X-valued Functions Research Articles

A note on some classes of operators on C(K,X)

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K , C (K, X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T : C (K, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y). We show that if T is a strongly bounded operator and : B(K, X) → Y is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 ≤ p < ∞) if and only if has the same property.

Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1&lt;3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX&gt;1 and TT−1&lt;λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

On integration in banach spaces and total sets

Let X be a Banach space and Γ ⊆ X∗ a total linear subspace. We study the concept of Γ-integrability for X-valued functions f defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions ⟨x∗, f ⟩ for x∗ ∈ Γ. We show that Γ-integrability and Pettis integrability are equivalent whenever X has Plichko’s property () (meaning that every w∗ - sequentially closed subspace of X∗ is w∗ -closed). This property is enjoyed by many Banach spaces including all spaces with w∗ -angelic dual as well as all spaces which are w∗ -sequentially dense in their bidual. A particular case of special interest arises when considering Γ = T ∗ (Y ∗ ) for some injective operator T : X → Y . Within this framework, we show that if T : X → Y is a semi-embedding, X has property () and Y has the Radon-Nikodým property, then X has the weak Radon-Nikodým property. This extends earlier results by Delbaen (for separable X) and Diestel and Uhl (for weakly -analytic X).

Generalized Numerical Index of Function Algebras

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

Operators and multipliers on weighted Bergman spaces

In 1992, Blasco considered a weighted Bergman space using Dini-weight function. He proved that a linear operator T on weighted Bergman space to any Banach space X is bounded if and only if certain one parameter fractional derivative of a single X-valued analytic function satisfy some growth condition. In this paper, we use a three parameters fractional derivative of a single X-valued analytic function and provide an equivalent condition. Our technique uses the Gaussian hypergeometric functions. Furthermore we supply some conditions on the parameters a, b and c under which the Gaussian hypergeometric functions F(a, b; c; z) are Dini-weight.

An isomorphic property in spaces of compact operators and some classes of operators on C(K,X)

Let $${K_{w^{*}}(X^{*},Y)}$$ denote the set of all w*−w continuous compact operators from X* to Y. We investigate whether the space $${K_{w^{*}}(X^{*},Y)}$$ has property RDP * ( $${1\le p < \infty}$$ ) when X and $${Y}$$ have the same property. Suppose X and Y are Banach spaces, K is a compact Hausdorff space, $${\Sigma}$$ is the $${\sigma}$$ -algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and $${T: C(K,X)\to Y}$$ is a strongly bounded operator with representing measure $${m: \Sigma \to L(X,Y)}$$ . We show that if T is a strongly bounded operator and $${\hat{T}: B(K, X) \to Y}$$ is its extension, then T* is p-convergent if and only if $${\hat{T}^{*}}$$ is p-convergent, for $${1\le p < \infty}$$ .

Quasi uniform convexity—Revisited

Quasi uniform convexity (QUC) is a geometric property of Banach spaces, introduced in 1973 by J.R. Calder et al., which implies existence of Chebyshev centers for bounded sets. We extend and strengthen some known results about this property. We show that (QUC) is equivalent to existence and continuous dependence (in the Hausdorff metric) of Chebyshev centers of bounded sets. If X is (QUC) then the space C(K;X) of continuous X-valued functions on a compact K is (QUC) as well. We also show that a sufficient condition introduced by L. Pevac already implies (QUC), and we provide a couple of new sufficient conditions for (QUC). Together with Chebyshev centers, we consider also asymptotic centers for bounded sequences or nets (of points or sets).

Best N-Simultaneous Approximation in Lp(μ,X)

Let X be a Banach space. Let 1≤p&lt;∞ and denote by Lp(μ,X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω,Σ,μ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(μ,W) is N-simultaneously proximinal in Lp(μ,X) for any arbitrary monotonous norm N in Rn.

An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate.

Vector-valued invariant means revisited once again

Banach spaces that are complemented in the second dual are characterised precisely as those spaces X which enjoy the property that for every amenable semigroup S there exists an X-valued analogue of an invariant mean defined on the Banach space of all bounded X-valued functions on S. This was first observed by Bustos Domecq (2002) [5], however the original proof was slightly flawed as remarked by Lipecki. The primary aim of this note is to present a corrected version of the proof. We also demonstrate that universally separably injective spaces always admit invariant means with respect to countable amenable semigroups, thus such semigroups are not rich enough to capture complementation in the second dual as spaces falling into this class need not be complemented in the second dual.

Boundary values of vector-valued Hardy spaces on nonsmooth domains and the Radon–Nikodym property

We define Hardy spaces of functions taking values on a Banach space X over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space X in terms of the existence of nontangential limits of X-valued functions u in the corresponding Hardy space with index p≥1, (b) to identify the function of the boundary values of u in the Hardy space with index p>1 with an element in the space VXp of measures of p-bounded variation in the absence of the Radon–Nikodym property of X. This extends similar results already known on the unit disk of C and the semispace Rn×(0,∞).

On metric semigroups-valued functions of bounded Riesz- $$\Phi $$ Φ -variation in several variables

Using classical techniques related to the so-called Hardy–Vitali variation, we present the class of X-valued functions of bounded \(\Phi \)-variation in several variables, where \((X,d,+ )\) is a metric semigroup. We exhibit some of the main properties of this class; among them, we show that this class can be made into a normed space and present a counterpart of the renowned Riesz’s Lemma for the case in which \(X=\mathbb {R}\) with its usual metric.

Generalized eigenspaces of generators of evolution semigroups

A periodic evolutionary process {U(t,s)}t≥s with period τ on a Banach space X gives an evolution semigroup with the generator L on the space of τ-periodic continuous X-valued functions. We prove that the generalized eigenspaces NL=N((αI−L)n) and NU=N((eατI−U(τ,0))n) are isomorphic. The algebraic representation of a function u∈NL is given by the initial value u(0)=w∈NU.

On the isomorphic classification of [formula omitted] spaces

We provide isomorphic classifications of some C(K,X) spaces, the Banach spaces of all continuous X-valued functions defined on infinite compact metric spaces K, equipped with the supremum norm. We first introduce the concept of ω1-quotient of Banach spaces X. Thus, we prove that if X has some ω1-quotient which is uniformly convex, then for all K1 and K2 the following statements are equivalent:(a)C(K1,X) is isomorphic to C(K2,X).(b)C(K1) is isomorphic to C(K2). This allows us to classify, up to an isomorphism, some C(K,Y⊕lp(Γ)) spaces, 1<p≤∞, and certain C(S) spaces involving large compact Hausdorff spaces S.

Weak forms of Banach–Stone theorem for C0(K,X) spaces via the αth derivatives of K

Let X be a Banach space and S be a locally compact Hausdorff space. By C0(S,X) we will stand the Banach space of all continuous X-valued functions on S endowed with the supremum norm.Suppose that C0(S,X) contains a copy of some C0(K) space with K infinite. Does it follow that the cardinality of the αth derivative of K is less than or equal to the αth derivative of S, for every ordinal number α? In general the answer is no, even when α=0.In the present paper we prove that the answer is yes whenever X contains no copy of c0 and α=0. Moreover, in the case where α>0 and the αth derivative of S is infinite, we show that the existence an isomorphism from C0(K) into C0(S,X) with distortion ‖T‖‖T−1‖ strictly less than 3 provides also a positive answer to this question.As a consequence, we improve a classical Cengiz theorem and a recent result on isomorphisms between spaces of vector-valued continuous functions by obtaining two weak forms of Banach–Stone theorem for C0(S,X) spaces via the αth derivatives of S.

Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ-periodic evolutionary process {U(t,s)}t≥s on the space Pτ(X) of all τ-periodic continuous X-valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−ϵF(p,ϵ) under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0). The proof is based on a new decomposition of the space Pτ(X) by constructing a right inverse of L.

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S l ) l⩾1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c 0, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.

Linear bounds for Calderón–Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calderón–Zygmund singular integral operators can be extended to X-valued functions if and only if the Banach space X has the UMD property. The dependence of the norm of an X-valued Calderón–Zygmund operator on the UMD constant of the space X is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderón–Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.

How does the distortion of linear embedding of [formula omitted] into [formula omitted] spaces depend on the height of [formula omitted

Let C0(K) denote the space of all continuous scalar-valued functions defined on the locally compact Hausdorff space K which vanish at infinity, provided with the supremum norm. Let Γ be an infinite set endowed with discrete topology and X a Banach space. We denote by C0(Γ,X) the Banach space of X-valued functions defined on Γ which vanish at infinity, provided with the supremum norm. In this paper, we prove that, if X has non-trivial cotype and there exists a linear isomorphism T from C0(K) into C0(Γ,X), then K has finite height ht(K), and the distortion ‖T‖‖T−1‖ is greater than or equal to 2ht(K)−1. The statement of this theorem is optimal and improves a 1970 result of Gordon.

Eberlein-weakly almost periodic (in Stepanov-like sense) functions and application

In this article, we prove a number of properties concerning a (new) class of (Stepanov-like) Eberlein-weakly almost periodic (S P -E.w. a. p.) functions with values in a Banach space. We use the results obtained to study the asymptotic behaviour of solutions to the evolution equation: where A is the generator of an integral resolvent family in a Banach space 𝕏, a ∈ L 1(ℝ), and f is a given 𝕏-valued function on ℝ. The objective is to deduce Eberlein-weakly almost periodicity (in Stepanov-like sense) of the solution u from corresponding properties on the part f.