Werner J. Ricker

Optimal Domains for Kernel Operators via Interpolation

The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval (0,1) is considered. The techniques used are based on interpolation theory and integration with respect to C((0,1))-valued measures.

Banach lattices with the Fatou property and optimal domains of kernel operators

Abstract New features of the Banach function space L 1 w ( v ), that is, the space of all v -scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operators taking their values in a Banach function space.

Spectral measures and the Bade reflexivity theorem

Let B be a strongly equicontinuous Boolean algebra of projections on the quasi-complete locally convex space X and assume that the space L(X) of continuous linear operators on X is sequentially complete for the strong operator topology. Methods of integration with respect to spectral measures are used to show that the closed algebra generated by B in L(X) consists precisely of those continuous linear operators on X which leave invariant each closed B-invariant subspace of X.

On Mean Ergodic Operators

Aspects of the theory of mean ergodic operators and bases in Frechet spaces were recently developed in [1]. This investigation is extended here to the class of barrelled locally convex spaces. Duality theory, also for operators, plays a prominent role.

Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).

Optimal extension of the Hausdorff-Young inequality

Abstract Given 1 < p < 2, we construct a Banach function space with σ-order continuous norm which contains and has the property that the Fourier transform map has a continuous ℓ p′ (ℤ)-valued extension to . Moreover, is maximal with these properties and satisfies with both containments proper. Each turns out to be a weakly sequentially complete, translation invariant, homogeneous Banach space and consists precisely of those functions such that for every Borel set . This answers a question of R. E. Edwards posed some 40 years ago.

Separability of the L 1 -space of a vector measure

Let Σ be a σ-algebra of subsets of some set Ω and let μ:Σ→[0,∞] be a σ-additive measure. If Σ(μ) denotes the set of all elements of Σ with finite μ-measure (where sets equal μ-a.e. are identified in the usual way), then a metric d can be defined in Σ(μ) by the formula here E Δ F = ( E \ F ) ∪ ( F \ E ) denotes the symmetric difference of E and F . The measure μ is called separable whenever the metric space (Σ(μ), d ) is separable. It is a classical result that μ is separable if and only if the Banach space L 1 (μ), is separable [8, p.137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If Σ happens to be the σ-algebra of μ-measurable sets constructed (via outer-measure μ*) by extending μ defined originally on merely a semi-ring of sets Γ ⊆ Σ, then it is also classical that the countability of Γ guarantees the separability of μ and hence, also of L 1 (μ), [8, p. 69].

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=∫fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that σ(T) is precisely the closedP-essential range of the functionf or equivalently, that σ(T) is equal to the support of the (unique) equicontinuous spectral measureQ* defined on the Borel sets of the extended complex plane ℂ* such thatQ*({∞})=0 andT=∫zdQ*(z). This result is then used to prove a spectral mapping theorem; namely, thatg(σ(T))=σ(g(T)) for anyQ*-integrable functiong: ℂ* → ℂ* which is continuous on σ(T). This is an improvement on previous results of this type since it covers the case wheng(σ(T))/{∞} is an unbounded set inℂ a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.

Montel resolvents and uniformly mean ergodic semigroups of linear operators

Abstract For C 0-semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the infinitesimal generator A being a closed subspace or, whether 0 belongs to the resolvent set of A or is a simple pole of the resolvent map . It is shown in the setting of locally convex spaces (even in Fréchet spaces), that neither of these criteria remain equivalent to uniform ergodicity of the semigroup (i.e., the averages should now converge for the topology of uniform convergence on bounded sets). Our aim is to exhibit new results dealing with uniform mean ergodicity of C 0-semigroups in more general spaces. A characterization of when a complete, barrelled space with a basis is Montel, in terms of uniform mean ergodicity of certain C 0-semigroups acting in the space, is also presented.

The Cesàro operator in the Fréchet spaces l^p+ and L^p-

The research of the first two authors was partially supported by the projects MTM2010-15200, MTM2013-43540-P and GVA Prometeo II/2013/013 (Spain).