On Complemented Subspaces of Non-Archimedean Power Series Spaces

2011 ◽  
Vol 63 (5) ◽  
pp. 1188-1200 ◽  
Author(s):  
Wiesław Śliwa ◽  
Agnieszka Ziemkowska
Keyword(s):  
Power Series ◽  
Double Sequence ◽  
Schauder Basis ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Complemented Subspaces ◽  
Complemented Subspace ◽  
Power Series Space ◽  
Series Space

Abstract The non-archimedean power series spaces, A1(a) and A∞(b), are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from Ap(a) to Aq(b) has a Schauder basis if either p = 1 or p = ∞ and the set Mb,a of all bounded limit points of the double sequence (bi/aj )i, j∈ℕ is bounded. It follows that every complemented subspace of a power series space Ap(a) has a Schauder basis if either p = 1 or p = ∞ and the set Ma,a is bounded.

scholarly journals DUALITY FOR SOME LARGE SPACES OF ANALYTIC FUNCTIONS

2001 ◽  
Vol 44 (3) ◽  
pp. 571-583 ◽  
Author(s):  
H. Jarchow ◽  
V. Montesinos ◽  
K. J. Wirths ◽  
J. Xiao
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Nuclear Power ◽  
Orlicz Spaces ◽  
Primary 46E10 ◽  
Smirnov Class ◽  
Power Series Space ◽  
Secondary 30D55 ◽  
Series Space ◽  
Spaces Of Analytic Functions

AbstractWe characterize the duals and biduals of the $L^p$-analogues $\mathcal{N}_\alpha^p$ of the standard Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\ge-1$ and $1\le p\lt \infty$. We adopt the convention to take $\mathcal{N}_{-1}^p$ to be the classical Smirnov class $\mathcal{N}^+$ for $p=1$, and the Hardy–Orlicz space $LH^p$ $(=(\text{Log}^+H)^p)$ for $1\lt p\lt\infty$. Our results generalize and unify earlier characterizations obtained by Eoff for $\alpha=0$ and $\alpha=-1$, and by Yanigahara for the Smirnov class.Each $\mathcal{N}_\alpha^p$ is a complete metrizable topological vector space (in fact, even an algebra); it fails to be locally bounded and locally convex but admits a separating dual. Its bidual will be identified with a specific nuclear power series space of finite type; this turns out to be the ‘Fréchet envelope’ of $\mathcal{N}_\alpha^p$ as well.The generating sequence of this power series space is of the form $(n^\theta)_{n\in\mathbb{N}}$ for some $0\lt\theta\lt1$. For example, the $\theta$s in the interval $(\smfr12,1)$ correspond in a bijective fashion to the Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\gt-1$, whereas the $\theta$s in the interval $(0,\smfr12)$ correspond bijectively to the Hardy–Orlicz spaces $LH^p$, $1\lt p\lt \infty$. By the work of Yanagihara, $\theta=\smfr12$ corresponds to $\mathcal{N}^+$.As in the work by Yanagihara, we derive our results from characterizations of coefficient multipliers from $\mathcal{N}_\alpha^p$ into various smaller classical spaces of analytic functions on $\Delta$.AMS 2000 Mathematics subject classification: Primary 46E10; 46A11; 47B38. Secondary 30D55; 46A45; 46E15\vskip-3pt

scholarly journals The existence of a factorized unbounded operator between Fréchet spaces

2018 ◽  
Vol 13 (01) ◽  
pp. 2050017
Author(s):  
Ersin Kızgut ◽  
Murat Yurdakul
Keyword(s):  
Unbounded Operator ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Infinite Dimensional ◽  
Locally Convex ◽  
Convex Spaces

For locally convex spaces [Formula: see text] and [Formula: see text], the continuous linear map [Formula: see text] is called bounded if there is a zero neighborhood [Formula: see text] of [Formula: see text] such that [Formula: see text] is bounded in [Formula: see text]. Our main result is that the existence of an unbounded operator [Formula: see text] between Fréchet spaces [Formula: see text] and [Formula: see text] which factors through a third Fréchet space [Formula: see text] ends up with the fact that the triple [Formula: see text] has an infinite dimensional closed common nuclear Köthe subspace, provided that [Formula: see text] has the property [Formula: see text].

Spaces of operators between Fréchet spaces

1994 ◽  
Vol 115 (1) ◽  
pp. 133-144 ◽  
Author(s):  
José Bonet ◽  
Mikael Lindström
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Linear Maps ◽  
Spaces Of Operators ◽  
Space Of Compact Operators ◽  
Compact Subsets

AbstractMotivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces (E, F) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F. As a consequence of our results we characterize pairs of Köthe echelon spaces (E, F) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F.

Representations of the spaces Cm(Ω) ∩ Hk, p (Ω)

1996 ◽  
Vol 120 (3) ◽  
pp. 489-498 ◽  
Author(s):  
A. A. Albanese ◽  
G. Metafune ◽  
V. B. Moscatelli
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Inductive Limit ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Complemented Subspaces ◽  
Complemented Subspace ◽  
Strong Dual

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

Some Double Sequence Spaces Generated by Four-Dimensional Pascal Matrix

2021 ◽  
Vol 09 (08) ◽  
Author(s):  
Ahmadu Kiltho ◽  
Keyword(s):  
Double Sequence ◽  
Sequence Spaces ◽  
Schauder Basis ◽  
Topological Properties ◽  
Matrix Domain ◽  
Double Sequence Spaces ◽  
Pascal Matrix

The purpose of this paper is to discover and examine a four-dimensional Pascal matrix domain on Pascal sequence spaces. We show that they are spaces and also establish their Schauder basis, topological properties, isomorphism and some inclusions.

Sign in / Sign up

Export Citation Format

Share Document