Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon-Nikodým property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented here can be applied to operators whose domains are not dense.

/pdf/vector-valued-laplace-transforms-and-cauchy-problems-4imt52dtio.pdf

Vector-Valued Laplace Transforms and Cauchy Problems

New updated! The latest book from a very famous author finally comes out. Book of positive operators, as an amazing reference becomes what you need to get. What's for is this book? Are you still thinking for what the book is? Well, this is what you probably will get. You should have made proper choices for your better life. Book, as a source that may involve the facts, opinion, literature, religion, and many others are the great friends to join with.

Positive Operators

In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

The main result is the following stability theorem: Let T = (T(t))tzo be a bounded Co-semigroup on a reflexive space X. Denote by A the generator of T and by a(A) the spectrum of A. If a(A) n iR is countable and no eigenvalue of A lies on the imaginary axis, then limt BOO T(t)x = O for all x E X.

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/downloads/pubbib/short/1988-AreBat-TbrThrStbOnPrmSmg.pdf

Tauberian theorems and stability of one-parameter semigroups

A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.

Predator-prey populations with parasitic infection.

Basic results on semigroups on banach spaces.- Characterization of semigroups on banach spaces.- Spectral theory.- Asymptotics of semigroups on banach spaces.- Basic results on spaces Co(X).- Characterization of positive semigroups on Co(X).- Spectral theory of positive semigroups on Co(X).- Asymptotics of positive semigroups on Co(X).- Basic results on banach lattices and positive operators.- Characterization of positive semigroups on banach lattices.- Spectral theory of positive semigroups on banach lattices.- Asymptotics of positive semigroups on banach lattices.- Basic results on semigroups and operator algebras.- Characterization of positive semigroups on w*-algebras.- Spectral theory of positive semigroups on w*-algebras and their preduals.- Asymptotics of positive semigroups on c*-and w*-algebras.

One-parameter Semigroups of Positive Operators

/pdf/extensions-and-liftings-of-positive-linear-mappings-on-kh9veaqjyk.pdf

Extensions and liftings of positive linear mappings on Banach lattices

Some characterizations of AM- and AL-spaces

It is a most implausible fact that a one-to-one operator from c0 into a Banach space which maps the unit ball of c0 onto a closed set is necessarily an isomorphism.In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semi-embeddings defined on C(X), X compact.

/pdf/semi-embeddings-of-banach-space-1bscfw6dcu.pdf

Semi-embeddings of Banach space

In this paper we prove some uniform convergence theorems for operators on a Grothendieck space with the Dunford-Pettis property. As consequences we obtain 1) that on such spaces (in particular on L ∞ and H ∞ (D)) every C 0 -semigroup is uniformly continous, 2) that on such spaces the strong ergodic theorem becomes a uniform ergodic theorem, and 3) Dean's result that such a space does not have a Schauder decomposition.

Heinrich P. Lotz

Papers

One-parameter Semigroups of Positive Operators

Extensions and liftings of positive linear mappings on Banach lattices

Some characterizations of AM- and AL-spaces

Semi-embeddings of Banach space

Tauberian Theorems for Operators on L∞ and Similar Spaces