Positive One-Parameter Semigroups on Ordered Banach Spaces

Batty, C. J. K.; Robinson, Derek W.

doi:10.1007/978-94-009-6484-6_2

Cited by 33 publications

(11 citation statements)

References 48 publications

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“…The canonical half‐norm on an ordered Banach space X is defined by

N false(x false) = d i s t false(- x, X_{+} false) = inf \{(‖‖ x + y), y \in X_{+}\}

or equivalently by

N false(x false) = inf \{(‖‖, z), z \in X, z ⩾ x\} .

In Banach lattices or on the self‐adjoint part of a

C^{*}

algebra

N false(x false) = ‖x_{+}‖

where

x_{+}

is the positive component of

x \in X

. For all these results above (and many others), we refer to the exhaustive survey .…”

Section: Reminders On Ordered Banach Spacesmentioning

confidence: 99%

“…

L^{1} false(μ false)

spaces or the space of Borel measures on a metric space endowed with the total variation norm), the duals of the hermitian part of

C^{*}

algebras or by the preduals of the hermitian part of von Neumann algebras. We refer to , for the details.…”

Section: Reminders On Ordered Banach Spacesmentioning

confidence: 99%

“…Notice that in a Banach lattice or on (hermitian elements of) a

C^{*}

algebra we have

N false(x false) = ‖x_{+}‖, x \in X,

where

x_{+}

is the positive component of x (see e.g. ).…”

Section: Introductionmentioning

confidence: 99%

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Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

Mokhtar-Kharroubi

2016

Mathematische Nachrichten

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This paper provides various “contractivity” results for linear operators of the form I−C where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces Lpfalse(μfalse), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. Afalse(λ−Afalse)−1x≤x for all x∈L+pfalse(μfalse)false(λ>0false), provided that p⩾2. We show also that this result is not true for 1 ⩽ p<2. We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X+. We deduce from this result that, in such spaces, I−C is a contraction on X+ for any positive projection C with norm 1. We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base‐norm spaces X (e.g. in real L1false(μfalse) spaces or in preduals of hermitian part of von Neumann algebras), we show that Nfalse(u−Cufalse)≤u for all u∈X where N is the canonical half‐norm in X. For any positive contraction C on order‐unit spaces X (e.g. on the hermitian part of a C* algebra), we show that I−C is a contraction on X+. Applications to relative operator bounds, ergodic projections and conditional expectations are given.

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“…The canonical half‐norm on an ordered Banach space X is defined by

N false(x false) = d i s t false(- x, X_{+} false) = inf \{(‖‖ x + y), y \in X_{+}\}

or equivalently by

N false(x false) = inf \{(‖‖, z), z \in X, z ⩾ x\} .

In Banach lattices or on the self‐adjoint part of a

C^{*}

algebra

N false(x false) = ‖x_{+}‖

where

x_{+}

is the positive component of

x \in X

. For all these results above (and many others), we refer to the exhaustive survey .…”

Section: Reminders On Ordered Banach Spacesmentioning

confidence: 99%

“…

L^{1} false(μ false)

spaces or the space of Borel measures on a metric space endowed with the total variation norm), the duals of the hermitian part of

C^{*}

algebras or by the preduals of the hermitian part of von Neumann algebras. We refer to , for the details.…”

Section: Reminders On Ordered Banach Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

Mokhtar-Kharroubi

2016

Mathematische Nachrichten

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“…and divergence-free, then the operator ( (D), D) defined by (18) is the generator of a strongly continuous stochastic semigroup ( D ( )) ≥0 , given by…”

Section: Theorem 2 If the Function Is Globally Lipschitz Continuousmentioning

confidence: 99%

“…Inequality (41) is therefore valid for any nonnegative ∈ X . Using the fact that any arbitrary element of X (equipped with the pointwise order almost everywhere) can be written in the form = + − − , where + , − ∈ (X ) + , the positive element approach [18,19] or [5,Theorem 2.64], allows us to extend the right inequality of (41) to all X so as to have…”

Section: Perturbed Transport-fragmentation Problemsmentioning

confidence: 99%

Global Solvability of a Continuous Model for Nonlocal Fragmentation Dynamics in a Moving Medium

Noutchie

Goufo

2013

Mathematical Problems in Engineering

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Existence of global solutions to continuous nonlocal convection-fragmentation equations is investigated in spaces of distributions with finite higher moments. Under the assumption that the velocity field is divergence-free, we make use of the method of characteristics and Friedrichs's lemma (Mizohata, 1973) to show that the transport operator generates a stochastic dynamical system. This allows for the use of substochastic methods and Kato-Voigt perturbation theorem (Banasiak and Arlotti, 2006) to ensure that the combined transport-fragmentation operator is the infinitesimal generator of a strongly continuous semigroup. In particular, we show that the solution represented by this semigroup is conservative.

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On an Unbounded Linear Operator Arising in the Theory of Growing Cell Population

Latrach

Mokhtar-Kharroubi

1997

Journal of Mathematical Analysis and Applications

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This paper deals with the spectral analysis of a class of unbounded linear operators corresponding to a partial differential equation originally proposed by Ž . J. L. Lebowitz and S. I. Rubinow J. Math. Biol. 1, 1974, 17᎐36 to model an age structured proliferating cell population. Individual cells are distinguished by age and cell cycle length. The cell cycle length is considered as an inherited property Ž determined at birth. After a detailed spectral analysis for general boundary conditions which model the process of cell division of mother cell and the . inherence of cycle length by daughter cells it is shown that the associated Cauchy problem is governed by a C 0 -semigroup. A spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given. ᮊ

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Positive One-Parameter Semigroups on Ordered Banach Spaces

Cited by 33 publications

References 48 publications

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

Global Solvability of a Continuous Model for Nonlocal Fragmentation Dynamics in a Moving Medium

On an Unbounded Linear Operator Arising in the Theory of Growing Cell Population

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