scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

scholarly journals Browder and Weyl spectra of upper triangular operator matrices

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 111-130 ◽  
Author(s):  
B.P. Duggal
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Weyl’S Theorem ◽  
Single Valued Extension Property ◽  
Browder’S Theorem ◽  
Subject Classifications ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Mathematics Subject Classifications ◽  
Necessary And Sufficient

Let MC = (A/0 C/B) ( B(X ( X ) be an upper triangulat Banach space operator. The relationship between the spectra of MC and M0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators MC and M0 satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type 'M0 satisfies a-Browder's (or a-Weyl's) theorem ( MC satisfies a-Browder's (resp., a-Weyl's) theorem' to hold. 2010 Mathematics Subject Classifications. Primary 47B47, 47A10, 47A11. .

scholarly journals A Highly D‑efficient Spring Balance Weighing Design for an Even Number of Objects

2019 ◽  
Vol 5 (344) ◽  
pp. 17-27
Author(s):  
Małgorzata Graczyk ◽  
Bronisław Ceranka
Keyword(s):  
Optimal Design ◽  
Sufficient Conditions ◽  
Optimality Criteria ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Optimal Designs ◽  
Efficient Design ◽  
Spring Balance ◽  
Experimental Errors ◽  
Necessary And Sufficient

The problem of determining unknown measurements of objects in the model of spring balance weighing designs is presented. These designs are considered under the assumption that experimental errors are uncorrelated and that they have the same variances. The relations between the parameters of weighing designs are deliberated from the point of view of optimality criteria. In the paper, designs in which the product of the variances of estimators is possibly the smallest one, i.e. D‑optimal designs, are studied. A highly D‑efficient design in classes in which a D‑optimal design does not exist are determined. The necessary and sufficient conditions under which a highly efficient design exists and methods of its construction, along with relevant examples, are introduced.

scholarly journals When do Projections Commute?

1980 ◽  
Vol 35 (4) ◽  
pp. 437-441 ◽  
Author(s):  
W. Rehder
Keyword(s):  
Hilbert Space ◽  
Conditional Probability ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Mechanical ◽  
Probability Operator ◽  
Quantum Mechanical Systems ◽  
Necessary And Sufficient ◽  
New Criteria

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

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 AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 309-315 ◽  
Author(s):  
ROBERTO CONTI
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Cuntz Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Matrix Algebras ◽  
The Family ◽  
The Matrix ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

scholarly journals Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms

1996 ◽  
Vol 28 (3) ◽  
pp. 784-801 ◽  
Author(s):  
I-Jeng Wang ◽  
Edwin K. P. Chong ◽  
Sanjeev R. Kulkarni
Keyword(s):  
Hilbert Space ◽  
Approximation Algorithms ◽  
Stochastic Approximation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Noise Sequences

We consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark's condition, Chen's condition, a decomposition condition, and Kulkarni and Horn's condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.

scholarly journals On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Karim Hedayatian ◽  
Lotfollah Karimi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Necessary And Sufficient

A bounded linear operatorTon a Hilbert spaceℋ, satisfying‖T2h‖2+‖h‖2≥2‖Th‖2for everyh∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

scholarly journals On the Option Pricing by the Binomial Model

2021 ◽  
Author(s):  
Nikos Halidias
Keyword(s):  
Sufficient Conditions ◽  
Fair Value ◽  
Point Of View ◽  
Binomial Model ◽  
No Arbitrage ◽  
Mathematical Arguments ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Probabilistic Point ◽  
Absence Of Arbitrage

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

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