical system in a state described by a holomorphic vector bundle with connection with logarithmic singularities. This bundle and its connection are constructed with the aid of unitary operators realizing the given algorithm using methods of the monodromic Riemann–Hilbert problem. Universality is understood in the sense that for any collection of unitary matrices there exists a connection with logarithmic singularities whose monodromy representation involves these matrices.

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