Series expansions in Fréchet spaces and their duals, construction of Fréchet frames

Frames for Fréchet spaces X_F with respect to Fréchet sequence spaces Θ_F are studied, and conditions implying series expansions in X_F and are determined. If is a Θ₀-frame for X₀ and Θ_F (resp. X_F) is given, we construct a sequence {X_s}_s∈N0, X_s⊂X_s−1, s∈N, (resp. {Θ_s}_s∈N0, Θ_s⊂Θ_s−1, s∈N), so that is a pre-F-frame or F-frame for X_F with respect to Θ_F under different assumptions given on X₀, Θ₀ and Θ_F (resp. X_F).     

Quasi-reflexive Fréchet spaces and mean ergodicity

We characterize quasi-reflexive Fréchet spaces with a basis in terms of the properties of this basis. As a consequence we prove that a Fréchet space with a basis is quasi-reflexive of order one if and only if for every power bounded operator T, either T or T^′ is mean ergodic.     

Continuity of mappings between Fréchet spaces

For all linear and many nonlinear operators between Fréchet spaces, several continuity results are established. Especially, a new closed graph theorem is given.     

On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

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)     

Almost periodicity in Fréchet spaces

In this paper we deal with almost periodic functions with values in a Fréchet space. We apply obtained results to prove the existence of solutions of the initial value problem as well as the Volterra integral equation in this class of functions. We also introduce and investigate asymptotically almost periodic functions with values in a Fréchet space.     

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C², or even C¹, or even Fréchet-differentiable.     

The density condition and distinguishedness for weighted Fréchet spaces of holomorphic functions

We consider weighted Fréchet spaces of holomorphic functions which are defined as countable intersections of weighted Banach spaces of type H^∞. We study when these spaces have Stefan Heinrich's density condition and when they are distinguished.     

The Fréchet Approximate Jacobian and Local Uniqueness in Variational Inequalities

We introduce the concept of Fréchet approximate Jacobian matrices for continuous vector functions and use it to establish some sufficient criteria for the local uniqueness of solutions to a variational inequality problem involving continuous, not necessarily locally Lipschitz functions. Examples are also given to illustrate the usefulness of our approach.     

Universal and chaotic multipliers on spaces of operators

We use tensor product techniques to study universality, hypercyclicity and chaos of multipliers defined on operator ideals and of multiplication operators on the space of all continuous and linear operators, thus continuing the work of Kit Chan. We also obtain the first examples of outer multipliers on a Banach algebra which are chaotic in the sense of Devaney, and prove sufficient conditions for the existence of closed subspaces of universal vectors for operators between Fréchet spaces.     

Workshop lecture on products of Fréchet spaces

The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α₄ Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α₄, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α₃ Fréchet spaces and its subclass of ℵ₀-bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α₃ and Fréchet and being ℵ₀-bisequential.     

On some relationships between biorthogonal systems,linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ?₁‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.     

Remarks on quasi-variational inequalities and fixed points in locally convex topological vector spaces

In this note, we first prove existence theorems for noncompact generalized quasivariational inequalities. As applications, two fixed point theorems for upper or lower semicontinuous multivalued mappings without compact domains are given in locally Hausdorff topological vector spaces. These results generalize or improve corresponding results in the literature.     

Local means,wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces.