Echelon Spaces Research Articles

Mean Ergodic Weighted Shifts on Köthe Echelon Spaces

Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on Köthe echelon spaces in terms of the weight sequence and the Köthe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.

Some results about diagonal operators on Köthe echelon spaces

This research was partially supported by MINECO Project MTM2016-76647-P and the grant PAID-01-16 of the Universitat Politecnica de Valencia.

Characterizing Fréchet–Schwartz spaces via power bounded operators

We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known cha

Toeplitz matrices, ergodicity and Fréchet spaces

Abstract Yosida´s version of the mean ergodic theorem is extended to positive Toeplitz matrices and locally convex spaces. This is used to provide extensions of the results of Albanese, Bonet and Ricker on the structure of Fréchet spaces and also Köthe echelon spaces.

Grothendieck spaces with the Dunford–Pettis property

Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Frechet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For $${p\in \{0\}\cup[1,\infty)}$$ it is shown that the classical co-echelon spaces k p (V) and $${K_p(\overline{V})}$$ are GDP-spaces if and only if they are Montel. On the other hand, $${K_\infty(\overline{V})}$$ is always a GDP-space and k ∞(V) is a GDP-space whenever its (Frechet) predual, i.e., the Kothe echelon space λ 1(A), is distinguished.

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

It is shown that every echelon space λ∞(A), with A an arbitrary Kothe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ∞(A) is Montel if and only if it coincides with λ0(A), this identifies an extensive class of non-normable, non-Montel Frechet spaces having these two properties. Even though the canonical unit vectors in λ∞(A) fail to form an unconditional basis whenever λ∞(A) ≠ λ0(A), it is shown, nevertheless, that in this case λ∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given.

(DN)-type properties and projective tensor products

The aim of this note is to investigate the projective tensor products of Fréchet spaces satisfying a “dominating norm” condition. It is shown that the class of these spaces is not closed under completed projective tensor products. Positive inheritance results are obtained in the setting of Köthe echelon spaces and of Fréchet spaces of type 2.

Polynomials on K�the echelon spaces

We give conditions under which the space of n-homogeneous continuous polynomials on a Kothe echelon space, endowed with the topology of uniform convergence on bounded subsets, is barrelled. We also characterize when it is barrelled for every positive integer n in terms of the space order. A characterization of the reflexivity of spaces of homogeneous continuous polynomials on Kothe echelon spaces is also obtained.

On unconditional bases in tensor products of Köthe echelon spaces.

On unconditional bases in tensor products of Köthe echelon spaces.

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Frechet spaces. Our main result is the following one: LetE be a Frechet space which is neither Montel nor isomorphic to a closed subspace ofX × e, withX a Banach space, also assume thatE can be written asF⊕G withF andG infinite dimensional closed subspaces ofE not isomorphic to e, thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Kothe echelon space of orderp, 1<p<∞, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.

Spaces of operators between Fréchet spaces

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 fromEintoFis Montel, i.e. it maps bounded subsets ofEinto relatively compact subsets ofF. As a consequence of our results we characterize pairs of Köthe echelon spaces (E,F) such that the space of Montel operators fromEintoFis complemented in the space of all continuous linear maps fromEintoF.

Köthe echelon spaces à la Dieudonné

Let ( g n ) be a sequence of locally integrable functions defined on a Radon measure space. The echelon space associated to ( g n ) was defined by J. Dieudonné as the Köthe-dual of ( g n ), i.e. the space Λ of all locally integrable functions f such that all the integrals ∫ | f· g n | are finite. Denote by Λ x the Köthe-dual of Λ. We prove that Λ(β(Λ,Λ x)) is a Fréchet space with dual Λ x. This result gives its correct sense to a wrong affirmation of J. Dieudonné and validates those instances where it has been used. As a tool to prove this result, we study the problem of when the strong dual of a perfect space coincides with its Köthe-dual and give some necessary and sufficient conditions.

Distinguishedness of weighted Fréchet spaces of continuous functions

In this paper, we prove that ifis an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff spaceXsuch thatthen the Fréchet spaceC(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequencesatisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

The Structure of the Sequence Spaces of Maddox

AbstractThe sequence of spaces of Maddox, c0(p), c(p)and l∞(p), are investigated. Here, p — (pk) is a bounded sequence of strictly positive numbers. It is observed that C0(P) is an echelon space of order 0 and that l∞(p) is a co-echelon space of order ∞, while clearly c(p) = c0(p) ⊗ 〈 (1,1,1,…) 〉. This sheds a new light on the topological and sequence space structure of these spaces: Based on the highly developed theory of (co-) echelon spaces all known and various new structural properties are derived.

Lebesgue’s theorem of differentiation in Fréchet lattices

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some l 1 ( Γ ) {l^1}\left ( \Gamma \right ) .

Locally Bounded Noncontinuous Linear Forms on Strong Duals of Nondistinguished Kothe Echelon Spaces

In this note it is proved that if Al (A) is any nondistinguished Kothe echelon space of order one and K. ,0 (AI (A))' is its strong dual, then there is even a linear form : K C which is locally bounded (i.e. bounded on the bounded sets) but not continuous. It is also shown that every nondistinguished Kothe echelon space contains a sectional subspace with a particular structure.

Locally bounded noncontinuous linear forms on strong duals of nondistinguished Köthe echelon spaces

In this note it is proved that if λ 1 ( A ) {\lambda _1}(A) is any nondistinguished Köthe echelon space of order one and K ∞ ≃ ( λ 1 ( A ) ) b ′ {K_\infty } \simeq {({\lambda _1}(A))’_b} is its strong dual, then there is even a linear form : K ∞ → C {K_\infty } \to {\mathbf {C}} which is locally bounded (i.e. bounded on the bounded sets) but not continuous. It is also shown that every nondistinguished Köthe echelon space contains a sectional subspace with a particular structure.

Dual Density Conditions in (DF)— spaces, I

We define the two “dual density conditions” (DDC) and (SDDC) for locally convex topological vector spaces and study them in the setting of the class of (DF)- spaces (originally introduced by A. Grothendieck [14]). We show that for a (DF)- space E, (DDC) is equivalent to the metrizability of the bounded subsets of E, and prove that such a space E has (DDC) resp. (SDDC) if and only if the space l∞(E) of all bounded sequences in E is quasibarrelled resp. bornological. As a consequence, we can then characterize the barrelled spaces % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! ${\cal L}_b(\lambda_1, E)$ of continuous linear mappings from a Kothe echelon space λ1 into a locally complete (DF)- space E; for purposes of a comparison, we also provide the corresponding characterization of the quasibarrelled resp. bornological (DF)- tensor products (λ1)b ′ ⊗e E. Our results on the (DF)- spaces of type % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! ${\cal L}_b(\lambda_1, E)$ and (λ1)b ′) ⊗e E are of special interest in view of the recent negative solution, due to J. Taskinen (see [25]), of Grothendieck’s “probleme des topologies” ([15]). — In part II of the article, we will treat weighted inductive limits of spaces of continuous functions and their projective hulls (cf. [6]) as an application. In his study of ultrapowers of locally convex spaces, S. Heinrich [16] had found it necessary to introduce the “density condition”. Our article [2] investigated this condition, mainly in the setting of Frechet spaces, and with applications to distinguished echelon spaces λ1. However, on the way to the main theorems of [2], it became apparent that the “right” setting for most of this material was a dual reformulation of the density condition in the context of (DF)- spaces, and this observation prompted the present research.

Stefan Heinrich's Density Condition for Fréchet Spaces and the Characterization of the Distinguished Köthe Echelon Spaces

Mathematische NachrichtenVolume 135, Issue 1 p. 149-180 Article Stefan Heinrich's Density Condition for Fréchet Spaces and the Characterization of the Distinguished Köthe Echelon Spaces† Klaus D. Bierstedt, Klaus D. Bierstedt F B 17, Mathematik Universität-GH-Paderborn Warburger Straße 100 Postfach 1621 D - 4790 Paderborn BRDSearch for more papers by this authorJosé Bonet, José Bonet Departamento de Matemáticas, Aplicada E.T.S. Arquitectura Universidad Politécnica de Valencia Camino de Vera s.n. E - 46022 Valencia EspañaSearch for more papers by this author Klaus D. Bierstedt, Klaus D. Bierstedt F B 17, Mathematik Universität-GH-Paderborn Warburger Straße 100 Postfach 1621 D - 4790 Paderborn BRDSearch for more papers by this authorJosé Bonet, José Bonet Departamento de Matemáticas, Aplicada E.T.S. Arquitectura Universidad Politécnica de Valencia Camino de Vera s.n. E - 46022 Valencia EspañaSearch for more papers by this author First published: 1988 https://doi.org/10.1002/mana.19881350115Citations: 46 † Dedicated to Professor Gottfried Köthe on the occasion of his 80th birthday AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume135, Issue11988Pages 149-180 RelatedInformation

A New Class of Perfect Fréchet Spaces

This paper deals with a new class of perfect Frèchet spaces which can be obtained by interpolation of echelon spaces: lp,q[am,n]. We determine the reflexive, Montel, Schwartz, totally reflexive, totally Montel and nuclear spaces lp,q[am,n]. We also derive results on closed subspaces on the spaces (lp,q)(N).