scholarly journals Convex semigroups on $$L^p$$-like spaces

2021 ◽  
Author(s):  
Robert Denk ◽  
Michael Kupper ◽  
Max Nendel
Keyword(s):  
Semigroup Theory ◽  
Banach Lattices ◽  
Heat Equations ◽  
Well Posedness ◽  
Strong Continuity ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Typical Application ◽  
Convex Case ◽  
Order Continuous

AbstractIn this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ L p -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ C 0 -semigroup theory extend to the convex case. We prove that the generator of a convex $$C_0$$ C 0 -semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $$C_0$$ C 0 -semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

Converses for the Dodds–Fremlin and Kalton–Saab theorems

1996 ◽  
Vol 120 (1) ◽  
pp. 175-179 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Positive Operator ◽  
Banach Lattices ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Full Generality ◽  
Order Continuous

The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].

M-embedded symmetric operator spaces and the derivation problem

2019 ◽  
Vol 169 (3) ◽  
pp. 607-622
Author(s):  
JINGHAO HUANG ◽  
GALINA LEVITINA ◽  
FEDOR SUKOCHEV
Keyword(s):  
Symmetric Spaces ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Symmetric Operator Spaces ◽  
Order Continuous

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

scholarly journals Triangularizing semigroups of positive operators on an atomic normed Riesz Space

2000 ◽  
Vol 43 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Roman Drnovšek
Keyword(s):  
Positive Operator ◽  
Riesz Space ◽  
Positive Operators ◽  
Closed Ideal ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Multiplicative Semigroup ◽  
A Chain ◽  
Order Continuous

AbstractIn the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band.The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.

scholarly journals Kadec–Klee properties of Orlicz–Lorentz sequence spaces equipped with the Orlicz norm

Positivity ◽  
2021 ◽  
Author(s):  
Yunan Cui ◽  
Paweł Foralewski ◽  
Henryk Hudzik ◽  
Radosław Kaczmarek
Keyword(s):  
Sufficient Conditions ◽  
Sequence Spaces ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Orlicz Norm ◽  
Full Characterization ◽  
Orlicz Functions ◽  
Necessary And Sufficient ◽  
Order Continuous

AbstractThe necessary and sufficient conditions for both the Kadec–Klee property as well as the Kadec–Klee property with respect to the coordinatewise convergence in Orlicz–Lorentz sequence spaces equipped with the Orlicz norm and generated by arbitrary Orlicz functions as well as any non-increasing weight sequences are given. Moreover, for their subspaces of elements with an order continuous norm the full characterization of the Kadec–Klee property with respect to the coordinatewise convergence is presented. Some tools useful in the proofs of the main results are also provided.

scholarly journals On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Barış Akay ◽  
Ömer Gök
Keyword(s):  
Linear Span ◽  
Compact Operators ◽  
Approximation Properties ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Weakly Compact ◽  
Domination Property ◽  
Weakly Compact Operators ◽  
Lattice Approximation ◽  
Order Continuous

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Well-Posedness of the Gnedenko System with Multiple Vacations

2012 ◽  
Vol 522 ◽  
pp. 902-909
Author(s):  
Bilikiz Yunus ◽  
Abdukerim Haji
Keyword(s):  
Semigroup Theory ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiple Vacations ◽  
Dynamic Solution ◽  
Multiple Vacation ◽  
Gnedenko System

We investigate the solution of the Gnedenko system with multiple vacation of a repairman. By using-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

scholarly journals Hyperbolic Maxwell variational inequalities of the second kind

2020 ◽  
Vol 26 ◽  
pp. 34 ◽  
Author(s):  
Irwin Yousept
Keyword(s):  
Variational Inequalities ◽  
Existence Result ◽  
Semigroup Theory ◽  
Regularity Property ◽  
Well Posedness ◽  
Test Functions ◽  
Local Boundedness ◽  
Subdifferential Calculus ◽  
Faraday’S Law ◽  
Yosida Regularization

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.

scholarly journals An age-dependent population equation with diffusion and delayed birth process

2005 ◽  
Vol 2005 (20) ◽  
pp. 3273-3289 ◽  
Author(s):  
G. Fragnelli
Keyword(s):  
Asymptotic Behavior ◽  
Semigroup Theory ◽  
New Age ◽  
Well Posedness ◽  
Birth Process ◽  
Age Dependent ◽  
Population Equation

We propose a new age-dependent population equation which takes into account not only a delay in the birth process, but also other events that may take place during the time between conception and birth. Using semigroup theory, we discuss the well posedness and the asymptotic behavior of the solution.

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