scholarly journals On a Chaotic Weighted Shiftzpdp+1/dzp+1of Orderpin Bargmann Space

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Abdelkader Intissar
Keyword(s):  
Differential Operators ◽  
Unbounded Operators ◽  
Bargmann Space ◽  
Backward Shift ◽  
Creation Operators

This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operatorsHp=A*pAP+1;p=0,1,…realized as differential operators in Bargmann space, whereAandA*are the standard Bose annihilation and creation operators such that[A,A*]=I.

Green's Functions for Singular Ordinary Differential Operators

1967 ◽  
Vol 19 ◽  
pp. 571-582 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Hilbert Space ◽  
Green's Function ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Good Deal ◽  
Linear Operators ◽  
Spectral Theorem ◽  
Unbounded Operators ◽  
Expansion Theorem ◽  
Ordinary Differential Operators

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.

scholarly journals Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

2020 ◽  
Vol 2020 ◽  
pp. 1-27
Author(s):  
Yoritaka Iwata
Keyword(s):  
Banach Algebra ◽  
Algebraic Structure ◽  
Differential Operators ◽  
Mathematical Physics ◽  
Rotation Group ◽  
Module Structure ◽  
Unbounded Operators ◽  
Infinitesimal Generators ◽  
Rigorous Formulation

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.

scholarly journals An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3415-3425 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Unbounded Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Normal Operator ◽  
Condition Numbers ◽  
Unbounded Operators ◽  
Similarity Condition ◽  
Von Neumann ◽  
Operator Functions

Let H be a linear unbounded operator in a separable Hilbert space. It is assumed the resolvent of H is a compact operator and H ? H* is a Schatten - von Neumann operator. Various integro-differential operators satisfy these conditions. Under certain assumptions it is shown that H is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS

2014 ◽  
Vol 97 (3) ◽  
pp. 331-342 ◽  
Author(s):  
MICHAEL GIL’
Keyword(s):  
Hilbert Space ◽  
Condition Number ◽  
Unbounded Operator ◽  
Differential Operators ◽  
Normal Operator ◽  
Condition Numbers ◽  
Unbounded Operators ◽  
Similarity Condition ◽  
Operator Functions ◽  
Schmidt Operator

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

scholarly journals Subnormality and generalized commutation relations of families of operators

1990 ◽  
Vol 32 (2) ◽  
pp. 231-238 ◽  
Author(s):  
Jerzy Bartłomiej Stochel
Keyword(s):  
Hilbert Space ◽  
Commutation Relation ◽  
Infinite Order ◽  
Dense Subset ◽  
Commutation Relations ◽  
Bargmann Space ◽  
Subnormal Operators ◽  
Families Of Operators ◽  
Special Case ◽  
Creation Operators

1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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