An F-space Sampler

Kalton, N. J.; Peck, N. T.; Roberts, James W.

doi:10.1017/cbo9780511662447

Cited by 396 publications

(339 citation statements)

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“…Indeed, note that if X = L p (Ω, A, μ) with 0 p < 1 and μ has no atoms, then X * = {0} (see e.g. [21]). In particular, L p [0, 1] for 0 p < 1 has trivial dual.…”

Section: Theorem Cs For Each Admissible Comparison Function γ the mentioning

confidence: 99%

Disjoint mixing operators

Bès¹,

Martin²,

Peris³

et al. 2012

Journal of Functional Analysis

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Chan and Shapiro showed that each (non-trivial) translation operator f (z) T λ → f (z + λ) acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C 0 -semigroups. We also provide an easy proof of the result of Salas that every infinitedimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T , T 2 ) is not d-mixing.

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“…Indeed, note that if X = L p (Ω, A, μ) with 0 p < 1 and μ has no atoms, then X * = {0} (see e.g. [21]). In particular, L p [0, 1] for 0 p < 1 has trivial dual.…”

Section: Theorem Cs For Each Admissible Comparison Function γ the mentioning

confidence: 99%

Disjoint mixing operators

Bès¹,

Martin²,

Peris³

et al. 2012

Journal of Functional Analysis

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show abstract

“…It is well-known, however, that L 0 is generally not a locally-convex topological vector space. In fact, when P is non-atomic, L 0 admits no non-trivial continuous linear functionals (see Kalton et al (1984), Theorem 2.2, p. 18). In addition to not being locally-convex, the space L 0 has poor compactness properties.…”

Section: A Characterization In Terms Of Generalized Sequencesmentioning

confidence: 99%

Convex compactness and its applications

Zitkovic

2010

Math Finan Econ

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Abstract. The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability.Two applications in infinite-dimensional optimization -attainment of infima and a version of the Minimax theorem -are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem.

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“…We say that X = (X, · ) is a quasi-Banach space if it is complete for this metric. To find out more about quasi-normed spaces (and about even more general -normed spaces) the reader is referred to [25].…”

Section: X Construction For (Quasi-)symmetric Spaces Xmentioning

confidence: 99%

Rotundity and monotonicity properties of selected Cesàro function spaces

Kiwerski

Kolwicz

2017

Positivity

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We study rotundity, strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some classes of Cesàro function spaces. We present full criteria of these properties in the Cesà ro-Orlicz function spaces Ces ϕ (under some mild assumptions on the Orlicz function ϕ). Next, we prove a characterization of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in the Cesàro-Lorentz function spaces C φ . We also show that the space C φ is never rotund. Finally, we will prove that Cesàro-Marcinkiewicz function space C M ( * ) φ is neither strictly monotone nor order continuous for any quasiconcave function φ.

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An F-space Sampler

Cited by 396 publications

References 0 publications

Disjoint mixing operators

Disjoint mixing operators

Convex compactness and its applications

Rotundity and monotonicity properties of selected Cesàro function spaces

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