scholarly journals Invariant subspaces for Fréchet spaces without continuous norm

2020 ◽  
pp. 1
Author(s):  
Quentin Menet
Keyword(s):  
Invariant Subspaces ◽  
Fréchet Spaces ◽  
Continuous Norm

scholarly journals THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−

2016 ◽  
Vol 59 (2) ◽  
pp. 273-287 ◽  
Author(s):  
ANGELA A. ALBANESE ◽  
JOSÉ BONET ◽  
WERNER J. RICKER
Keyword(s):  
Banach Spaces ◽  
Point Spectrum ◽  
Fréchet Spaces ◽  
Cesàro Operator ◽  
Continuous Norm ◽  
Spectrum Point ◽  
Mean Ergodicity ◽  
Cesaro Operator

AbstractThe classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

scholarly journals Some results on asymptotically normable Frechet spaces

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 39-43
Author(s):  
GK Palei ◽  
NP Sah
Keyword(s):  
Banach Spaces ◽  
Tensor Products ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Köthe Spaces ◽  
Köthe Space

In this paper, it is shown that the asymptotically normable spaces are the smallest class of Frechet spaces which contains the nuclear Kothe spaces with continuous norm, the Banach spaces and is closed under e-tensor products and sub-spaces. Again our main aim will be to construct an example of a Kothe space which is Montel, admits a continuous norm, but still is not asymptotically normable. Keywords: Asymptotically normable; Frechet space; Kothe space DOI: 10.3126/bibechana.v7i0.4043BIBECHANA 7 (2011) 39-43

scholarly journals Characterization of Frechet spaces with nuclear Kothe quotients

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 161-164
Author(s):  
Satya Narayan Sah
Keyword(s):  
Fréchet Spaces ◽  
Continuous Norm

In this paper our result based on the characterization of Frechet spaces with nuclear Kothe quotients is in terms of the following condition which labeled as ?. We show that a Frechet spaces E satisfies condition ? if and only if it has a quotient which admits continuous norm and satisfies condition ?. For the condition ?, there exists ? such that for every k there exists j such that the ?. ?kclosure of E′? is not closed in E′j. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10397 BIBECHANA 11(1) (2014) 161-164

scholarly journals Fréchet AL-spaces have the Dunford-Pettis property

1998 ◽  
Vol 58 (3) ◽  
pp. 383-386 ◽  
Author(s):  
J.C. Díaz ◽  
A. Fernández ◽  
F. Naranjo
Keyword(s):  
Banach Spaces ◽  
Positive Cone ◽  
Weak Order ◽  
Fréchet Spaces ◽  
Order Unit ◽  
Continuous Norm

A Fréchet lattice E is an AL-space if its topology can be defined by a family of lattice seminorms that are additive in the positive cone of E. Grothendieck proved that AL-Banach spaces have the Dunford-Pettis property. This result was recently extended by Fernández and Naranjo to AL-Fréchet spaces with a continuous norm and weak order unit. In this note we show how to remove both hypotheses.

scholarly journals Every Separable Complex Fréchet Space with a Continuous Norm is Isomorphic to a Space of Holomorphic Functions

2020 ◽  
pp. 1-5
Author(s):  
José Bonet
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Fréchet Space ◽  
Dense Subspace ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Infinite Dimensional ◽  
Spaces Of Holomorphic Functions

Abstract Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

scholarly journals Montel subspaces of Fréchet spaces of Moscatelli type

1997 ◽  
Vol 39 (3) ◽  
pp. 345-350
Author(s):  
Angela A. Albanese
Keyword(s):  
General Idea ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Finite Dimensional

In this note we show that every complemented Montel subspace F of a Fréchet space E of Moscatelli type is isomorphic to ω or is finite–dimensional; the last case always occurs when E has a continuous norm. To do this, we first study the topology induced by E on its Montel subspaces, extending a result on Fr6chet-Montel spaces of Moscatelli type in [4].We recall that the Fréchet spaces of Moscatelli type were introduced and studied by J. Bonet and S. Dierolf in [4]; the general idea behind the construction of such spaces was due to V. B. Moscatelli [7].

scholarly journals Construction of operators with prescribed orbits in Fréchet spaces with a continuous norm

2011 ◽  
Vol 109 (1) ◽  
pp. 147 ◽  
Author(s):  
Angela A. Albanese
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Fréchet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Continuous Norm ◽  
Infinite Dimensional ◽  
Linearly Independent ◽  
Dense Set

Let $X$ be a separable, infinite dimensional real or complex Fréchet space admitting a continuous norm. Let $\{v_n:\ n\geq 1\}$ be a dense set of linearly independent vectors of $X$. We show that there exists a continuous linear operator $T$ on $X$ such that the orbit of $v_1$ under $T$ is exactly the set $\{v_n:\ n\geq 1\}$. Thus, we extend a result of Grivaux for Banach spaces to the setting of non-normable Fréchet spaces with a continuous norm. We also provide some consequences of the main result.

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