Function Spaces

PREFACE.

CONTENTS.

No abstract received.

No abstract received.

I am not going to give here a detailed history of mathematics in Wrocław. I would like only to point out some scattered facts, names and documents, more or less interesting for a wider audience.

This paper presents the definition and some properties of (NC⁽ⁿ⁾)-almost periodic functions, i.e. uniformly almost periodic functions in the sense of Levitan with all derivatives of order n.

For every closed set G ⊂ [0, 1] there exists an injective linear operator T = T_G bounded on L₁[0, 1] and L_∞[0, 1] such that T is normally solvable in an Orlicz space L_M[0, 1] if and only if G ∩ [α_M, β_M] = ∅ (α_M and β_M are the Boyd indices of the space L_M). Analogous result holds also for isomorphisms of spaces of functions defined on the semiaxis [0, ∞).

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Köthe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions ||f||_X : Ω → [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p-variation (for E = L^p), Φ-variation (for E = L^Φ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation V_E(X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X))* = V_E′ (X*), that V_E(X) can be identified with space of cone absolutely summing operators from E′ into X and that E(X) = V_E(X) if and only if X has the RNP property.

We prove a Stone-Weierstrass theorem for subspaces of C₀(X, ℝ). If an ultraseparating subspace B of C₀(X, ℝ) has a continuous operating function that is neither odd nor affine in every neighbourhood of 0 then B is uniformly dense in C₀(X, ℝ).

In this paper we deal with hyperconvex metric spaces. We compare a certain class of hyperconvex metric spaces with another one: complete ℝ-trees. An elementary proof of hyperconvexity of “finite” ℝ-trees is also presented. In the second part of the paper we introduce and investigate an index which measures how far a subset of a given metric space is from having the small ball property introduced by Behrends and Kadets.

Using relations between the Zygmund–Calderon Taylor spaces and those of approximation space theory we establish several new facts of these and more general spaces which include interpolation and duality.

The concept of w-discontinuous mappings is defined. We consider the set w − C(A, Y) of all w-discontinuous mappings from a compact subset A of metric space Y to metric space Y. It can be shown that space w − C(A, Y) is not separable and that if Y is complete space then w − C(A, Y) is also complete space. We establish some necessary and sufficient conditions for relatively compactness of families of w-discontinuous mappings in space w − C(A, Y).

In this paper we investigate the measurable transformations that induce composition operators on L^ϕ-1 spaces and study some properties of these operators.

Tauberian operators appeared in a problem of summability and were studied by Kalton and Wilansky and other authors. Moreover they have received some attention because they form a broader class than that of isomorphisms (into) but yet they preserve some isomorphic properties of Banach spaces.

The aim of this paper is to generalize the definition of tauberian operators by using the notion of weakly conditionally compact set.

As consequence, some properties of the classic tauberian operators have counterparts in this new context, taking into account that the roll played by the reflexivity in the tauberian operators is played in the case of weakly conditionally compact tauberian operators by the fact of not containing copies of l¹.

We suggest a new approach to prove the dominating norm property for spaces of Whitney functions, based on the estimation of least deviation of polynomials on Cantor-type sets. In this way we prove that the generalized Cantor sets of finite type and logarithmic dimension 1 have the extension property, since by Tidten-Vogt characterization a compact set K has the extension property iff the space has the property DN.

First we will show that property (β) of Rolewicz can be equivalently considered in order continuous Köthe sequence spaces only for nonnegative elements. Notice that the respective result for Köthe function spaces is different ²¹. Next we study a property (β), strict and uniform orthogonal convexity ((SC^⊥), (UC^⊥)) in generalized Calderón-Łozanowskiĭ sequence spaces. Finally we investigate properties (UC^⊥) and (β) in Orlicz-Lorentz sequence spaces (the obtained result extends the respective ones from ⁸ and ²²).

Under group symmetry conditions of the original nonlinear equation the necessary and sufficient conditions of the simultaneous reduction for the corresponding bifurcation systems by unknowns and equations are obtained. Connections between the reduction possibilities of the branching equation (BEq) and its potentiality properties are discussed. The equivalence of Lyapounov’s and Schmidt’s bifurcation equations in the root – subspace and the intertwining properties (group and non-group symmetry) inheritance theorem are proved.

We consider a fairly large class of Hilbert spaces of holomorphic functions in one complex variable which contains the Hardy, the Fock and the Bergman space. The purpose of our paper is to study linear bounded operators on these spaces by means of their matrix elements 〈Te_k, e_j〉 where e_k are the normalized monomials. To this end we introduce a “Fourier series expansion” of T into special operators and discuss how the properties of T are reflected in its “Fourier summands”. We characterize, among others, the essential spectrum and the essential norm for a large class of T including many Toeplitz operators.

Lindenstrauss and Pełczyński¹³ in 1968 proved that for a compact Hausdorff space K every bounded linear operator on C(K) with values in some L_p(μ), 2 ≤ p < ∞ is absolutely (p, 2)-summing. More generally, Maurey¹⁷ in 1973/74 showed that this is also valid for any Banach space X of cotype p instead of L_p(μ), and Defant and the author⁶ in 2000 proved the case where X is a p/2-Hilbert space instead of ℓ_p. In this note, we investigate refinements of the above results within the more general setting of absolutely (E, 2)-summing operators and uniformly convex Banach lattices. We apply our results to obtain lower estimate for Gelfand numbers of certain finite-dimensional identity maps.

Let E be an ideal of L⁰ over a complete σ-finite rneasure space (Ω, Σ, μ), E′-the Köthe dual of E with supp E = supp E′ = Ω. Let (X,|| · ||_x) be a real Banach space, X*-the topological dual of X. Let E(X) be subspaces of the space L⁰(X) of equivalence classes of strongly Σ-measurable function f : Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ||f(·)||_x belongs to E(X). We give a criterion for relative sequential σ(E(X), E′(X*))-compactness in E(X). We characterize Banach spaces X having the Radon-Nikodym Property in terms of relatively sequentially σ(E(X), E′(X*))-compact subsets of E(X). Moreover, we show that E(X) is sequentially σ(E(X), E′(X*))-complete if and only if E is sequentially σ(E, E′)-complete (i.e., E is perfect), X is sequentially weakly complete and X has the Radon-Nikodym Property (with respect to ^μ|_A∩Σ for every A ∈ Σ with μ(A) < ∞ and χ_A ∈ E′).

Minimal elements in the family of closed bounded convex sets in a topological vector space X were studied in many papers (see e.g [1], [2], [3], [5], [7]). In this paper we give positive answer to the question asked by R. Urbański concerning the existence of minimal elements in with respect to certain partial orders ≤* and ≤_*.

We propose a general method for extending various interpolation theorems for Banach function spaces to corresponding quasi-Banach spaces. The method consists in a special measure transformation which states a homeomorphism between the cones of decreasing functions in considered spaces. We also give a quasi-Banach analog of the Krein-Semenov theorem on weak interpolation.

Let X be a Banach space with dual X*, and let J : X → 2^X* be the duality mapping defined by . We prove that if X is a function space so that for every positive simple function x ∈ X there exists a scalar k_x so that k_x · x ∈ J(x) then X is isometric to a Hilbert space. This result is valid in both real and complex spaces.

A convex subset Q of a Hausdorff topological vector space (HTVS) is called locally nonconical (LNC) if for every two points x, y ∈ Q there is a relative neighborhood U of x in Q such that U + ½(y − x) ⊂ Q. This paper includes the following: (1) a summary of the history of the LNC concept, (2) examples of LNC sets, (3) a geometric characterization of closed LNC sets with nonempty interior in a locally convex HTVS, (4) a result (Theorem 2.2) on the preservation of the LNC property by bounded linear maps between Hilbert spaces, (5) a list of Banach spaces whose closed unit balls possess the LNC property, and (6) a list of those that lack the property. A corollary of Theorem 2.2 is that the sum A + B of any two closed LNC subsets A and B of ℝⁿ is LNC. An example is given to show that Theorem 2.2 does not hold if the target space is not a Hilbert space. Finally, it is shown that the preimage of an LNC set under a continuous linear map is LNC.

In this work we generalize a number of results from¹ on extension of measurable functions and local operators. In comparasion with ¹ we assume neither measure to be defined on considered σ– algebras, nor the metric space on functions values is complete. As well in contrast to ¹ here we do not make use Boolean algebras. This circumstance made possible essentially to simplify the presentation.

A quasi-normed function space A in ℝⁿ is said to have the positivity property if any real function f ∈ A can be represented as f = f₁ − f₂, where f₁ ∈ A and f₂ ∈ A are non-negative functions such that ||f |A|| and ||f₁ |A|| + ||f₂ |A|| are equivalent quasi-norms on A. The paper deals with this problem in case of and .