Negative results on the Nash-Moser theorem for Köthe sequence spaces and for spaces of ultradifferentiable functions

1996 ◽  
Vol 90 (1) ◽  
pp. 465-478 ◽  
Author(s):  
Markus Poppenberg
Keyword(s):  
Sequence Spaces ◽  
Ultradifferentiable Functions ◽  
Negative Results ◽  
Köthe Sequence Spaces

On Köthe sequence spaces and linear logic

2002 ◽  
Vol 12 (5) ◽  
pp. 579-623 ◽  
Author(s):  
THOMAS EHRHARD
Keyword(s):  
Simple Structure ◽  
Linear Logic ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Cartesian Closed ◽  
Linear Category ◽  
Köthe Sequence Spaces ◽  
Locally Convex ◽  
Standard Concept

We present a category of locally convex topological vector spaces that is a model of propositional classical linear logic and is based on the standard concept of Köthe sequence spaces. In this setting, the ‘of course’ connective of linear logic has a quite simple structure of a commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting in which typed λ-calculus and differential calculus can be combined; we give a few examples of computations.

scholarly journals Chaos and frequent hypercyclicity for weighted shifts

2020 ◽  
pp. 1-37
Author(s):  
STÉPHANE CHARPENTIER ◽  
KARL GROSSE-ERDMANN ◽  
QUENTIN MENET
Keyword(s):  
Difference Sets ◽  
Weighted Shift ◽  
Sequence Spaces ◽  
Hypercyclic Operators ◽  
Weighted Shifts ◽  
Banach Sequence Spaces ◽  
Frequently Hypercyclic Operators ◽  
Köthe Sequence Spaces ◽  
The Relationship ◽  
Banach Sequence

Abstract Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

scholarly journals Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Aparajita Dasgupta ◽  
Michael Ruzhansky
Keyword(s):  
Hilbert Spaces ◽  
Sequence Spaces ◽  
Compact Manifolds ◽  
Tensor Structure ◽  
Eigenfunction Expansions ◽  
Ultradifferentiable Functions ◽  
Perfect Sequence ◽  
In Trans

AbstractIn this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

scholarly journals The Modulus of Nearly Uniform Smoothness in Orlicz Sequence Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Shaoyong Zhang ◽  
Meiling Zhang ◽  
Yujia Zhan
Keyword(s):  
Fixed Point ◽  
Sequence Spaces ◽  
Point Property ◽  
Luxemburg Norm ◽  
Continuous Norm ◽  
Absolutely Continuous ◽  
Uniform Smoothness ◽  
Orlicz Sequence Spaces ◽  
Köthe Sequence Spaces ◽  
Equivalent Conditions

It is well known that the modulus of nearly uniform smoothness related with the fixed point property is important in Banach spaces. In this paper, we prove that the modulus of nearly uniform smoothness in Köthe sequence spaces without absolutely continuous norm is ΓX(t)=t. Meanwhile, the formula of the modulus of nearly uniform smoothness in Orlicz sequence spaces equipped with the Luxemburg norm is given. As a corollary, we get a criterion for nearly uniform smoothness of Orlicz sequence spaces equipped with the Luxemburg norm. Finally, the equivalent conditions of R(a,l(Φ))<1+a and RW(a,l(Φ))<1+a are given.

Generalized Köthe sequence spaces and decompositions

1977 ◽  
Vol 113 (1) ◽  
pp. 287-301 ◽  
Author(s):  
Manjul Gupta ◽  
P. K. Kamthan ◽  
K. L. N. Rao
Keyword(s):  
Sequence Spaces ◽  
Köthe Sequence Spaces
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