Displaying similar documents to “On an integral of fractional power operators”

Fermi Golden Rule, Feshbach Method and embedded point spectrum

Jan Dereziński (1998-1999)

Séminaire Équations aux dérivées partielles

Similarity:

A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak s ˇ ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

Similarity:

Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , S p ( A , B ) = S p ( A ) × S p ( B ) , where Sp denotes the joint Taylor spectrum.

Spectrum of L

W. Marek, K. Rasmussen

Similarity:

CONTENTS0. Motivation, results to be used in the sequel ................51. Slicing L α ’s ..........................................................102. Hereditarily countable, definable elements ................133. Spectrum of L.............................................................154. The width of elements of spectrum ............................195. Non-uniform strong definability ..................................266. Solution to a problem of Wilmers................................327....

Generalized spectral perturbation and the boundary spectrum

Sonja Mouton (2021)

Czechoslovak Mathematical Journal

Similarity:

By considering arbitrary mappings ω from a Banach algebra A into the set of all nonempty, compact subsets of the complex plane such that for all a A , the set ω ( a ) lies between the boundary and connected hull of the exponential spectrum of a , we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

The norm spectrum in certain classes of commutative Banach algebras

H. S. Mustafayev (2011)

Colloquium Mathematicae

Similarity:

Let A be a commutative Banach algebra and let Σ A be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by σ ( f ) = f · a : a A ¯ Σ A , where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

Ascent spectrum and essential ascent spectrum

O. Bel Hadj Fredj, M. Burgos, M. Oudghiri (2008)

Studia Mathematica

Similarity:

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if σ a s c e ( T + F ) = σ a s c e ( T ) for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued...

Fractional q -difference equations on the half line

Saïd Abbas, Mouffak Benchohra, Nadjet Laledj, Yong Zhou (2020)

Archivum Mathematicum

Similarity:

This article deals with some results about the existence of solutions and bounded solutions and the attractivity for a class of fractional q -difference equations. Some applications are made of Schauder fixed point theorem in Banach spaces and Darbo fixed point theorem in Fréchet spaces. We use some technics associated with the concept of measure of noncompactness and the diagonalization process. Some illustrative examples are given in the last section.

The single-point spectrum operators satisfying Ritt's resolvent condition

Yu. Lyubich (2001)

Studia Mathematica

Similarity:

It is shown that an operator with the properties mentioned in the title does exist in the space L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. The maximal sector for the extended resolvent condition can be prescribed a priori jointly with the corresponding order of the exponential growth of the resolvent in the complementary sector.

On the norm-closure of the class of hypercyclic operators

Christoph Schmoeger (1997)

Annales Polonici Mathematici

Similarity:

Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if f ( σ W ( T ) ) z : | z | = 1 is connected, where σ W ( T ) denotes the Weyl spectrum of T.

Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter Trebels, Ursula Westphal (2010)

Banach Center Publications

Similarity:

Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to ( X , D ( ( - A ) α ) ) , α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, | | λ R ( λ ; A ) f | | Y c φ ( 1 / λ ) | | f | | X , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has...

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

Similarity:

We introduce a fractional Langevin equation with α-stable noise and show that its solution Y κ ( t ) , t 0 is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of Y κ ( t ) via the measure of its codependence r(θ₁,θ₂,t). We prove that Y κ ( t ) is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type ( D α - i = 1 n - 1 a i D α i ) x ( t ) F ( t , x ( φ ( t ) ) ) , a.e. on (0,1), I 1 - α x ( 0 ) = c , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

Similarity:

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ - d / d t ( 1 / 2 0 D t - σ ( u ' ( t ) ) + 1 / 2 t D T - σ ( u ' ( t ) ) ) - λ β ( t ) f ( u ( t ) ) - μ γ ( t ) g ( u ( t ) ) = 0 , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where 0 D t - σ and t D T - σ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

Similarity:

We give a necessary and a sufficient condition for a subset S of a locally convex Waelbroeck algebra 𝒜 to have a non-void left joint spectrum σ l ( S ) . In particular, for a Lie subalgebra L 𝒜 we have σ l ( L ) if and only if [ L , L ] generates in 𝒜 a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

Similarity:

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form u ( x , t ) = t α f ( | x | t β ) with suitable and β . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...

On operators with the same local spectra

Aleksandar Torgašev (1998)

Czechoslovak Mathematical Journal

Similarity:

Let B ( X ) be the algebra of all bounded linear operators in a complex Banach space X . We consider operators T 1 , T 2 B ( X ) satisfying the relation σ T 1 ( x ) = σ T 2 ( x ) for any vector x X , where σ T ( x ) denotes the local spectrum of T B ( X ) at the point x X . We say then that T 1 and T 2 have the same local spectra. We prove that then, under some conditions, T 1 - T 2 is a quasinilpotent operator, that is ( T 1 - T 2 ) n 1 / n 0 as n . Without these conditions, we describe the operators with the same local spectra only in some particular cases.

A spatially sixth-order hybrid L 1 -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng (2021)

Applications of Mathematics

Similarity:

We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an L 1 strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid L 1 -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order 2 - γ in time, where 0 < γ < 1 is the order of the Caputo fractional derivative...

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

Similarity:

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts a / n n = 1 is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length c N - 0 . 475 contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.