Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation

SynopsisIn the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

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Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2015-041-7 ◽

2016 ◽

Vol 68 (4) ◽

pp. 876-907 ◽

Cited By ~ 1

Author(s):

Mikhail Ostrovskii ◽

Beata Randrianantoanina

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Metric Spaces ◽

Euclidean Spaces ◽

Dimensional Banach Space ◽

Low Distortion ◽

Bounded Distortion ◽

Finite Metric Spaces ◽

Uniformly Bounded

AbstractFor a fixed K > 1 and n ∈ ℕ, n ≫ 1, we study metric spaces which admit embeddings with distortion ≤ K into each n-dimensional Banach space. Classical examples include spaces embeddable into log n-dimensional Euclidean spaces, and equilateral spaces.We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that n-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension log n.The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension n. This partially answers a question of G. Schechtman.

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Cohomology of deformations

Journal of Topology and Analysis ◽

10.1142/s1793525315500041 ◽

2014 ◽

Vol 07 (01) ◽

pp. 81-104 ◽

Cited By ~ 3

Author(s):

Uri Bader ◽

Piotr W. Nowak

Keyword(s):

Fixed Point ◽

Banach Spaces ◽

Fixed Point Theorems ◽

Linear Part ◽

Quantitative Estimates ◽

Fixed Point Properties ◽

Vanishing Of Cohomology ◽

Uniformly Bounded ◽

Affine Actions

In this paper we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.

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Uniformly bounded projections and semi-groups of operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100048428 ◽

1974 ◽

Vol 75 (2) ◽

pp. 199-217

Author(s):

G. O. Okikiolu

Keyword(s):

Banach Spaces ◽

Characteristic Functions ◽

Semi Group ◽

Trans Form ◽

Fourier Trans ◽

Special Cases ◽

Dirichlet Integrals ◽

Multiplier Operators ◽

Uniformly Bounded

Classes of operators called uniformly bounded projections defined on Banach spaces have been formally introduced (see for example (2), 8·9·28; 8·9·92), and their connexions with semi-groups of operators considered. Notable cases of such operators include the Dirichlet integrals which are connected with the semi-group of Poisson operators in Lp In the main conclusions of this paper, there is an exposition of the connexion between classes of uniformly bounded projections and semi-groups of operators. In particular, it is shown that certain uniformly bounded analytic semi-groups of operators give rise to classes of uniformly bounded projections. Various of the known special cases in Lp are also considered as Fourier trans form multiplier operators with characteristic functions of intervals as multipliers.

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The form of locally defined operators in waterman spaces

Mathematica Slovaca ◽

10.1515/ms-2021-0069 ◽

2021 ◽

Vol 71 (6) ◽

pp. 1529-1544

Author(s):

Małgorzata Wróbel

Keyword(s):

Banach Spaces ◽

Bounded Variation ◽

Composition Operators ◽

Representation Formula ◽

Continuous Functions ◽

Functions Of Bounded Variation ◽

Spaces Of Continuous Functions ◽

Local Operators ◽

Uniformly Bounded

Abstract A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

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