Banach Lattices

Schaefer, Helmut

doi:10.1007/978-3-642-65970-6_2

Cited by 96 publications

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“…By Example 5, page 337 in [10], it results that condition (C) is satisfied. From Remark 2.3 it is easy to see that condition (βf) is also satisfied.…”

Section: J Oomentioning

confidence: 99%

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Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels

Beznea¹

1988

Nagoya Mathematical Journal

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Intro ductionLet (X, 0) be a measurable space and f be a submarkovian resolvent of kernels (with the initial kernel V proper) on X which is absolutely continuous and has a dual resolvent (with the same properties) with respect to a σ-finite measure.A positive numerical function s on X is called V-ultrapotentίal if it is ^-excessive (in particular ^-a.e. finite) and if the following condition is fulfilled: for every integer n >, 1, there exists a positive ^-measurable function f n on X such that s = V n (f n ), where V n is the n-th iteration of the kernel V.The main purpose of this paper (see Theorem 3.5 and Corollary 3.6) is to prove that, under a "regularity" condition (which will be discussed in the last part of Section 2) on the resolvent iΓ, for each F-ultrapotential s there exist a finite positive Borel measure σ on the open interval ]0, oo[ and a family (s x \ <λ 0, such that for each is o -measurable and s(x)= [s λ (x)dσ(λ).In fact, this type of representation is given for a slightly more general class of excessive functions, as the F-ultrapotentials.An uniqueness of the representation and a converse statement are also proved.These results aiv analogous, in this context, with those obtained by M. Itό and N. Suzuki in [7] (see also [6]) for the set up of diffusion semi- Recerred May 6, 1987. 125 https://www.cambridge.org/core/terms. https://doi

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“…By Example 5, page 337 in [10], it results that condition (C) is satisfied. From Remark 2.3 it is easy to see that condition (βf) is also satisfied.…”

Section: J Oomentioning

confidence: 99%

“…Schaefer, and from Theorem 6.6, ch. V in [10] it follows that there exists exactly one eigenvalue r > 0 for V which has positive eigenvectors and the corresponding eigenspace is one dimensional. Remarking that every element of E(V, λ\ is m-integrable for 0 < λ < oo, we conclude that E(V, r\ is one dimensional and E[(V, X) = {0} if λ Φ r, λeR.…”

Section: J Oomentioning

confidence: 99%

Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels

Beznea¹

1988

Nagoya Mathematical Journal

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“…Instead, we refer to another paper of ours [11], where a similar eigenvalue problem has been treated, and the reader will have no difficulty in seeing that many results of that paper can be carried over to our present case. operators we refer to the monograph of Schaefer [19].…”

Section: Ta and Va Define Completely Continuous Operatorsmentioning

confidence: 99%

On the stable size distribution of populations reproducing by fission into two unequal parts

Heijmans

1984

Mathematical Biosciences

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“…Let us very briefly recall basic notions necessary to formulate our results (regarding the theory of Banach lattices and other facts from functional analysis the reader is referred to [2] or [27]). A Banach space (F, · ) equipped with a partial order ≤ is a Banach lattice if the lattice operations (the infimum x ∧ y and the supremum x ∨ y) are well defined in F, satisfy axioms of Riesz spaces and are compatible with the norm topology (cf.…”

Section: Introductionmentioning

confidence: 99%