Bounded factorization property for Fréchet spaces

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by L^G(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ???) if L^G(E, F) ? LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ??? is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K?othe spaces the property ??? is characterized in terms of their K?othe matrices. As an application we prove that in certain cases the relations L(E, G₁) = LB(E, G₁) and L(G₂, F) = LB(G₂, F) imply (E, G, F) ∈ ??? where G is a tensor product of G₁ and G₂.     

Fréchet Spaces of Holomorphic Functions without Copies of l1

Let X be a Banach space. Let H_w*(X*) the Fréchet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of H_w*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l₁ when this space contains no copy of l¹. We also show that if X is an Asplund space, then H_w*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.     

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: l_p,q[a_m,n]. We determine the reflexive, Montel, Schwartz, totally reflexive, totally Montel and nuclear spaces l_p,q[a_m,n]. We also derive results on closed subspaces on the spaces (l_p,q)(N).     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.     

A characterization of distinguished Fréchet spaces

Bierstedt and Bonet proved in 1988 that if a metrizable locally convex space E satisfies the Heinrich's density condition, then every bounded set in the strong dual (E ′, β (E ′, E)) of E is metrizable; consequently E is distinguished, i.e. (E ′, β (E ′, E)) is quasibarrelled. However there are examples of distinguished Fréchet spaces whose strong dual contains nonmetrizable bounded sets. We prove that a metrizable locally convex space E is distinguished iff every bounded set in the strong dual (E ′, β (E ′, E)) has countable tightness, i.e. for every bounded set A in (E ′, β (E ′, E)) and every x in the closure of A there exists a countable subset B of A whose closure contains x. This extends also a classical result of Grothendieck. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the bidual of quasinormable Fréchet algebras

It is known that the bidual of a quasinormable Fréchet space E with local Banach spaces $(E_n)_{n\in {\mathbb N}}$ is topologically isomorphic to the inverse limit of $\big (E_n^{\prime \prime }\big )_{n\in {\mathbb N}}$. With the aid of the Arens product and by homological means, we prove that the previous result is equally valid for quasinormable Fréchet m‐convex algebras. This allows showing that the bidual of a σ‐C*‐algebra equipped with the Arens product is a σ‐C*‐algebra and presenting a new direct proof of a result on acyclic spectra due to Palamodov.     

Sequential order of product of Fréchet spaces

We construct, assuming the continuum hypothesis (CH), two (strongly) Fréchet spaces whose product is sequential and its sequential order is for any given ω₁.     

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

Summary. In part II the evolution of large populations of infinitesimal particles is studied, in which one can follow the path of any particle surviving up to time t. To construct the distribution of the totality of these paths — the so-called backward tree —, we need a general extension theorem for random measures. The present part gives such a theorem under very natural conditions, and its variante for product spaces.     

Fréchet differentiability for a damped sine-Gordon equation

We consider a damped sine-Gordon equation with a variable diffusion coefficient. The goal is to derive necessary conditions for the optimal set of parameters minimizing the objective function J. First, we show that the solution map is continuous under a weak assumption on the topology of the admissible set P. Then the solution map is shown to be weakly Gâteux differentiable on P, implying the Gâteux differentiability of the objective function. Finally we show the Fréchet differentiability of J. The optimal set of parameters is shown to satisfy a bang–bang control law.     

Bases and quasi‐reflexivity in Fréchet spaces

A Fréchet space E is quasi‐reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Fréchet space E is totally quasi‐reflexive if every separated quotient is quasi‐reflexive. In this paper we show, using Schauder bases, that E is totally quasi‐reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi‐reflexive Banach spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Inverse Function Theorem for Fréchet Spaces Satisfying a Smoothing Property and (DN)

Classical inverse function theorems of Nash-Moser type are proved for Fréchet spaces that admit smoothing operators as introduced by Nash. In this note an inverse function theorem is proved for Fréchet spaces which only have to satisfy the condition (DN) of Vogt and the smoothing property (S_Ω)_t; for instance, any Fréchet-Hilbert space which is an (Ω)-space in standard form has property (S_Ω)_t. The main result of this paper generalizes a theorem of Lojasiewicz and Zehnder. It can be applied to the space C^∞(K) if the compact K ? ?^N is the closure of its interior and subanalytic; different from classical results the boundary of K may have singularities like cusps. The growth assumptions on the mappings are formulated in terms of the weighted multiseminorms [ ]m,k introduced in this paper; nonlinear smooth partial differential operators on C^∞(K) and their derivatives satisfy these formal assumptions.     

On quotients of non‐Archimedean Fréchet spaces

It is proved when a non‐Archimedean Fréchet space E of countable type has a quotient isomorphic to ??^?, c^?₀ or c₀ × ??^?. It is also shown when E has a non‐normable quotient with a continuous norm. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing the Fréchet distance between piecewise smooth curves

We consider the Fréchet distance between two curves which are given as a sequence of m+n curved pieces. If these pieces are sufficiently well-behaved, we can compute the Fréchet distance in O(mnlog(mn)) time. The decision version of the problem can be solved in O(mn) time. The results are based on an analysis of the possible intersection patterns between circles and arcs of bounded curvature.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

Entire Functions on Infinite Dimensional Spaces

In this work we study partial differential operators of infinite order on spaces of entire functions, generalizing some of the results presented in [2] and [5] to the new spaces introduced in [3].     

Invariant Spaces of Entire Functions

Mathematical Notes - Subspaces of the space of analytic functions on a convex domain in the complex plane that are invariant with respect to the differentiation operator are studied. The problem of...