scholarly journals A class of strongly stable approximation for unbounded operators

2019 ◽  
pp. 218-234
Author(s):  
Ammar Khellaf ◽  
Sarra Benarab ◽  
Hamza Guebbai ◽  
Wassim Merchela
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Background ◽  
Spectral Approximation ◽  
Unbounded Operators ◽  
Numerical Application ◽  
Spectrum Method ◽  
Pollution Problem ◽  
Generalized Spectrum ◽  
Spectral Pollution ◽  
Discretization Process

We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schr¨odinger’s operator where the discretization process based upon the Kantorovich’s projection.

scholarly journals NEW SUFFICIENT CONDITIONS IN THE GENERALIZED SPECTRUM APPROACH TO DEAL WITH SPECTRAL POLLUTION

2018 ◽  
pp. 595-604
Author(s):  
Ammar Khellaf
Keyword(s):  
Theoretical Foundation ◽  
Sufficient Conditions ◽  
Numerical Results ◽  
Spectral Approach ◽  
Spectrum Method ◽  
Generalized Spectrum ◽  
Spectral Pollution

In this work, we propose new sufficient conditions to solve the spectralpollution problem by using the generalized spectrum method. We give the theoretical foundation of the generalized spectral approach, as well as illustrate its effectivenessby numerical results.

scholarly journals On unbounded commuting Jacobi operators and some related issues

2019 ◽  
Vol 6 (1) ◽  
pp. 82-91
Author(s):  
Andrey Osipov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Matrices ◽  
Unbounded Operators ◽  
Jacobi Operators ◽  
Necessary And Sufficient

Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.

FROM QUANTUM FIELDS TO LOCAL VON NEUMANN ALGEBRAS

1992 ◽  
Vol 04 (spec01) ◽  
pp. 15-47 ◽  
Author(s):  
H.J. BORCHERS ◽  
JAKOB YNGVASON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Energy Bounds ◽  
Local Algebras ◽  
Quantized Fields ◽  
Necessary And Sufficient

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

scholarly journals Two conditions for subnormality of unbounded operators

2001 ◽  
Vol 43 (1) ◽  
pp. 23-28
Author(s):  
Jan Niechwiej
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
The Real ◽  
Completely Monotonic ◽  
Binomial Expansion ◽  
Hilbert Space Operators ◽  
The One

We give two new sufficient conditions for unbounded Hilbert space operators to be subnormal. The first assumes that the sequence //Tnf//2 on a suitable subset of the domain is completely monotonic, the second is similar to the one given by Lambert in [3] for bounded operators and involves the sequence of binomial expansion of the real part of the operator.

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

Eigenvalues computation by the generalized spectrum method of Schrödinger’s operator

2018 ◽  
Vol 37 (5) ◽  
pp. 5965-5980 ◽  
Author(s):  
Ammar Khellaf ◽  
Hamza Guebbai ◽  
Samir Lemita ◽  
Mohamed Zine Aissaoui
Keyword(s):  
Spectrum Method ◽  
Generalized Spectrum

Prediction Error Statistics in Deterministic Linear Ship Motion Forecasting

2018 ◽  
Author(s):  
Fabio Fucile ◽  
Gabriele Bulian ◽  
Claudio Lugni
Keyword(s):  
Prediction Error ◽  
Theoretical Background ◽  
Ship Motion ◽  
Irregular Waves ◽  
Motion Prediction ◽  
Ship Motions ◽  
Error Statistics ◽  
Numerical Application ◽  
Analytical Methodology ◽  
Wave Radar

Deterministic ship motions predictions methodologies represent a promising emerging approach, which could be embedded in decision support systems for certain types of operation. The typically envisioned prediction chain starts from the remote sensing of the wave elevation through wave radar technology. An estimated wave field is then fitted to the data, it is propagated in space and time, and it is finally fed to a ship motion prediction model. Prediction time horizons, typically, are practically limited to the order of minutes. Deterministic predictions are, however, inevitably associated with prediction uncertainty which is seldom quantified. This paper, therefore, presents a semi-analytical methodology for the estimation of ship motion prediction error statistics in ensemble domain as function of the forecasting time, assuming linear Gaussian irregular waves and stationary linear ship motions. This information can be used, for instance, to supplement deterministic forecasting with corresponding confidence intervals. The paper describes the theoretical background of the developed methodology and reports some numerical application examples.

Spectral representation of local semigroups associated with Klein-Landau systems

1984 ◽  
Vol 95 (1) ◽  
pp. 93-100 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Hilbert Space ◽  
Spectral Representation ◽  
Sufficient Conditions ◽  
Real Spectrum ◽  
Spectral Operator ◽  
Unbounded Operators ◽  
Reflexive Banach Spaces ◽  
Scalar Type ◽  
Necessary And Sufficient ◽  
Local Semigroup

In their paper [5], Klein and Landau prove that given a symmetric ‘local semigroup’ of unbounded operators {T(t); t ≥ 0} on a Hilbert space, there exists a unique selfadjoint operator T such that T(t) is a restriction of e−tT, for each t ≥ 0. A similar representation theorem was proved earlier by Nussbaum [8]. The result of Klein and Landau was recently extended to the setting of reflexive Banach spaces by Kantorovitz ([4], theorem 2–3). More precisely, Kantorovitz presented necessary and sufficient conditions for a local semigroup of unbounded operators {T(t); t ≥ 0} to consist of restrictions of e−tT, t ≥ 0, for some unbounded spectral operator of scalar-type T with real spectrum (cf. [1] for the terminology).

Egorov measurability and generator cores

1986 ◽  
Vol 100 (1) ◽  
pp. 137-143
Author(s):  
Brian Jefferies
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Unbounded Operators ◽  
Locally Convex

AbstractSufficient conditions are given for a set to be a core for the generator of a weakly integrable semigroup on a locally convex space. The conditions are illustrated by semigroups of unbounded operators on a Banach space.

scholarly journals Sufficient conditions for the optimality of the greedy algorithm in greedoids

2021 ◽  
Author(s):  
Dávid Szeszlér
Keyword(s):  
Greedy Algorithm ◽  
Greedy Algorithms ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Theoretical Background ◽  
Theoretical Computer Science ◽  
Special Cases ◽  
Original Works ◽  
Optimum Solution ◽  
The Greedy Algorithm

AbstractGreedy algorithms are among the most elementary ones in theoretical computer science and understanding the conditions under which they yield an optimum solution is a widely studied problem. Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. One of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Indeed, many well-known algorithms of a greedy nature that cannot be interpreted in a matroid-theoretical context are special cases of the greedy algorithm on greedoids. Although this algorithm turns out to be optimal in surprisingly many cases, no general theorem is known that explains this phenomenon in all these cases. Furthermore, certain claims regarding this question that were made in the original works of Korte and Lovász turned out to be false only most recently. The aim of this paper is to revisit and straighten out this question: we summarize recent progress and we also prove new results in this field. In particular, we generalize a result of Korte and Lovász and thus we obtain a sufficient condition for the optimality of the greedy algorithm that covers a much wider range of known applications than the original one.

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