On splittings and integration of almost periodic functions with and without geometry

Abstract

Recently, the authors introduced the notion of weighted semigroups which apply to sun-dual semigroups and especially to the translation semigroup on the space of left continuous functions with values in dual spaces. In this article, we will show that it is sufficient that we either assume geometry on the Banach, or an abelian structure on the minimal subgroup to prove almost periodicity. This yields a different approach to the almost periodicity of semigroups and integrals, by extending Basit generalized Kadets result to general groups, to obtain almost periodicity.

1 Introduction

In this paper, we will investigate conditions for initial values of abstract Cauchy problems on Banach spaces, which allow unique solution splittings into an almost periodic part and a noisy part. In particular, we study problems of the following form

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x}(t)=Ax(t),&{}{}\quad t\ge 0,\\ x(0)=x_0\in X.&{}{} \end{array}\right. } \end{aligned}$$

We will be able to do this, even if the well-known Jacobs-deLeeuw-Glicksberg splitting, cf. [9], does not apply. Moreover, we extend the methods worked out by Kadets who studied the integration of almost periodic functions with values in Banach spaces, cf. [26]. More on the theory and applications of almost periodic functions can for example be found in the research monographs by Levitan [29], Hino, Naito, Nguyen and Shin [23] or Kostić [27]. We will also see how the theory of operator semigroups contributes in this case.

In 1969, Kadets studied integration of almost periodic functions with values in Banach spaces, cf. [26]. As a matter of fact, also the theory of operator semigroups plays an important role in this framework. In particular, the translation semigroup $\left\{ {{T}(t)} \right\} _{t\in {\mathbf{\mathbb R}} }$ on the space $\textrm{BUC}(\mathbb {R},X)$ of bounded and uniformly continuous functions, as well as the corresponding difference operators $T(t+s)-T(t)$, are crucial ingredients. An operator theoretical approach to Kadets integration theorem of almost periodic functions with values in Banach spaces can for example be found in the monograph by Arendt et al. [1, Thm. 4.6.11]. A generalization of Kadets’ integration theorem for general groups has been investigated by Basit [5]. Applications of Kadets’ theorem have been studied by Ruess and Summers [34, 35] and more recently by Farkas and Kreidler [16], just to mention a few.

In this article, we will show that the difference theorems by [5, 21, 26] also very fruitful in the context of weighted semigroups, which were introduced in [7]. By making an extension of these works we will be able to obtain an almost periodic part, in an absence of an abelian structure. However, we have to restrict ourselves to a given underlying Banach space geometry. One of the classical assumptions is that the Banach space does not contain an isomorphic copy of the sequence space $\textrm{c}_0$. A characterisation of Banach spaces containing $\textrm{c}_0$ is for example given by [1, Thm. 4.6.14]. It is noteworthy that the Banach space $\textrm{c}_0$ as well as related properties have intensively been studied in the past, see for example [2, 15, 19, 32, 33]. If we skip the assumption on the geometry of the Banach space but restrict ourselves to an abelian structure, our assumption reduces to a norm-separability condition of the closure with respect to the locally convex topology coming with the definition of weighted semigroups.

As an application we give an abstract condition for the reversible part of a bounded integral of an Eberlein weakly almost periodic function to be almost periodic, whereby the integral need not to be Eberlein weakly almost periodic.

2 Preliminaries and notations

The translation semigroup on $\textrm{BUC}(\mathbf{\mathbb R},X)$ is defined as follows

$$\begin{aligned} \begin{array}{ccccc} {T(t)} &{} : &{} {\textrm{BUC}(\mathbf{\mathbb R},X)} &{} \longrightarrow &{} {\textrm{BUC}(\mathbf{\mathbb R},X)} \\ &{} &{} {f} &{} \longmapsto &{} \displaystyle { \left\{ {s\mapsto f_t(s):=f(s+t)} \right\} } \end{array} . \end{aligned}$$

Let $U\in \left\{ {\mathbf{\mathbb R},\mathbf{\mathbb R^+}} \right\} .$ Then $f\in \textrm{BUC}(U,X)$ if, and only if, the translation semigroup

$$\begin{aligned} \begin{array}{ccccc} {T} &{} : &{} {U} &{} \longrightarrow &{} {\textrm{BUC}(U,X)} \\ &{} &{} {t} &{} \longmapsto &{} \displaystyle {T(t)f} \end{array} \end{aligned}$$

is continuous. So one natural generalisation of uniform continuity in a semigroup ${\mathscr {S}}$ is the continuity of the left or right translation semigroup. In order to avoid confusions, we follow an extended vector-valued notation given in [3], and quote $\textrm{BUC}({\mathscr {S}},X)$ only if a uniformity on ${\mathscr {S}}$ is given.

The orbit of a function $f\in \textrm{BUC}(\mathbf{\mathbb R},X)$ under the translation semigroup is defined by

$$\begin{aligned} \mathscr {O}(f):= \left\{ {T(t)f:\ t\in \mathbf{\mathbb R}} \right\} . \end{aligned}$$

A function $f\in \textrm{BUC}(\mathbf{\mathbb R},X)$ is called almost periodic if its orbit is relatively compact. By $\textrm{AP}(\mathbf{\mathbb R},X)$ we denote the set of all almost periodic functions, i.e., one has

$$\begin{aligned} \textrm{AP}(\mathbf{\mathbb R},X):= \left\{ {f\in \textrm{BUC}(\mathbf{\mathbb R},X):\ \mathscr {O}(f) \text{ relatively } \text{ compact }} \right\} . \end{aligned}$$

The above can be generalised to a general semigroup setting. We define for a semigroup ${\mathscr {S}},$ with a Hausdorff topology

$$\begin{aligned} \mathscr {O}_L(f):= \left\{ { \left\{ {s\mapsto f(rs)} \right\} :\ r \in {\mathscr {S}}} \right\} \text{ and } \mathscr {O}_R(f):= \left\{ { \left\{ {s\mapsto f(sr)} \right\} :\ r \in {\mathscr {S}}} \right\} . \end{aligned}$$

In some cases relative compactness of orbits imply uniform continuity. Considering the left $( \left\{ {{L}(t)} \right\} _{t\in {{\mathscr {S}}} })$ and right $( \left\{ {{R}(t)} \right\} _{t\in {{\mathscr {S}}} })$ translation semigroups, we define:

$$\begin{aligned} \textrm{LC}({\mathscr {S}},X):= & {} \left\{ {f\in \textrm{C}_\textrm{b}({\mathscr {S}},X):\ {\mathscr {S}}\ni t\mapsto L(t)f \in \textrm{C}_\textrm{b}({\mathscr {S}},X) \text{ is } \text{ continuous } } \right\} \\ \textrm{LC}_{\textrm{wrc}}({\mathscr {S}},X):= & {} \left\{ {f\in \textrm{LC}({\mathscr {S}},X):\ f({\mathscr {S}}) \text{ weakly } \text{ relatively } \text{ compact }} \right\} \\ \textrm{RC}({\mathscr {S}},X):= & {} \left\{ {f\in \textrm{C}_\textrm{b}({\mathscr {S}},X):\ {\mathscr {S}}\ni t\mapsto R(t)f \in \textrm{C}_\textrm{b}({\mathscr {S}},X) \text{ is } \text{ continuous } } \right\} \\ \textrm{LRC}({\mathscr {S}},X):= & {} \textrm{LC}({\mathscr {S}},X)\cap \textrm{RC}({\mathscr {S}},X) \end{aligned}$$

Throughout the whole paper we assume U to be a Hausdorff abelian semitopological semigroup. For U one may think about an additive semigroup in n-dimensional reals with the natural topology. Whereby ${\mathscr {S}}$ mostly is a possibly non-abelian semigroup of bounded operators in $\mathscr {L}(Z),$ equipped with some locally convex operator topology. Whereby, the locally convex topology is given on the Banach space Z. A function $f\in \textrm{C}_\textrm{b}(U,X)$ is called Eberlein-weak almost periodic, or E.-wap for short, if the orbit with respect to the translation semigroup is relatively $\sigma (\textrm{C}_\textrm{b}(U,X),\textrm{C}_\textrm{b}(U,X)^*)$-compact. Furthermore, we call $f\in \textrm{C}_\textrm{b}(U,X)$ to be asymptotically almost periodic, or aap for short, if the orbit is relatively compact with respect to the uniform topology. We also recall the definition of the following function spaces, cf. [7, Def. 3.3], as they are crucial for our theory.

$$\begin{aligned} \textrm{W}(U,X):= & {} \left\{ {f\in \textrm{LC}(U,X): \ f \text{ is } \text{ E.-wap }} \right\} , \\ \textrm{AAP}(U,X):= & {} \left\{ {f\in \textrm{C}_\textrm{b}(U,X):\ f \text{ is } \text{ asymptotically } \text{ almost } \text{ periodic }} \right\} . \end{aligned}$$

We note that the function spaces defined above are Banach spaces with respect to the uniform topology if X is. This dues to the uniform boundedness of the translation semigroup as well as an $\varepsilon /3$-argument. Also $\textrm{W}(U,X)$ is Banach space due the Grothendieck weak compactness condition [14, Lemma 13.32, p.591]. Again, by an $\varepsilon /3$-argument, $\textrm{AAP}(U,X)$ also becomes a Banach space. As the weak operator topology compactification of the translation semigroup ${\mathscr {T}}:=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{WOT}$ on the space $\textrm{W}(U,X)$ is a compact abelian semitopological semigroup, see for example [28, Thm. 4.4, p.105], we obtain the topological splitting

$$\begin{aligned} \textrm{W}(U,X)=\textrm{W}(U,X)_{\textrm{rev}}\oplus \textrm{W}(U,X)_0 \end{aligned}$$

with

$$\begin{aligned} \textrm{W}_0(U,X):= & {} \left\{ {f\in \textrm{W}(U,X):\ f_{t_{\alpha }} \rightarrow 0 \text{ weakly } \text{ for } \text{ some } \left\{ { {t}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset U } \right\} ,\\ \textrm{W}(U,X)_{\textrm{rev}}:= & {} \left\{ {f\in \textrm{W}(U,X): g\in \overline{\mathscr {O}(f)}^{weak} \text{ then } f\in \overline{\mathscr {O}(g)}^{weak}} \right\} \end{aligned}$$

and that the restriction of ${\mathscr {T}}$ to $\textrm{W}(U,X)_{\textrm{rev}}$ is a group. Let G denote a topological group. It is obvious that if $U=G$, the notions of asymptotically almost periodic and almost periodic coincide.

Last but not least, for a Banach space X, an abelian semitopological semigroup U and a norming and norm closed space $\Lambda \subset \textrm{LC}(U,X)^*$ (for more details we refer the reader to [7, Def. 4.1]) we define a Hausdorff locally convex topology $\tau _X$ on X by saying that a net $(x_\alpha )_{\alpha \in A}$ in X converges to some $x\in X$ if, and only if, for some $t\in U$ one has that $x_{\alpha }= f_{\lambda }(t)\rightarrow f(t)= x$ with respect to the topology $\sigma (\textrm{LC}(U,X),\Lambda ).$ As the constant functions are left continuous, the topology is well defined. Moreover, we observe that the new topology is also norming on X as $\Lambda$ is assumed to be norming. With this we define

$$\begin{aligned}{} & {} \textrm{WW}_{\Lambda }(U,X)\\{} & {} \quad := \left\{ {f\in \textrm{LC}(U,X):\ \left\{ {s \mapsto \left\langle x^*,f_s\right\rangle } \right\} \in W(U) \text{ for } \text{ all } x^*\in \Lambda \text{ and } \overline{f(U)}\ \tau _X\text{-complete } } \right\} . \end{aligned}$$

The requirement for completeness of the image can be motivated by the need for an extension of the evaluation operator

$$\begin{aligned} \begin{array}{ccccc} {\delta _x} &{} : &{} {(\mathscr {O}(f),\sigma (\textrm{LC}(U,X),\Lambda ))} &{} \longrightarrow &{} {\overline{f(U)}^{\tau _X}} \\ &{} &{} {g} &{} \longmapsto &{} \displaystyle {g(x)} \end{array} \end{aligned}$$

to

$$\begin{aligned} \begin{array}{ccccc} {\widetilde{\delta _x}} &{}{} : &{}{} {(\overline{\mathscr {O}(f)}^{\sigma (\text {LC}(U,X),\Lambda ))},\sigma (\text {LC}(U,X),\Lambda ))} &{}{} \longrightarrow &{}{} {\overline{f(U)}^{\tau _X}} \\ {} &{}{} &{}{} {g} &{}{} \longmapsto &{}{} \displaystyle {g(x).} \end{array} \end{aligned}$$

We observe that assuming quasi-completeness for the whole space is too strong. For this, we just consider as an example the space $\textrm{BUC}(\mathbf{\mathbb R},X^*)$ and for given $f\in \textrm{BUC}(\mathbf{\mathbb R},X^*)$ the subset $\overline{\mathscr {O}(f)}^{pointwise}$ equipped with the pointwise topology. While assuming only completeness and not compactness, for the above definition linearity and completeness of the space still needs to be verified explicitly.

For unexplained terminology we refer for example to the work of Ruppert [37], Jarchow [25], Ellis [12], just to mention a few. As this work is a follow up on the work of Budde and Kreulich we also refer to [7] as one of the main references. The following definition is crucial for our work. In the upcoming section we discuss the notion of so-called weighted semigroups. Those objects are called weighted as they keep the compactification in $\mathscr {L}(Z).$ Hence, the minimal idempotent found by the general semigroup theory [37] serves for a splitting, but ${\mathscr {T}}:=\overline{ \left\{ {T(t):\ t\in U} \right\} }^{\tau \text {-}OT}\subset \mathscr {L}(Z)$ is not necessarily abelian. In this study Z,  depending on the viewpoint, may be general Banach spaces $Y,X,X^*,$ or some closed and translation invariant subspace of $\textrm{LC}(U,X),\textrm{LC}(U,X^*).$

3 On weighted semigroups

The definition of weighted semigroups collects the properties for a bounded set of operators in $\mathscr {L}(X)$ in order to obtain a compact left-semitopological semigroup. In consequence, it weights the continuity properties of the operators of the semigroup against the topology of needed or given compactness of the orbit. This observation results in the naming of such semigroups. In doing so, the results on right-semitopological semigroups of [37] become the starting point for the splittings.

Definition 3.1

For a Banach space X, a set of operators $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset \mathscr {L}(X)$ is called a weighted semigroup representation on X with respect to a locally convex topology $\tau$ if the following properties are satisfied:

• (1) For some $M\in \mathbf{\mathbb R^+},$ we have $\displaystyle \left\| {T(t)} \right\| \le M$ for all $t\in U.$

• (2) The topology $\tau$ on X is weaker than the norm-topology.

• (3) The norm is lower-semicontinuous with respect to $\tau$-topology, i.e., $x_{\lambda }\rightarrow x$ in the topology $\tau ,$ implies $\displaystyle \left\| {x} \right\| |\le \liminf _{\lambda } \displaystyle \left\| {x_{\lambda }} \right\|$.

• (4) $\left\{ {T(s):\ X\rightarrow X} \right\} _{s\in U}\subset \mathscr {L}(X)$ is a semigroup representation.

• (5) For all $y\in X$,

  $$\begin{aligned} \begin{array}{ccccc} {T_y} &{} : &{} {U} &{} \longrightarrow &{} {X} \\ &{} &{} {t} &{} \longmapsto &{} \displaystyle {T(t)y} \end{array} \end{aligned}$$

  is $\tau$-continuous.

• (6) $\overline{ \left\{ {T(t)y:\ t\in U} \right\} }^{\tau }\subset X$ is quasi-complete for all $y\in X$.

• (7) For all $s\in U$ and $y\in X,$

  $$\begin{aligned} \begin{array}{ccccc} {T(s)} &{} : &{} {(\overline{ \left\{ {T(t)y:\ t\in U} \right\} }^{\tau },\tau )} &{} \longrightarrow &{} {(\overline{ \left\{ {T(t)y:\ t\in U} \right\} }^{\tau },\tau )} \\ &{} &{} {z} &{} \longmapsto &{} \displaystyle {T(s)z} \end{array} \end{aligned}$$

  is continuous.

The additional locally convex topology $\tau$ appearing in Definition 3.1 yields an additional operator topology. In fact, we say that a net $\left\{ { {T}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset \mathscr {L}(Y,X)$ is $\tau \text {-}OT$ convergent to some $T\in \mathscr {L}(Y,X)$ if $Ty=\tau {\text {-}}\lim _{ {\alpha }\in {A}} T_{\alpha }y$ for all $y\in Y$. Let U throughout this paper be an abelian semitopological semigroup and $\left\{ {{T}(t)} \right\} _{t\in {U} }$ a weighted semigroup representation on X.

For the sake of completeness we give the following two results for norming topologies on Banach spaces. First, we show that norming topologies automatically lead to lower-semicontinuous locally convex topology on Banach spaces.

Proposition 3.2

[14], Ex. 3.90,p.161] Let X be a Banach space with a norming topology $\sigma (X,\Gamma )\subset \sigma (X,X^*)$, in particular, we have that

$$\begin{aligned} \sup _{\begin{array}{c} x^*\in \Gamma \\ \displaystyle \left\| {x^*} \right\| \le 1 \end{array}} \left| {\left\langle x,x^*\right\rangle } \right| = \displaystyle \left\| {x} \right\| , \end{aligned}$$

for all $x\in X$. Then the norm is $\sigma (X,\Gamma )$-lower-semicontinuos.

Next, we show that the Grothendieck lemma [14, Lem. 13.32, p. 591] on weak compactness extends to norming topologies.

Lemma 3.3

Let $\tau$ be a norming locally convex topology on X which is weaker than the norm. Then, whenever for sets $K\subset X,$ and $\left\{ { K_{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset X$ one has that $K\subset K_n+2^{-n}B_X$, the $\tau$-compactness of the sets $\left\{ { K_{n} } \right\} _{{n} \in \mathbf{\mathbb N}}$ imply the $\tau$-compactness of K.

Proof

Let $\left\{ { {x}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset K$. Then we find $\left\{ {x_{\alpha ,n}:n\in \mathbf{\mathbb N}} \right\} \subseteq K_n,$ such that $\displaystyle \left\| {x_{\alpha }-x_{\alpha ,n}} \right\| \le 2^{-n}$ for all $\alpha >\alpha _0$. Using Tychonov’s compactness theorem for $\prod _{n\in \mathbf{\mathbb N}} K_n$, we are able to find a subnet such that $\lim _{ {\lambda }\in {\Lambda }} x_{{\alpha }_{\lambda },n}=\gamma _n,$ for all $n \in \mathbf{\mathbb N}$. By $\Gamma$ we denote the collection of linear functionals which are weak continuous with respect to $\tau$. Then $\Gamma \subset X^*$, as $\tau$ is by assumption weaker than the norm and $\Gamma ^*$ is a Banach space. While $\Gamma$ is norming, the embedding

$$\begin{aligned} \begin{array}{ccccc} {\iota } &{} : &{} {X} &{} \longrightarrow &{} {\Gamma ^*} \\ &{} &{} {x} &{} \longmapsto &{} \displaystyle { \left\{ {\gamma \mapsto \left\langle \gamma ,x\right\rangle } \right\} } \end{array} \end{aligned}$$

yields that X is isometrically embedded into $\Gamma ^*$. Furthermore, $\sigma (X,\Gamma )$ is the restriction of the $w^*$-topology on $\Gamma ^*$ to X. Consequently, we may assume that $\sigma (\Gamma ^*,\Gamma )\text {-}\lim _{ {\lambda }\in {\Lambda }} x_{{\alpha }_{\lambda }}=\gamma _0\in \Gamma ^*$ and $\sigma (X,\Gamma )\text {-}\lim _{ {\lambda }\in {\Lambda }} x_{{\alpha }_{\lambda },n}=\gamma _n\in X$. Hence, in $\Gamma ^*$ we have

$$\begin{aligned} \iota (x_{{\alpha }_{\lambda }})-\gamma _0= & {} \iota (x_{{\alpha }_{\lambda }}-x_{{\alpha }_{\lambda },n}+x_{{\alpha }_{\lambda },n})-\gamma _0. \end{aligned}$$

We conclude, that $\displaystyle \left\| {(\iota (x_{{\alpha }_{\lambda }})-\gamma _0)+(\iota (x_{{\alpha }_{\lambda },n})-\gamma _0)} \right\| _{\Gamma ^*}\le \displaystyle \left\| {x_{{\alpha }_{\lambda }}-x_{{\alpha }_{\lambda },n}} \right\| _{\Gamma ^*}$. Using the lower semicontinuity of $\sigma (\Gamma ^*,\Gamma )$, see also Proposition 3.2, as well as the fact that $\iota : (X,\sigma (X,\Gamma ))\rightarrow (\Gamma ^*,\sigma (\Gamma ^*,\Gamma ))$ is continuous, we obtain

$$\begin{aligned} \displaystyle \left\| {0+(\gamma _n-\gamma _0)} \right\| _{\Gamma ^*}\le \liminf _{\lambda \in \Lambda } \displaystyle \left\| {x_{{\alpha }_{\lambda },n}-x_{{\alpha }_{\lambda }}} \right\| <2^{-n}. \end{aligned}$$

As X is complete, $\iota (X)$ is complete and we obtain $\gamma _0\in X$. Hence, $\lim _{ {n} \rightarrow \infty } \gamma _n=\gamma _0$ and we conclude by using the triangle inequality that starting with $\displaystyle \left\| {x_{{\alpha }_{\lambda }}-x_{{\alpha }_{\lambda },n}} \right\| \le 2^{-n} < \varepsilon /2$ for all $\lambda >\lambda _0$. $\square$

Next we give the main examples of weighted semigroups, which we will use in the framework of this paper.

Proposition 3.4

Let U be an abelian semitopological semigroup, G a semitopological and metric group and X, Y both Banach spaces. Furthermore, for $\left\{ {y_s} \right\} _{s\in U}\subset X$ we set $\textrm{supp} \left\{ {y} \right\} := \left\{ {t\in U, y_t\not =0} \right\}$ and consider the space

$$\begin{aligned} \ell ^1(U,X):= \left\{ { \left\{ {y_s} \right\} _{s\in U}\subset X: \textrm{supp} \left\{ {y} \right\} \text{ countable } \text{ and } \sum _{s\in \textrm{supp} \left\{ {y} \right\} } \displaystyle \left\| {y_s} \right\| <\infty } \right\} \end{aligned}$$

equipped with the canonical norm. Moreover, if U is a uniform space let

$$\begin{aligned} \textrm{BUC}_{\textrm{wrc}}(U,X):= \left\{ {f\in \textrm{BUC}(U,X):\ f(U) \text{ weakly } \text{ relatively } \text{ compact }} \right\} \end{aligned}$$

and let

$$\begin{aligned} \begin{array}{ccccc} {T(t)} &{} : &{} {U} &{} \longrightarrow &{} {E} \\ &{} &{} {s} &{} \longmapsto &{} \displaystyle { \left\{ {s\mapsto f(s+t)} \right\} } \end{array} \end{aligned}$$

be the translation semigroup on functions defined on U with values in a given Banach space E. Finally, let $\left\{ {{S}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }\subset L(X)$ be a uniformly bounded $C_0$-semigroup. By specifying U and Z, the following semigroups form weighted semigroups on Z with respect to some specified topology $\tau$:

• (1) $U=\mathbf{\mathbb N},$ $Z:=Y^*,$ $T^*\in L(Y^*)$ dual and power-bounded with $\tau =w^*$

• (2) $U=\mathbf{\mathbb R^+},$ $Z=Y^{\odot },$ $\left\{ {{S^{\odot }}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} },$ with $\tau =w^*.$

• (3) $U=\mathbf{\mathbb R^+},$ $Z=Y,$ $\left\{ {{S}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} },$ with $\mathscr {O}(x)$ weakly relatively compact for all $x\in X,$ and $\tau =\sigma (Y,Y^*).$

• (4) U,  $Y=X^*,$ $Z:=\textrm{LC}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$ with $\tau =\sigma (\textrm{LC}(U,Y),\ell ^1(U,X))$

• (5) $U\subset G$ closed, $Y=X^*,$ $Z=\textrm{BUC}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$ with $\tau =\sigma (\textrm{BUC}(U,Y),\ell ^1(U,X))$

• (6) U,  $Z:=\textrm{LC}_{wrc}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$with $\tau =\sigma (\textrm{LC}_{\textrm{wrc}}(U,Y),\ell ^1(U,Y^*))$

• (7) $U\subset G$ closed $Z:=\textrm{BUC}_{wrc}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$with $\tau =\sigma (\textrm{BUC}_{\textrm{wrc}}(U,Y),\ell ^1(U,Y^*))$

• (8) U,  $Z:=\textrm{WW}_{\Lambda }(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$with $\tau =\sigma (\textrm{WW}(U,Y),\Lambda )$ where $\ell ^1(U,Y^*)\subset \Lambda \subset \textrm{LC}(U,Y)^*$ and $0\in U.$

• (9) U,  $Z:=\textrm{W}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$with $\tau =\sigma (\textrm{W}(U,Y),\textrm{W}(U,Y)^*)$

• (10) U,  $Z:=\textrm{AAP}(U,Y),$ $\left\{ {{T}(t)} \right\} _{t\in {U} },$with $\tau = \displaystyle \left\| { {\cdot } } \right\| _{\infty }$.

Consequently, all these spaces admit a splitting $Z=Z_a\oplus Z_0$ with respect to a minimal projection $P\in \overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{\tau \text {-}OT}$.

Remark 3.5

In view of the previous proposition and the known isometric embeddings $j:X\rightarrow X^{\odot \odot },$ and $i:X\rightarrow X^{**}$ we always obtain a splitting in $X^{\odot \odot },$ for elements in $x\in X$ (i.e. $jx=x^{\odot \odot }_a\oplus x^{\odot \odot }_0$ with $x^{\odot \odot }_a,x^{\odot \odot }_0\in X^{\odot \odot },$ or $f\in \textrm{LC}(U,X^{**})$, respectively whenever $f\in \textrm{LC}(U,X)$ (i.e. $if=f^{**}_a\oplus f^{**}_0)$ with $f^{**}_a, f^{**}_0\in \textrm{LC}(U,X^{**})$.

Proof

The example (1) we refer to [4], (2) and (3) are discussed in [7]. Let us continue with (4). Note that the topology $\tau =\sigma (\textrm{LC}(U,Y^*),\ell ^1(U,Y))$ is stronger that the pointwise-$w^*$-topology. We claim that they coincide on bounded sets. To see this, let $\left\{ { {f}_{\alpha } } \right\} _{{\alpha } \in {A}}$ be a uniformly bounded net which is pointwise-$w^*$ convergent to f. Then for given $y\in \ell ^1(U,Y)$ we find a countable set $U_N:= \textrm{supp} \left\{ {y} \right\}$, such that $y(t)=0$ for all $t\in U\backslash U_N.$ and

$$\begin{aligned} \left\langle y,f_{\alpha }-f\right\rangle= & {} \sum _{t\in U_N}\left\langle y(t),f_{\alpha }(t)-f(t)\right\rangle \\= & {} \sum _{t\in \left\{ {t_1,\dots ,t_n} \right\} }\left\langle y(t),f_{\alpha }(t)-f(t)\right\rangle +\sum _{t\in U_N\backslash \left\{ {t_1,\dots ,t_n} \right\} }\left\langle y(t),f_{\alpha }(t)-f(t)\right\rangle \end{aligned}$$

where the second sum is small due to the summability of $\left\{ {y(t)} \right\} _{t\in U_N}$ and uniformly boundedness of $\left\{ { {f}_{\alpha } } \right\} _{{\alpha } \in {A}}$. Hence, the finite sum becomes small due to pointwise convergence. Furthermore, we notice that $\textrm{LC}(U,Y^*) \subset \ell ^1(U,Y)^*$ which yields the lower semicontinuity. To verify (7) we have to check all properties mentioned in Definition 3.1. Observe that the properties (2)–(4) are straightfoward. To verify (4) note that

$$\begin{aligned} \overline{ \left\{ {T(s)f:s\in U} \right\} }^{\sigma (\text {LC}(U,Y^*),\ell ^1(U,Y))}\subset \overline{ \left\{ {T(s)f:s\in U} \right\} }^{pointwise\text {-}w^*} \end{aligned}$$

Now, let $\left\{ { {t}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset U$. Without loss of generality we set $g(t):= \lim _{ {\alpha }\in {A}} f(t+t_{\alpha })$, where the limit is taken pointwise-$w^*$. Then, we find by the uniform continuity of f that

$$\begin{aligned} \displaystyle \left\| {g(t+r)-g(s+r)} \right\| \le \liminf _{\alpha } \displaystyle \left\| {f(t+r+t_{\alpha })-f(s+r+t_{\alpha })} \right\| <\varepsilon \end{aligned}$$

Consequently, $g\in \textrm{LC}(U,Y^*)$. Now, let $f_{t_{\alpha }}$ be a bounded Cauchy net with respect to $\tau$. Then, g becomes the limit and $g\in \textrm{LC}(U,Y^*)$. In order to investigate (7), let $g=\tau$-$\lim _{ {\alpha }\in {A}} f(\cdot +t_{\alpha })$ and $y\in \ell ^1(U,Y)$. Then, for $y_s(r):=y(t)$ with $r=t-s$ we have

$$\begin{aligned} \left\langle y,f(\cdot +s+t_{\alpha })\right\rangle =\sum _{t\in \textrm{supp} \left\{ {y} \right\} }\left\langle y(t),f(t+s+t_{\alpha })\right\rangle =\sum _{r\in \textrm{supp} \left\{ {y_s} \right\} }\left\langle y_s(r),f(r+t_{\alpha })\right\rangle , \end{aligned}$$

which yields $\tau$-convergence. To obtain the splitting it leaves to verify that $\left\{ {T(s)f:\, s\in U} \right\}$ is $\tau$-compact. This becomes a consequence of its pointwise-$w^*$ compactness and the fact that $\tau$ and the pointwise-$w^*$-convergence coincide on bounded sets. The claim (6) can be verified by using (4) as well as the canonical embedding $\textrm{LC}(U,X)\subset \textrm{LC}(U,X^{**}).$

In the metric subgroup case, observe that $\textrm{BUC}(U,X^*)\subset \textrm{LC}(U,X^*)$. Hence it is sufficient to show that for $f\in \textrm{BUC}(U,X^*)$ one has $\overline{\mathscr {O}(f)}^{w^*}\subset \textrm{BUC}(U,X^*)$. However, this is a consequence of the $w^*$-lower-semicontinuity. Note that $\textrm{BUC}(U,X^*)\subset \ell ^1(U,X)^*$ and that for $g(t)=\sigma (X^*,X)\text {-}\lim _{\lambda \in A}f_{t_{\lambda }}(t)$ one has

$$\begin{aligned} \displaystyle \left\| {g(s)-g(t)} \right\| \le \liminf _{\lambda \in A} \displaystyle \left\| {f(t_{\lambda }+s)-f(t_{\lambda }+t)} \right\| \end{aligned}$$

(7) can be obtained similarly to (6).

To verify (8) we claim that $\overline{f(U)}^{\tau _X}$ is $\tau _X$-compact. To see this, we observe that the map

$$\begin{aligned} \begin{array}{ccccc} {\delta _x} &{} : &{} {(\textrm{LC}(U,X),\sigma (\textrm{LC}(U,X),\Lambda ))} &{} \longrightarrow &{} {(X,\tau _X)} \\ &{} &{} {g} &{} \longmapsto &{} \displaystyle {g(x)} \end{array} \end{aligned}$$

is linear and by definition continuous, therefore uniformly continuous. Consequently, the restriction to $\mathscr {O}(f)$ is uniformly continuous as well. Hence, by assumption on the completeness we see that the map

$$\begin{aligned} \begin{array}{ccccc} {\tilde{\delta _x}} &{} : &{} {(\overline{\mathscr {O}(f)}^{\sigma (\textrm{LC}(U,X),\Lambda )},\sigma (\textrm{LC}(U,X),\Lambda ))} &{} \longrightarrow &{} {\overline{f(U)}^{\tau _X}} \\ &{} &{} {g} &{} \longmapsto &{} \displaystyle {g(x)} \end{array} \end{aligned}$$

is continuous and surjective, due to the fact that $0\in U$. Therefore, $\textrm{WW}(U,X)$ is linear. For the completeness apply Lemma 3.3. Therefore let $\mathscr {O}(f)$ relatively $\sigma (\textrm{LC}(U,X),\Lambda )$-compact, then it suffices to show that $\overline{\mathscr {O}(f)}^{\tau }\subset \textrm{WW}_{\Lambda }(U,X)$. To do so, by the previous observation, we $g\in \overline{\mathscr {O}(f)}^{\tau }$ as the pointwise limit of translates. We observe that for every $y^*\in \Lambda$ one has that $\overline{\mathscr {O}(\left\langle y^*,f(\cdot )\right\rangle )}^w$ is weakly compact in $\textrm{W}(U)$. Hence, we find $\left\langle y^*,g\right\rangle \in \overline{\mathscr {O}(\left\langle y^*,f(\cdot )\right\rangle )}^w$. The translation invariance of the closed orbit concludes the proof.

Now, assertion (9) comes with $\Lambda =\textrm{W}(U,X)^*)$. In fact, the condition Definition 3.1(7) can be obtained from the equivalence of $\sigma (\textrm{WW}_{\Lambda }(U,X),\ell ^1(U,X^*))$ and $\sigma (\textrm{WW}_{\Lambda }(U,X),\Lambda ))$ and the weak continuity of the scalar translation semigroup on $\textrm{W}(U).$

To obtain (10) reuse the arguments with the norm-topology. $\square$

However, as described in [7], we are able to find a minimal projection in this operator semigroups, which eventually yields the decomposition into a reversible and flight vector. In order to apply the results from [5], we use the following construction for a group if an underlying semigroup is given. Notice, that the following example has been previously investigated by the authors, see [7, Ex. 2.7]. However, as it plays an important role for this work we recall it for the sake of completeness.

Remark 3.6

To verify Lemma 3.4(8) avoiding $0\in U$ one may assume $\overline{\mathscr {O}(f)}^{\tau }$ to be compact. Here, the $\tau _X$-sequential completeness enters the picture, whenever $\overline{\mathscr {O}(f)}^{\sigma (\textrm{LC}(U,X),\Lambda )}$ is angelic, see also [18, p. 30, Def.]. For a discussion on weak sequential completeness, we refer the reader to [30, p. 33-37]. An angelic orbit is given, if U is separable with a uniformity and $\textrm{LC}(U,X)=\textrm{BUC}(U,X).$ As a matter of fact, f(U) becomes $\displaystyle \left\| {\cdot } \right\|$-separable. However, we need $\overline{f(U)}^{\tau _X}$ to be norm separable. To see this, consider $\displaystyle \left\| {\cdot } \right\|$-dense sequences $\left\{ { d_{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset U$ and $\left\{ { x_{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset \overline{f(U)}^{\tau _X}-\overline{f(U)}^{\tau _X}$. As the topology $\tau _X$ is norming for X, for $m,n\in \mathbf{\mathbb N}$ we have the non-empty sets defined by

$$\begin{aligned} A_{m,n}:= \left\{ {x^*\in (X,\tau _X)^*: \displaystyle \left\| {x^*} \right\| =1, \text{ and } \left| {x^*(x_n)} \right| \ge \displaystyle \left\| {x_n} \right\| -\frac{1}{m}} \right\} . \end{aligned}$$

Choose $x^*_{n,m,}\in A_{n,m}$, which are norming for elements of $\overline{f(U)-f(U)}^{\tau _X}$ due to their $\tau _X$-continuity. For $x,y\in \overline{f(U)}^{\tau _X},$ we find $\displaystyle \left\| {\cdot } \right\| - \lim _{ {l} \rightarrow \infty } x_{n_l}= x-y,$ and

$$\begin{aligned} \left| {x^*_{m,n_l}(x-y)} \right| +\varepsilon \ge \left| {x^*_{m,n_l}(x_{n_l})} \right| \ge \displaystyle \left\| {x_{n_l}} \right\| -\frac{1}{m}. \end{aligned}$$

For $l\rightarrow \infty$

$$\begin{aligned} \liminf _{l\rightarrow \infty } \left| {x^*_{m.n_l}(x-y)} \right| +\varepsilon \ge \lim _{ {l} \rightarrow \infty } \displaystyle \left\| {x_{n_l}} \right\| -\frac{1}{m}= \displaystyle \left\| {x-y} \right\| -\frac{1}{m}. \end{aligned}$$

Now, consider the Hausdorff metric

$$\begin{aligned} \begin{array}{ccccc} {d} &{} : &{} {\overline{\mathscr {O}(f)}^{\sigma (\textrm{LC}(U,X),\Lambda )}\times \overline{\mathscr {O}(f)}^{\sigma (\textrm{LC}(U,X),\Lambda )}} &{} \longrightarrow &{} {\mathbf{\mathbb R^+}} \\ &{} &{} {(g,h)} &{} \longmapsto &{} \displaystyle {\sum _{j,k,l=1}^{\infty }2^{-j-k-l}\min \left\{ {1, \left| {x^*_{j,k}(g(d_l)-h(d_l))} \right| } \right\} .} \end{array} \end{aligned}$$

This metric induces a weaker Hausdorff topology on the closure of the orbit of f. For a more general setting a helpful result will be [38, Prop. 1.6(a)].

The following two results will be a useful tool later on.

Definition 3.7

• (1) Let $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset L(X)$ be a weighted representation of a semigroup S. An element $y\in X$ is called reversible if for every $\tau \text {-}OT$ convergent net $\left\{ {s_{\alpha }} \right\} _{\alpha \in A}\subset U$, there exists a net $\left\{ { {t}_{\gamma } } \right\} _{{\gamma } \in {\Gamma }} \subset U$ such that $\tau \text {-}\lim _{ {\gamma }\in {\Gamma }} \tau \text {-}\lim _{ {\alpha }\in {A}} T(t_{\gamma })T(s_{\alpha })y=y$. In what follows, we will denote by set of reversible vectors by $X_{rev}$.

• (2) A reversible vector $x\in X_{rev}$ is called almost periodic iff $\mathscr {O}(x)$ is relatively compact.

• (3) $y\in X$ is called a flight vector if for a net $\left\{ { {s}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset U,$ we have $\tau \text {-} \lim _{ {\alpha }\in {A}} T(s_{\alpha })y=0$. Let $Y_{0}$ be the set of flight vectors.

Remark 3.8

Observe, that if $x\in X$ is a flight vector and reversible, then $x=0$.

Definition 3.9

Let U be a semigroup and $E(U):=\left\{ e\in U:\ e^2=e\right\}$ the set of idempotents of U. Then we define the following relations:

• (1) $e\le _R f$ if $fe=e$

• (2) $e \le _L f$ if $ef=e$

An element $e\in U$ is right (left) minimal idempotent if $f\in U$ with $f\le _R e$ ($f\le _L e)$ implies that $e \le _R f$ $(e\le _L f)$. An element $e\in U$ is called a minimal idempotent if it left and right minimal.

By recalling [37, p. 14, p. 21], we obtain the following result.

Corollary 3.10

[7], Cor. 2.4] Let $\left\{ {{T}(t)} \right\} _{t\in {U} }$ be a weighted semigroup representation such that the orbit $\mathscr {O}(y):= \left\{ {T(t)y:t\in U} \right\}$ is $\tau$-relative-compact for all $y\in X$. Then

• (1) $\overline{ \left\{ {T(t)} \right\} _{t\in U}}^{\tau \text {-}OT}$ has minimal idempotents P, which are left and right minimal, such that $P{\mathscr {T}}P$ is an algebraic group.

• (2) The minimal idempotents lead to splittings $X=X_{a}^P\oplus X_0^P$, where $X_a^P:=PX\subset X_{rev}$ and $X_0^P:=(I-P)X\subset X_{fl}$, for every minimal idempotent P.

Next we extend [7, Thm. 5.12] to the semitopological semigroup setting $\left\{ {{T}(t)} \right\} _{t\in {U} }$ by applying [7, Cor. 4.12]. A simple approach to the cited corollary is to observe that in fact of the abelian structure given by assumption (1) beyond, the compact left-semitopological semigroup becomes compact semitopological, and one may follow the proofs given in [28, Thm. 4.1, Thm. 4.4, Thm. 4.5] up to the missing continuity of ${S\rightarrow Sx}$ at the end of the proof. This is made up by the separability. The corresponding separability arguments are similar to ones at the end of the proof of [7, Lem. 4.11].

Theorem 3.11

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset \mathscr {L}(Y)$ be a weighted semigroup with respect to $\tau$ on $Y\subset X^*$. Assume that $\overline{\mathscr {O}(y)}^{\tau }$ is $\tau$-compact and that for a norming and closed space $\Lambda \subset Y^*$ one has that

$$\begin{aligned} \left\{ {U\ni t \mapsto \left\langle T(t)z,x\right\rangle } \right\} \in \textrm{W}(U), \text{ for } \text{ all } z\in Y,\ x\in \Lambda . \end{aligned}$$

(1)

Under the assumption that $w^* \subset \tau$ and $y_a\in Y_a$, the following assertions are equivalent.

• (1) $\overline{\mathscr {O}(y_a)}^{\tau }$ is norm separable.

• (2) $y_a\in Y_{\text {fin}}:={\overline{\textrm{span}}} \left\{ {y\in Y:\ \overline{\text {span}}\mathscr {O}(y) \text { finite } \text { dimensional } } \right\}$

• (3) $\mathscr {O}(y_a)$ is relatively norm compact.

Proof

Note that in fact of condition (1) the left-semitopological semigroup ${\mathscr {T}}=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{\tau \text {-}OT}\subset \mathscr {L}(Y)$ becomes abelian due to Proposition 4.2 and therefore semitopological. We proceed by showing that we are in the situation of [7, Cor. 4.12]. To do so, we restrict our semigroup to

$$\begin{aligned} {\tilde{Y}}:=\overline{\textrm{span}}^{ \displaystyle \left\| {\cdot } \right\| } \left\{ {\overline{O(y_a)}^{\tau }} \right\} . \end{aligned}$$

which is $\left\{ {{T}(t)} \right\} _{t\in {U} }$- invariant due to Definition 3.1 (7). Moreover, assume that $T(t_{\alpha })\rightarrow R$ with respect to ${\tau \text {-}OT}$ and let $\left\{ {z_i^k} \right\} _{i,k\in \mathbf{\mathbb N}}\subset \overline{O(y_a)}^{\tau }$ and $\left\{ { \lambda _{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset \mathbf{\mathbb C}$ such that $\displaystyle \left\| {\cdot } \right\| \text {-}\lim _{k}\sum _{i=1}^{n_k}\lambda _i^kz_i^k= z$. Then $\left\{ {Rz_i^k} \right\} _{i,k\in \mathbf{\mathbb N}}\subset \overline{O(y_a)}^{\tau },$ due to ${\mathscr {T}}$ semitopological, and

$$\begin{aligned} {\tilde{Y}}\ni \sum _{i=1}^{n_k}\lambda _i^kRz_i^k =R\sum _{i=1}^{n_k}\lambda _i^kz_i^k\rightarrow Rz. \end{aligned}$$

This shows the space is invariant for ${\mathscr {T}}:=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{\tau \text {-}OT}$. Now, let $G:=(P{\mathscr {T}}P_{|{\tilde{Y}}}).$ By assumption the function

$$\begin{aligned} \begin{array}{ccccc} {f} &{} : &{} {G} &{} \longrightarrow &{} {{\tilde{Y}}} \\ &{} &{} {S} &{} \longmapsto &{} \displaystyle {Sy_a} \end{array} \end{aligned}$$

has separable range. Moreover, by [10, Cor. 4, pp.42-43] (by taking $\Gamma =X$) it is measurable. By its boundedness [10, Thm.2, p. 45] it is also Bochner-integrable with respect to the Haar measure $\rho$, cf. [10, Thm 2. p. 45]. Consequently, it becomes Pettis-integrable, cf. [10, Def. 2, pp.52-53]. Notice, that [31, Thm. 4.1] characterizes a function $g:\Omega \rightarrow X$ which is scalarly integrable (i.e., $x^*g\in L^1(\Omega )$) as Pettis-integrable if

$$\begin{aligned} \begin{array}{ccccc} {T_g} &{} : &{} {X^*} &{} \longrightarrow &{} {L^1(\Omega )} \\ &{} &{} {x^*} &{} \longmapsto &{} \displaystyle {x^*g} \end{array} \end{aligned}$$

is $w^*$-weak-continuous. By taking $\Omega :={\mathscr {G}}$ equipped with the Haar measure, this yields that for $r\in R(G,\mathbf{\mathbb C})$ one has that

$$\begin{aligned} \begin{array}{ccccc} {f} &{} : &{} {G} &{} \longrightarrow &{} {{\tilde{Y}}} \\ &{} &{} {S} &{} \longmapsto &{} \displaystyle {r(S)Sy_a} \end{array} \end{aligned}$$

is Pettis-integrable, cf. [24, Def. 3.3, p. 53]. Let $T\in {\mathscr {T}}$ and $z=Ty_a$. Then by [7, Prop. 4.4] and Assumption (1) together with [7, Thm. 3.4] we observe that

$$\begin{aligned} \begin{array}{ccccc} {f} &{} : &{} {G} &{} \longrightarrow &{} {{\tilde{Y}}} \\ &{} &{} {S} &{} \longmapsto &{} \displaystyle {r(S)STy_a} \end{array} \end{aligned}$$

is Pettis-integrable. (Note that this shows that for the splitting $y=y_a\oplus y_0$ the Pettis-integrability for y implies the Pettis-integrability for $y_a=Py$ and $y_0=(I-P)y$ whenever the operators commute on y.) As the uniform limits of Pettis-integrable functions are Pettis-integrable, cf. [31, Thm. 5.3,p. 551], for every element in $z\in {\tilde{X}}$ the functions $\left\{ {S\rightarrow r(S)Sz} \right\}$ are Pettis-integrable. It is also straightforward that $\left\{ {{T}(t)} \right\} _{t\in {U} }$ becomes a weighted semigroup on ${\tilde{Y}}.$ By [24, Lem. 3.50, p.83]

$$\begin{aligned} M:= \left\{ {S_{r}y_a:=(\text {Pettis}-)\int _Gr(S)Sy_ad\rho (S):r \in R(G,\mathbf{\mathbb C})} \right\} \subset {\tilde{Y}}_{fin}, \end{aligned}$$

By construction ${\tilde{Y}}$ is separable. The continuity of the quotient map $q:{\tilde{Y}}\rightarrow {\tilde{Y}}/M$ leads to the separability of image, which implies by [7, Cor. 4.12, Claim(3)] that ${\tilde{Y}}=M\subset {\tilde{Y}}_{ap}$. Moreover, $M={\tilde{Y}}_{fin}$ due to [24, Def. 3.1, Def. 3.40,]. Hence, we proved the implications $(1)\implies (2)\implies (3)$. Finally, we notice that the implication $(3)\implies (1)$ is a compactness argument. $\square$

Let us recall a common topological property.

Definition 3.12

A topological space $(X,\tau )$ is called Lindelöf if X is regular and every open cover of X has a countable subcover.

The following result connects the Lindelöf property with the Radon–Nikodym property (RNP). For more information regarding the Radon–Nikodym property, we refer for example to [6].

Corollary 3.13

Let U be separable ($U={\overline{D}}$, with D countable) and suppose that the assumptions of Theorem 3.11 are satisfied. Then the following assertions are equivalent:

• (1) $\overline{\mathscr {O}(y_a)}^{\tau }$ weakly Lindelöf.

• (2) $\overline{\textrm{co}}^{w^*}\mathscr {O}(y_a)$ has the Radon–Nikodym property.

Proof

We show that if $\overline{\mathscr {O}(y_a)}^{\tau }$ is weakly Lindelöf, that this implies that $\overline{\mathscr {O}(y_a)}^{w^*}$ has the RNP, and subsequently, that if D is countable with $U={\overline{D}}$, that this implies the separability of $\overline{\mathscr {O}(y_a)}^{w^*}$. As $\overline{\mathscr {O}(y_a)}^{\tau }$ is assumed to be $\tau$-compact and $w^*\subset \tau$ the $w^*$-topology is an equivalent topology on $\overline{\mathscr {O}(y_a)}^{\tau }$. Due to [8, Thm. 4.5 & Cor. 4.6, pp. 176-177] we obtain that $\overline{\textrm{co}}^{w^*}\mathscr {O}(y_a)$ is weakly Lindelöf. Hence, it is a set with the Radon–Nikodym property by [6, Cor. 4.2.17, p. 98]. Now, let D be a countable dense subset of U. From Definition 3.1 (5) we find T(D)x is $\tau$-dense in $\overline{\mathscr {O}(y_a)}^{\tau }$. The equation

$$\begin{aligned} \overline{\textrm{co}}^{w^*}O(y_a)=\overline{\textrm{co}}^{w^*} \left\{ {(T(d)y_a:d\in D} \right\} \end{aligned}$$

completes the argument by applying [6, Cor. 4.1.8, p. 76]. $\square$

4 A Review on the Günzler-Kadets-Basit Difference Theorems

In this section we discuss one of our main results for possibly non-abelian groups and semigroups. To avoid confusions we consider ${\mathscr {G}}$ for a possibly non-abelian group, and ${\mathscr {S}}$ for a possibly non-abelian semigroup, respectively. If U is mentioned, it is an abelian semitopological semigroup. To do so, we define almost periodicity by means of left and right almost periodicity. This indeed coincides in the case when the underlying group ${\mathscr {G}}$ carries a topology, cf. [3, Rem. 1.5, p. 131], by applying the identity $\textrm{C}(K,X)=\textrm{C}_{\textrm{b}}(K){\tilde{\otimes }}_{\varepsilon }X,$ for K compact and Hausdorff. Before we start into the construction, we recall some small proposition.

Proposition 4.1

Let $x\in X_{rev},$ $P\in {\mathscr {T}}$ a minimal idempotent. If $TPx=PTx$ for all $T\in {\mathscr {T}},$ then $x= Px.$

Proof

As $x\in X_{rev}$ for $P\in {\mathscr {T}},$ by definition there is a $T\in {\mathscr {T}}$ such that $x=TPx=PTx.$ Consequently, $x\in X_a.$ $\square$

The next proposition shows, that Grothendieck’s double limits condition will carry the abelian property of U to ${\mathscr {T}},$ and vice versa, if the orbits are relatively $\tau$-compact.

Proposition 4.2

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }$ be a weighted representation on X,  for separating set $Y\subset (X,\tau )^*$ the space of continuous functionals regarding to $\tau$, and $\mathscr {O}(z)$ be relatively $\tau$-compact for all $z\in X,$. Then $WVx=VWx$ for all $V,W\in {\mathscr {T}}$ if and only if

$$\begin{aligned} \begin{array}{ccccc} {i_x} &{} : &{} {U} &{} \longrightarrow &{} {W(U)} \\ &{} &{} {t} &{} \longmapsto &{} \displaystyle {\left\langle y,T(\cdot +t)x\right\rangle ,} \end{array} \end{aligned}$$

for a set of separating vectors $y\in Y$.

Proof

Let $\left\{ { {t}_{\gamma } } \right\} _{{\gamma } \in {\Gamma }} , \left\{ { {s}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset U$ be net such that $R=\tau \text {-}OT\text {-}\lim _{\gamma }T(t_{\gamma })$ and $S=\tau \text {-}OT\text {-}\lim _{\alpha }T(s_{\alpha })$. For $y\in Y$ and $x\in X$ the following identity proves both implications:

$$\begin{aligned} \left\langle y,RSx\right\rangle = \lim _{ {\alpha }\in {A}} \lim _{ {\alpha }\in {A}} \left\langle y, T(t_{\gamma }+s_{\alpha })x\right\rangle = \lim _{ {\alpha }\in {A}} \lim _{ {\alpha }\in {A}} \left\langle y, T(t_{\gamma }+s_{\alpha })x\right\rangle =\left\langle y,SRx\right\rangle \end{aligned}$$

Indeed, if $\left\langle y,T(\cdot +t)x\right\rangle \in W(U)$ then the double limit criteria hold and the operators commute. For the converse, we observe that if they commute, the double limit criteria [20] imply that $\langle y,T(\cdot +t)x\rangle \in W(U)$. The converse argument starts with given nets $\left\{ { {s}_{\alpha } } \right\} _{{\alpha } \in {A}} , \left\{ { {t}_{\gamma } } \right\} _{{\gamma } \in {|\Gamma }}$ and, for which we may pass to $\tau$-convergent subnets $R:=\tau \text {-} OT\text {-} \lim _{\gamma }T(t_{\gamma })$ and $S:=\tau \text {-}OT\text {-}\lim _{\alpha }T(s_{\alpha }).$ Now [20] applies to obtain weak compactness of $\mathscr {O}( \left\{ {t\mapsto \langle y,T(t)x\rangle } \right\} ),$ which concludes the proof. $\square$

Corollary 4.3

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }$ be a weighted representation on X and $Y\subset (X,\tau )^*$ the space of continuous functionals regarding $\tau$-compact and $\mathscr {O}(z)$ be relatively $\tau$-compact for all $z\in X.$ Then for every minimal $P\in {\mathscr {T}}$ we have $PX_{ap}=X_{ap}.$

Proof

As $X_{ap}\subset X_{rev},$ and for almost periodic vectors

$$\begin{aligned} \begin{array}{ccccc} {i_x} &{} : &{} {U} &{} \longrightarrow &{} {AAP(U)\subset W(U)} \\ &{} &{} {t} &{} \longmapsto &{} \displaystyle {\left\langle y,T(\cdot +t)x\right\rangle ,} \end{array} \end{aligned}$$

we may apply Proposition 4.1 and Proposition 4.2 to conclude the proof. $\square$

Observation 4.4

By the previous we are ready to enhance to non-abelian compactifications, by forthcoming extension of the Basit–Kadets theorem due to the fact that the compactification ${\mathscr {T}}$ of $\left\{ {{T}(t)} \right\} _{t\in {U} }$ lead by [37] subgroups of the form $P{\mathscr {T}}P.$ Consequently, $(P{\mathscr {T}})_{|Z_a}$ forms a group on $Z_a$. Moreover, the translation-semigroup commutes with P due to its assumed $\tau$-continuity. Hence, if we consider the difference of $Pf_a$ to be almost periodic, i.e. in view of Corollary 4.3, we have

$$\begin{aligned} P(T(t+s)f-T(t)f)=T(t+s)f_a-T(s)f_a=PT(t)g_s=T(t)g_s \end{aligned}$$

Letting $R=\tau \text {-}OT\text {-} \lim _{ {\alpha }\in {A}} T(t_{\alpha })$ we obtain

$$\begin{aligned} RT(s)f-Rf=Rg_s \text{ with } R\in {\mathscr {T}}, \text{ and } s\in U. \end{aligned}$$

Further, using $P^2=P,$ $f_a=Pf_a,$ by definition, and $Pg_s=g_s$ for every $g_s$ almost periodic and P minimal, we observe for $R:=PRP$

$$\begin{aligned} PRT(s)f_a-PRf_a= & {} (PRP)(PT(s)P)f_a-PRPf_a\nonumber \\= & {} PRPg_s \text{ with } R\in {\mathscr {T}}, \text{ and } s\in U. \end{aligned}$$

(2)

for $f,g_s:U\rightarrow X$. Moreover, if we restrict the compactification ${\mathscr {T}}$ to the space of almost periodic functions $\textrm{AP}(U,X)\subset Z_a$, it becomes an abelian semitopological group. By applying $RSg=SRg$ for $g\in AP(U,X)\subset \textrm{W}(U,X).$ Nonetheless, it is possible that it yields a non-abelian left-semitopological group inside $\mathscr {L}(Z_a)$. However, for the topology $\tau$ and $f_a\in Z_a$ we have the continuity of

$$\begin{aligned} \begin{array}{ccccc} {f_a^T} &{}{} : &{}{} {({\mathscr {T}}_{|Z_a},\tau \text {-}OT)} &{}{} \longrightarrow &{}{} {(Z_a,\tau )} \\ {} &{}{} &{}{} {R} &{}{} \longmapsto &{}{} \displaystyle {Rf_a}. \end{array}\end{aligned}$$

The above observation motivates the extension of [5] to dense sub-semigroups, and we define.

Definition 4.5

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset \mathscr {L}(Z)$ be a weighted semigroup, $\mathscr {O}(x)$ relatively $\tau$-compact for all $x\in X$ and ${\mathscr {T}}$ the compactification. Then for a minimal $P\in {\mathscr {T}}$ we define with $Z_a:=PZ,$ the group

$$\begin{aligned} {\mathscr {T}}_{|Z_a}:= \left\{ {R\in \mathscr {L}(Z_a): R=PSP \text{ with } S\in {\mathscr {T}}} \right\} . \end{aligned}$$

Proposition 4.6

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }$ be a weighted semigroup. Then $P \left\{ {{T}(t)} \right\} _{t\in {U} }P$ is dense in ${\mathscr {T}}_{|Z_a}.$

Proof

From Definition 3.1 (7) we observe that P commutes with $\left\{ {{T}(t)} \right\} _{t\in {U} }.$ Consequently, $Z_a$ is $\left\{ {{T}(t)} \right\} _{t\in {U} }$-invariant. As $PSP\in {\mathscr {T}}$ there is a net such that $\tau \text {-}OT\text {-} \lim _{ {\alpha }\in {A}} T(t_{\alpha })= PSP.$ On $Z_a$ we have $T(s)f_a=PT(s)Pf_a.$ In sum, $P \left\{ {{T}(t)} \right\} _{t\in {U} }P$ is dense in ${\mathscr {T}}_{|Z_a}.$ $\square$

Having this in mind, we will assume in this section that $0\in U$, which actually allows an extension to the function space cases Z. We are particularly interested in the case where we have a function $f_a\in Z_a$ and we want to extend it to a function on ${\mathscr {T}}_{|Z_a}$ by

Definition 4.7

Let $0\in U$ and $\left\{ {{T}(t)} \right\} _{t\in {U} }$ be a weighted semigroup on $\textrm{LC}(U,X^*)$ with respect to $\tau =\sigma (\textrm{LC}(U,X^*),\ell ^1(U,X)).$ Then we define:

$$\begin{aligned} \begin{array}{ccccc} {\tilde{f_a}} &{} : &{} {{\mathscr {T}}_{|Z_a}} &{} \longrightarrow &{} {(X^*,w^*)} \\ &{} &{} {R} &{} \longmapsto &{} \displaystyle {\delta _0(Rf)} \end{array} . \end{aligned}$$

Surely, ${\tilde{f}}$ is then bounded and continuous. Note that $\delta _0(T(s)f)=\delta _0(f(\cdot +s))=f(s)$. In view of the difference equation (2), we have

$$\begin{aligned} \tilde{f_a}(RS)-\tilde{f_a}(R)=h_S(R), \text{ with } R\in {\mathscr {G}}, \text{ and } S\in {\mathscr {S}}. \end{aligned}$$

where ${\mathscr {G}}$ denotes a group and ${\mathscr {S}}\subset {\mathscr {G}}$ is a dense sub-semigroup. Hence, this view gives an application to possibly non-abelian semigroups. To do this, we have to investigate some Banach space geometry, before we are able to use the abelian methods. We start recalling the definitions from [5].

Definition 4.8

A set E in a group G is called relatively complete if G contains a countable number of elements $\left\{ {t_1,\dots ,t_k,s_1,\dots ,s_k} \right\} \subset G$, such that

$$\begin{aligned} G= \bigcup _{i=1}^k (t_iE), \text{ and } G=\bigcup _{i=1}^k ~ (Es_i). \end{aligned}$$

Definition 4.9

A function $f\in C_b(G,X)$ called almost periodic if every infinite sequence $f(t\alpha _n)$ and $f(\alpha _nt)$ contains a uniformly convergent subsequence.

$$\begin{aligned} \text {AP}(G,X):= \left\{ {f \in C_b(G,X): f \text { is } \text { almost } \text { periodic } } \right\} . \end{aligned}$$

Remark 4.10

It can be verified that $f\in \textrm{LRC}({\mathscr {G}},X)$ on a group ${\mathscr {G}}$ is almost periodic if and only if for every $\varepsilon >0$ there is a relatively complete set $E(\varepsilon ,f)$ such that

$$\begin{aligned} \sup _{t\in G} \displaystyle \left\| {f(tr)- f (t)} \right\| <\varepsilon \text{ for } \text{ all } r\in E(\varepsilon ,f). \end{aligned}$$

Moreover, if $r\in E(\varepsilon ,f)$, then $r^{-1}\in E(\varepsilon ,f)$ and $f\in \textrm{LRC}({\mathscr {G}},X)$. If ${\mathscr {S}}\subset {\mathscr {G}}$ dense, there exist $s_1\ldots ,s_N\in {\mathscr {S}}$ such that $G=\bigcup _{i=1}^k{E(\varepsilon ,f)s_i}$.

To actually see that the previous statement is true, let $r_1\ldots ,r_N\in {\mathscr {G}}$ be such that $G=\bigcup _{i=1}^k(E(\varepsilon /2,f)r_i).$ Then $G=\bigcup _{i=1}^k(r_i^{-1}E(\varepsilon ,f)),$ by applying the inverse on both sides, and choosing $t:=tr^{-1}.$ For $g\in {\mathscr {G}}$ we find $r_j$ such that $g \in r_j^{-1}E(\varepsilon /2,f)$. Hence, either $g=(r_j)^{-1}u$ or $r_jg=u\in E(\varepsilon /2,f)$. Now, let $s_j\in U(r_j)$ and set $v:=s_jg$ such that $\displaystyle \left\| {f(tr_jg)-f(ts_jg)} \right\| = \displaystyle \left\| {f(tv)-f(tu)} \right\| < \varepsilon /2$ uniformly for $t\in {\mathscr {G}}$. Hence, for $s_jg=:v$ we have

$$\begin{aligned} \displaystyle \left\| {f(tv)-f(t)} \right\| \le \displaystyle \left\| {f(tv)-f(tu)} \right\| + \displaystyle \left\| {f(tu)-f(t)} \right\| < \varepsilon , \end{aligned}$$

by the assumption that $f\in \textrm{LRC}({\mathscr {G}},X)$ Therefore $v\in E(\varepsilon ,f)$ and $g=s_j^{-1}v.$ Hence, $G=\bigcup _{i=1}^k(s_i^{-1}E(\varepsilon ,f))$, which concludes the proof.

In what follows let ${\mathscr {G}}$ be a group with a topology and identity element e and ${\mathscr {S}}$ a sub-semigroup. For functions $f,h_{\gamma }\in \textrm{C}_{\textrm{b}}({\mathscr {G}},X)$ we define the following sets

$$\begin{aligned} U(\varepsilon ):= {} \left\{ {R \in {\mathscr {G}}: \displaystyle \left\| {f(R)-f(id)} \right\| < \varepsilon } \right\} , \end{aligned}$$

(3)

$$\begin{aligned} V_{S}(\varepsilon ):= \left\{ {R \in {\mathscr {G}}: \displaystyle \left\| {h_{S}(R)-h_{S}(id))} \right\| <\varepsilon } \right\} . \end{aligned}$$

(4)

The next theorem applies to possibly non-abelian semitopological groups ${\mathscr {G}}$. In fact, we weight the prerequisite of the topology of the group ${\mathscr {G}}$ against the set where the difference equation

$$\begin{aligned} f(GS)-f(G)=h_S(G) \text{ for } \text{ all } G\in {\mathscr {G}}, \text{ and } S\in {\mathscr {S}}\end{aligned}$$

holds. Note, that we do not assume uniform continuity in s,  we will only assume boundedness for f. Next we fit [5, Lem. 1] to our setting.

Corollary 4.11

Let ${\mathscr {G}}$ be a group with a topology, ${\mathscr {S}}\subset {\mathscr {G}}$ a dense semigroup, and $c_0\not \subset X.$ Further, let $f:{\mathscr {G}}\rightarrow X,$ be bounded, and $h_S:{\mathscr {G}}\rightarrow X$ almost periodic, with $h_S\in \textrm{LC}({\mathscr {G}},X)$ for all $S\in {\mathscr {S}}$ such that

$$\begin{aligned} f(GS)-f(G)=h_s(G) \text{ for } \text{ all } G\in {\mathscr {G}}, \text{ and } S\in {\mathscr {S}}. \end{aligned}$$

Then for any $\varepsilon >0$ there exist $\delta >0$ and a finite set of elements $S_1,\ldots ,S_r$, such that

$$\begin{aligned} \emptyset \ne \bigcap _{i=1}^rV_{S_i}(\delta )\cap {\mathscr {S}}\subset U(\varepsilon ). \end{aligned}$$

Proof

From [5, Lemma 1, p. 183] we obtain

$$\begin{aligned} \emptyset \not =\bigcap _{i=1}^rV_{S_i}(\delta )\subset U(\varepsilon ). \end{aligned}$$

As $h_{S}\in LRC(G,X)$ the finite intersection $\bigcap _{i=1}^rV_{S_i}(\delta )$ is open, we may intersect with the dense set ${\mathscr {S}}$ to obtain

$$\begin{aligned} \emptyset \not =\bigcap _{i=1}^rV_{Si}(\delta )\cap {\mathscr {S}}\subset U(\varepsilon ). \end{aligned}$$

$\square$

Lemma 4.12

Let ${\mathscr {G}}$ be a group with a topology, ${\mathscr {S}}\subset {\mathscr {G}}$ a dense semigroup. Further, let $f:{\mathscr {G}}\rightarrow X$ be bounded and $h_S:{\mathscr {G}}\rightarrow X$ almost periodic, with $h_S\in \textrm{LC}({\mathscr {G}},X)$ for all $S\in {\mathscr {S}},$ such that

$$\begin{aligned} f(GS)-f(G)=h_s(G) \text{ for } \text{ all } G\in {\mathscr {G}}, \text{ and } S\in {\mathscr {S}}. \end{aligned}$$

If for all $\varepsilon >0,$ we find $\delta >0,$ there are $\left\{ {S_i} \right\} _{i=1}^N\subset {\mathscr {S}},$ such that

$$\begin{aligned} \bigcap _{i=1}^N V_{S_i}(\delta )\subset U(\varepsilon ), \end{aligned}$$

then f(G) is relatively compact.

Proof

Let $1\le i\le N,$ as $h_S$ is assumed to almost periodic, and ${\mathscr {G}}\ni s\rightarrow L(s)h_{S_i}\in C_b({\mathscr {G}},X)$ continuous. In view of Remark 4.10 we find a finite set $\left\{ {R_j} \right\} _{j=1}^M\subset {\mathscr {S}}$ such that

$$\begin{aligned} {\mathscr {G}}=\bigcup _{j=1}^M \left\{ {R: \displaystyle \left\| {h_{S_i}(R)-h_{S_i}(id)} \right\| \le \delta } \right\} R_j, \text{ for } \text{ all } i=1,\dots ,N, \end{aligned}$$

compare [3, Def. 10.1, pp. 206-207]. Consequently, for some (all) $1\le i\le N,$

$$\begin{aligned} G=\bigcup _{j=1}^M \left\{ {R: \displaystyle \left\| {h_{S_i}(R)-h_{S_i}(id)} \right\| \le \delta } \right\} R_j =\bigcup _{j=1}^M V_{S_i}R_j \subset \bigcup _{i=1}^N U(\varepsilon )R_j, \end{aligned}$$

and we are in the proof of [5, Lem.3]. $\square$

Next we show that the group property enhances from U to ${\mathscr {T}}_{|Z_a}$ if almost periodicity is considered.

Proposition 4.13

Let Y be a Banach space, Z a function space from Proposition 3.4 (4)-(10) U an abelian semitopological group. Then $\textrm{AP}(U,X)\subset Z_a,$ and for $f\in \textrm{AP}(U,X)$ we have ${\tilde{f}}\in \textrm{AP}({\mathscr {T}}_{|Z_a},X).$

Proof

Apply [7], Prop. 4.5]. $\square$

With the considerations above we obtain the following result.

Theorem 4.14

Let $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset \mathscr {L}(Z)$ with $\mathscr {O}(z)$ relatively $\tau$-compact for all $z\in Z$, and $0\in U.$ Further, let $c_0\not \subset Z$. Then $Z=Z_a\oplus Z_0,$ and ${\mathscr {T}}:=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{\tau -OT}.$ If for $\left\{ {h_s} \right\} _{s\in U}\subset AP({\mathscr {T}}_{|Z_a},Z)$ and $z=z_a\oplus z_0,$ we have

$$\begin{aligned} T(t+s)z_a-T(t)z_a=h_s(T(t)) \text{ for } \text{ all } t,s\in U, \end{aligned}$$

then

$$\begin{aligned} z_a \in Z_{ap}:= \left\{ {y\in Z: y\in Z_a, \text{ and } \mathscr {O}(y) \text{ relatively } \text{ compact } } \right\} . \end{aligned}$$

Proof

We may apply Corollary 3.10, to obtain $Z=Z_a\oplus Z_0.$ From the compactness of the orbits, we obtain

$$\begin{aligned} RT(s)z_a-Rz_a=h_s(R) \text{ for } \text{ all } R\in {\mathscr {T}}, \text{ and } s\in U, \end{aligned}$$

From the Lemmas 4.11, and  4.12 we obtain $\left\{ {Rz_a:R\in {\mathscr {T}}_{|Z_a}} \right\}$ is (relatively) compact, and as it is reversible, we obtain it is almost periodic, considering [9] on ${\overline{\textrm{span}}} \left\{ {Rz_a:R\in {\mathscr {T}}} \right\} .$ $\square$

Remark 4.15

Let G be a semitopological group, $Y\subset X^*,$ and $\left\{ {{T}(t)} \right\} _{t\in {U} }\subset \mathscr {L}(Y)$ a weighted semigroup with respect to $\tau =w^*.$ Then

$$\begin{aligned} Z_{ap}= \left\{ {y\in Z: \mathscr {O}(y) \text{ relatively } \text{ compact } } \right\} . \end{aligned}$$

For $y\in \left\{ {y\in Z: \mathscr {O}(y) \text{ relatively } \text{ compact } } \right\}$ we have to show it is reversible. Letting $R\in {\mathscr {T}}$ then from the compactness we have $Rx= \displaystyle \left\| {\cdot } \right\| \text {-} \lim _{ {\alpha }\in {A}} T(t_{\alpha })x.$ Without loss of generality $S= w^*\text {-}OT\text {-} \lim _{ {\alpha }\in {A}} T(\text {-}t_{\alpha }).$ Then for $y\in X,$ and $J_X:X\rightarrow X^{**}$ the canonical embedding

$$\begin{aligned} \left\langle x,y\right\rangle= & {} \lim _{ {\alpha }\in {A}}\left\langle T(-t_{\alpha })T(t_{\alpha })x,y\right\rangle = \lim _{ {\alpha }\in {A}}\left\langle T(t_{\alpha })x,T^*(-t_{\alpha })J_Xy\right\rangle \\ {}= & {} \left\langle Rx,S^*J_Xy\right\rangle =\left\langle SRx,y\right\rangle . \end{aligned}$$

This shows as well that almost periodic vectors are almost automorphic.

Surely, the previous theorem applies to the semigroups given in Proposition 3.4, but for the function spaces it is not suitable. For the function spaces we show how to restrict the $c_0$ condition to the range space.

Theorem 4.16

Let Y be a Banach space, Z a function space from Proposition 3.4 (4)-(10), such that $c_0\not \subset Y$ and $0\in U.$ If for an $f_a\in Z_a,$ and $\left\{ {h_s} \right\} _{s\in U}\subset Z$ with $\left\{ {\tilde{h_s}} \right\} _{s \in U}\subset AP({\mathscr {T}}_{|Z_a},Y),$ we have

$$\begin{aligned} T(t+s)f_a-T(t)f_a=\tilde{h_s}(T(t)) \text{ for } \text{ all } t,s\in U, \end{aligned}$$

then $\tilde{f_a}\in AP({\mathscr {T}}_{|Z_a},Y).$

Proof

An application of Cor 4.11, and then Lemma 4.12 gives the closure of the range of $\tilde{f_a}$ is relatively compact. Consequently, $\tilde{f_a}:({\mathscr {T}}_{|Z_a},\tau -OT) \rightarrow X$ becomes norm-continuous. For the semigroup ${\mathscr {S}},$ we have ${\tilde{f}}(RT)-{\tilde{f}}(R)=h_T(R)$ for $R,T\in {\mathscr {S}}.$ and $h_T$ almost periodic respect to ${\mathscr {S}}.$ We obtain $\tilde{f_a}:{\mathscr {S}}\rightarrow X$ is almost periodic by [21, Cor.1.a, or Lem. 2], in his sense due to $0\in U.$ That is, $\left\{ {T(s)f_a:s\in U} \right\}$ becomes relatively norm-compact. Consequently $f_a\in \textrm{AAP}(U,X),$ by the construction of the compactification, and $\tau$ weaker than the norm, we obtain $\left\{ {Rf_a:R\in {\mathscr {T}}_{|Z_a}} \right\}$ is compact in the SOT (strong operator topology) and $f_a$ reversible which concludes the proof. $\square$

5 Application: splittings of almost periodic functions

After the previous discussions, to find almost periodicity via difference equation, we will investigate geometric conditions on weighted semigroups, or certain function spaces, which eventually lead to the existence of an almost periodic reversible part. The condition that the semigroup on $\overline{\textrm{span}}\overline{\mathscr {O}(x)}^{\tau }$ or has to be abelian, can then be dropped. This will be substituted by a separability condition on the difference equation and the assumption that $\textrm{c}_0\not \subset Y$ for the underlying Banach space or range space. This lifts the well known deLeeuw–Glicksberg theory to non-abelian cases with still almost periodic reversible parts.

Corollary 5.1

Let U be an abelian semitopological semigroup with identity and let X be a Banach space such that $\textrm{c}_0\not \subset X$. Furthermore, let $\ell ^1(U,X)\subset \Lambda \subset \textrm{LC}(U,X)^*$ be a norming subspace such that the translation semigroup $\left\{ {{T}(t)} \right\} _{t\in {U} }$ is a weighted semigroup on $\textrm{WW}_{\Lambda }(U,X)$ with respect to $\tau =\sigma (\textrm{WW}_{\Lambda }(U,X),\Lambda ^*)$. Then

$$\begin{aligned} \textrm{WW}_{\Lambda }(U,X)=\textrm{WW}_{\Lambda }(U,X)_a\oplus \textrm{WW}_{\Lambda }(U,X)_0 \end{aligned}$$

If for $f\in \textrm{WW}_{\Lambda }(U,X)$ one has that

$$\begin{aligned} \overline{\mathscr {O}(f_t-f)}^{\tau }=\overline{ \left\{ {(T(s+t)-T(s))f:\ s\in U} \right\} }^{\tau } \text{ is } \text{ norm-separable, } \text{ for } \text{ all } t\in U, \end{aligned}$$

then for ${\mathscr {G}}:={\mathscr {T}}_{|WW_{\Lambda }(U,X)_a}$ we have $\tilde{f^a}\in \textrm{AP}({\mathscr {G}},X)$.

Proof

By Proposition 3.4 (8) we have that the translation semigroup on $\textrm{WW}_{\Lambda }(U,Y)$ is a weighted semigroup. Hence, let ${\mathscr {T}}=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{w^*}$ be its $w^*$-closure within the space $\mathscr {L}(\textrm{WW}_{\Lambda }(U,Y),\Lambda ^*)$. As it is a weighted semigroup the closure stays within $\mathscr {L}(\textrm{WW}_{\Lambda }(U,X))$. This set becomes a compact abelian semitopological semigroup, due to the assumption $\left\{ {s \mapsto \left\langle x^*,f_s\right\rangle } \right\} \in \textrm{W}(U)$, cf. [7, Thm. 4.4]. Moreover, we see that $\overline{\mathscr {O}(f^a_s-f^a)}^{w^*}\subset \overline{\mathscr {O}(f_s-f)}^{w^*}$. As $\overline{\mathscr {O}(f^a_s-f^a)}^{w^*}$ is norm-separable, we conclude by Theorem 3.11 that $f^a_s-f^a$ is almost periodic for all $s\in U$. Moreover, $f^a_s-f_a$ has compact range and

$$\begin{aligned} RT(s)f^a-Rf^a=g_s(R) \end{aligned}$$

where $\left\{ {R f^a} \right\} _{R\in {\mathscr {T}}}:U\rightarrow X$ are bounded and $g_s:{\mathscr {G}}\rightarrow X$ is almost periodic for all $s\in U$. Thus, we may apply Lemma 4.16 to obtain that $\left\{ {R f^a} \right\} _{R\in {\mathscr {T}}}$ compact, and $\tilde{f^a}:{\mathscr {G}}\rightarrow X$ is almost periodic. $\square$

Corollary 5.2

Let Y be a Banach spaces such that $\textrm{c}_0\not \subset Y^*$ and let $f\in \textrm{BUC}(\mathbf{\mathbb R^+},Y^*)$. If $\overline{\mathscr {O}(f_s-f)}^{w^*}$ is norm-separable and $(f_s-f)\in \textrm{WW}_{\Lambda }(\mathbf{\mathbb R^+},Y^*)$, for all $s\in \mathbf{\mathbb R^+}$ with $\ell ^1(U,Y)\subset \Lambda ,$ then for the corresponding splitting

$$\begin{aligned} \textrm{BUC}(\mathbf{\mathbb R^+},Y^*)=\textrm{BUC}(\mathbf{\mathbb R^+},Y^*)_a\oplus \textrm{BUC}(\mathbf{\mathbb R^+},Y^*)_a, f=f^a+f^0 \end{aligned}$$

we have $f^a\in \textrm{AP}(\mathbf{\mathbb R},Y^*).$

In view of [1, Lem. 4.6.13, p.300], we give sufficient condition. For the definition of $X^{\odot }$ we refer to [22, Def. 14.2.1, p. 422].

Corollary 5.3

Let X be a Banach space such that $\textrm{c}_0\not \subset X^{\odot \odot }$. Furthermore, let $\left\{ {{T}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }\subset \mathscr {L}(X)$ a uniformly bounded $C_0-$semigroup. Assume that for $x\in X$ one has that $\overline{\mathscr {O}(T(s)x-x)}^{w^*}$ is norm-separable and for a separating set $\Lambda \subset X^*,$

$$\begin{aligned} \left\{ {t\mapsto \left\langle T(t+s)x-T(t)x,x^*\right\rangle } \right\} \in W(\mathbf {\mathbb R^+},) \text { for } \text { all } s\in \mathbf {\mathbb R^+}, x^*\in \Lambda \end{aligned}$$

(5)

Let $j:X\rightarrow X^{\odot \odot },$ be the natural embedding. Then we have $jx=x_a^{\odot \odot }+x_0^{\odot \odot }$ and

$$\begin{aligned} \left\{ {t \mapsto T^{\odot \odot }(t)x_a^{\odot \odot }} \right\} \in \textrm{AP}(\mathbf{\mathbb R},X^{\odot \odot }). \end{aligned}$$

Proof

By Proposition 3.4 (2), $\left\{ {{T^{\odot \odot }}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }$ is a weighted semigroup on $X^{\odot \odot },$ with $\tau =w^*.$ Hence, if $j:X\rightarrow X^{\odot \odot }$ denotes the embedding, $jx=x_a^{\odot \odot }+x_0^{\odot \odot }$ due to Corollary 3.10. By assumption $\overline{\mathscr {O}(T(s)x-x)}^{w^*}$ is norm-separable, and that

$$\begin{aligned} \overline{\mathscr {O}(T^{ \odot \odot }(s)x_a^{\odot \odot }-x_a^{\odot \odot })}^{w^*}\subset \overline{\mathscr {O}(T(s)x-x)}^{w^*}\subset X^{\odot *}, \end{aligned}$$

we obtain norm-separability for the $w^*$-closure of the orbit $\mathscr {O}(T^{\odot \odot }(s)x_a^{\odot \odot }-x_a^{\odot \odot }),$ for all $s\in \mathbf{\mathbb R^+}.$ As (5) serves for ${\mathscr {T}}:=\overline{ \left\{ {{T^{\odot \odot }}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }}^{w^*-OT}$ abelian, on $E^s:={\overline{\textrm{span}}} \left\{ {\mathscr {O}(T^{\odot \odot }(s)x^{\odot \odot }-x^{\odot \odot })} \right\} ^{w^*}.$ Theorem 3.11 yields almost periodicity of

$$\begin{aligned} \left\{ {t\mapsto T^{\odot \odot }(t+s)x_a^{\odot \odot }-T^{\odot \odot }(t)x_a^{\odot \odot }} \right\} =:h_s(t) \text{ for } \text{ all } s\in \mathbf{\mathbb R^+}, \end{aligned}$$

and $E^s=E^s_{ap}=E^s_a=E^s_{rev}.$ Consequently, it leaves to apply Theorem 4.14. $\square$

Reapplying the above arguments in $X^{\odot }$ we obtain the following result.

Corollary 5.4

Let X be a Banach space such that $\textrm{c}_0\not \subset X^{\odot }.$ Furthermore, let $\left\{ {{T}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }\subset \mathscr {L}(X)$ a bounded $C_0$-semigroup. Assume that for $x^{\odot }\in X^{\odot }$ one has that $\overline{\mathscr {O}(T(s)x^{\odot }-x^{\odot })}^{w^*}$ is norm-separable and $\left\{ {t\mapsto T^{\odot }(t+s)x^{\odot }-T^{\odot }(t)x^{\odot }} \right\} \in \textrm{WW}_{\Lambda }(\mathbf{\mathbb R^+},X^{\odot })$ for all $s\in \mathbf{\mathbb R^+},$ and for some norming and closed space $\Lambda$. Then $x^{\odot }=x_a^{\odot }+x_0^{\odot }$ and $\left\{ {t \mapsto T^{\odot }(t)x_a^{\odot }} \right\} \in \textrm{AP}(\mathbf{\mathbb R},X^{\odot }).$

Example 5.5

Let $\left\{ { x_{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset X$ such that $w^*\text {-} \lim _{ {n} \rightarrow \infty } x_n=x^{**}\in X^{**}{\setminus } X$ and set $f(n):=x_n$. Then we have $f\in \ell ^{\infty }(\mathbf{\mathbb N},X)$ with a splitting $f(n)=f^a(n)+f^0(n)$ in $\ell ^{\infty }(\mathbf{\mathbb N},X^{**}),$ where $f^a(n)=x^{**}$ and $f^0(n)=x_n-x^{**}$. Note that

$$\begin{aligned} f(n+k)-f(n)=x_{n+k}-x_n, \text{ and } f^0(n+k)-f^0(n)=x_{n+k}-x_n. \end{aligned}$$

Hence, the difference is contained in X and $w^*\text {-} \lim _{ {n} \rightarrow \infty } (x_{n+k}-x_n)=0$ In order to keep its $w^*$-closure inside X, we observe that a condition on $\left\{ {x_{k+n}-x_n:\ k\in \mathbf{\mathbb N}} \right\}$ is needed. It is sufficient to assume that the set is pointwise weakly relatively compact, or its $w^*$-closure stays in X. In this case, $x^{**}\in X^{**}\setminus X$ becomes impossible. It is worth mentioning that the authors in [16] assumed the convergence of the sequence. Note that the $w^*$-closure of the orbit equals $\left\{ {k\mapsto x_{n+k}} \right\} _{n\in \mathbf{\mathbb N}}\cup \left\{ {x^{**}} \right\} .$ Hence, it is separable, and $\left\{ {n\mapsto x^{**}} \right\} ,$ is surely almost periodic.

Proposition 5.6

Let U be a metric, a semitopological semigroup with identity and $D\subset U$ dense. Moreover, let $f\in \textrm{LC}(U,X)$ and consider the corresponding splitting with $Pf=f^a$ and $(I-P)f=f^0,$ in $\textrm{LC}(U,X^{**}).$ If $\left\{ {(I-P)(f_s-f):\ s\in D} \right\}$ is pointwise weakly relatively compact in X and there exists a net $\left\{ { {t}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset U$ such that $w^*\text {-} \lim _{ {\alpha }\in {A}} f^0(t_{\alpha })=0$ then $f^a\in \textrm{LC}(U,X)$.

Proof

Let $\tau$ be the pointwise-$w^*$-topology. First canonically imbed $\textrm{LC}(U,X)$ into $\textrm{LC}(U,X^{**})$ which provides together with the translation semigroup a weighted semigroup. Note that by the definition of a flight vector $f^0$ we have $0\in \overline{\mathscr {O}(f^0)}^{pointwise\text {-}w*}$. Hence, we are able to find a net such that $\tau \text {-} \lim _{ {\alpha }\in {A}} f_{t_{\alpha }}=0$. The $w^*\text {-}w^*$-continuity of the translation semigroup yields $\tau \text {-} \lim _{ {\alpha }\in {A}} f_{t_{\alpha }+s}=0$. As $\delta _0\in \ell ^1(U,X^*)$ and $w^*\text {-} \lim _{ {\alpha }\in {A}} f(t_{\alpha }+s)=0$. Moreover,

$$\begin{aligned} -f^0(s)=w^*\text {-} \lim _{ {\alpha }\in {A}} (f^0((t_{\alpha }+s)-f^0(s))=w\text {-} \lim _{ {\alpha }\in {A}} (f^0(t_{\alpha }+s)-f^0(s))\in X.\end{aligned}$$

As $f^a(s)=f(s)-f^0(s)$, the proof concludes. $\square$

Corollary 5.7

Let U be an abelian semitopological semigroup with identity, and $D\subset U$ dense. If $f\in \textrm{LC}(U,X)$ and $(f_s-f)\in \textrm{LC}_{WRC}(U,X)$ for all $s\in D,$ is weakly relative compact, then $(f_s-f)_a\in \textrm{LC}_{WRC}(U,X),$ and for the splitting $f_a\in \textrm{LC}(U,X^{**}),$ we have $f_a\in \textrm{LC}(U,X).$ Consequently,

$$\begin{aligned} \textrm{LC}_D(U,X):= \left\{ {f\in \textrm{LC}(U,X): \left\{ {f_s-f} \right\} _{s\in D} \in \textrm{LC}_{WRC}(U,X)} \right\} \end{aligned}$$

splits like

$$\begin{aligned} \textrm{LC}_D(U,X)=\textrm{LC}_D(U,X)_a\oplus \textrm{LC}_D(U,X)_0 \end{aligned}$$

Proof

If P is a minimal idempotent on $\textrm{LC}(U,X^{**}),$ then it is minimal on the subset $\textrm{LC}_{WRC}(U,X),$ compare Example 3.4 (6). $(f_a(s+U)-f_a(U))_a\subset \overline{f(s+U)-f(U)}^{weak}.$ Appyling the uniform continuity, and

$$\begin{aligned}{} & {} f_0(s+U)-f_0(U)\subset \overline{f(s+U)-f(U)-f_a(s+U)-f(U)}^{weak}\\{} & {} \quad \subset \overline{f(s+U)-f(U)}^{weak}-\overline{f_a(s+U)-f(U)}^{weak} \end{aligned}$$

Now apply Prop. 5.6. $\square$

6 Application: a review of a result by Farkas & Kreidler

Theorem 6.1

Let U be an abelian metric semitopological semigroup with identity, and let D be dense in U. Assume that X is a Banach space not containing a copy of $\textrm{c}_0$. Note that the translation $\left\{ {{T}(t)} \right\} _{t\in {U} }$ on $\textrm{LC}(U,X^{**})$ equipped with $\tau =\sigma (\textrm{LC}(U,X^{**}),\ell ^1(U,X^*))$ is a weighted semigroup. Let ${\mathscr {T}}:=\overline{ \left\{ {{T}(t)} \right\} _{t\in {U} }}^{w^*}$ and choose $P:\textrm{LC}(U,X^{**})\rightarrow \textrm{LC}(U,X^{**})$ to be a minimal projection in ${\mathscr {T}}$. For a given $f\in \textrm{LC}(U,X)$ the following properties hold:

• (1) The translation semigroup $\left\{ {{T}(t)} \right\} _{t\in {U} }$ restricted to $\textrm{AAP}(U,X)$ equipped with the uniform topology becomes a weighted semigroup with a unique splitting.

• (2) Let $P\in {\mathscr {T}}$ be a minimal projection. Then

  $$\begin{aligned} P(\textrm{AAP}(U,X))= & {} \big \{f\in \textrm{LC}(U,X^{**})_a: f(U)\subset X,\\{} & {} \text{ and } \left\{ {Rf: R\in {\mathscr {T}}} \right\} \text{ relatively } \text{ compact } \big \}. \end{aligned}$$

• (3) If the mapping $\left\{ {s\mapsto (T(t+s)-T(s))f} \right\} \in \textrm{W}(U,X)$ pointwise for $t\in D$ and if $\left\{ {(I-P)(f_t-f)(s):\ s\in D} \right\}$ is weakly relatively compact in X, then

  $$\begin{aligned} f^a\in \left\{ {f\in \textrm{LC}(U,X^{**})_a: f(U)\subset X, \text{ and } \left\{ {Rf: R\in {\mathscr {T}}} \right\} \text{ relatively } \text{ compact } } \right\} . \end{aligned}$$

• (4) If the mapping $\left\{ {s\mapsto (T(t+s)-T(s))f} \right\} \in \textrm{W}(U,X)$ pointwise for $t\in U,$ and $\left\{ {(I-P)(f_s-f):\ s\in U} \right\}$ is weakly relatively compact in $\textrm{W}(U,X)$ for all $s\in U$, then f is Eberlein weakly almost periodic.

• (5) If $\left\{ {s\mapsto (T(t+s)-T(s))f} \right\} \in \textrm{AAP}(U,X)$ for all $s\in U$ and $(I-P)(S(t)-I)f$ is convergent, then f is aap.

Proof

To prove property (1) we note that $\mathscr {O}(f)$ is relative compact with respect to $\displaystyle \left\| { {\cdot } } \right\| _{\infty }$. As $\tau =\sigma (\textrm{BUC}(U,X^{**}),\ell ^1(U,X^*))$ is weaker than the uniform topology, the claim becomes straightforward.

For the verification of (2) note that $Pf= \displaystyle \left\| { {\cdot } } \right\| _{\infty } \text {-} \lim _{ {n} \rightarrow \infty } f(t+t_n)$ for some sequence $\left\{ { t_{n} } \right\} _{{n} \in \mathbf{\mathbb N}} \subset \mathbf{\mathbb R}$. Hence, we stay in X and the Jacobs–deLeeuw—Glicksberg decomposition yields the uniqueness, cf. [9].

To obtain (3) we observe that $f=f^a+f^0$ by the general splitting theorem in $\textrm{LC}(U,X^{**})$. Then, Proposition 5.6 yields $f_a\in \textrm{LC}(U,X)$. By the assumption that $f_{t+s}-f_s \in \textrm{W}(U,X)$ and by the Jacobs-deLeeuw-Glicksberg decomposition, see [9], we conclude that

$$\begin{aligned} (f(\cdot +s)-f(\cdot ))_a(t)=f_a(t+s)-f_a(t)=g_s(t) \end{aligned}$$

and $f^a\in \left\{ {f\in \textrm{LC}(U,X)_a: \left\{ {Rf: R\in {\mathscr {T}}} \right\} \text{ relatively } \text{ compact } } \right\}$ by Theorem 4.16.

In order to show that (4), we note that $f=f^a+f^0$ by the general splitting theorem in $\textrm{LC}(U,X^{**})$. From the previous we have

$$\begin{aligned} f^a\in \left\{ {f\in \textrm{LC}(U,X^{**})_a: f(U)\subset X, \text{ and } \left\{ {Rf: R\in {\mathscr {T}}} \right\} \text{ relatively } \text{ compact } } \right\} . \end{aligned}$$

It leaves to verify that the flight vector is weakly almost periodic. To do so, note that

$$\begin{aligned} f^0_s-f^0_t=(f^0_s-f^0)-(f^0_t-f^0)=(I-P)(T(s)f-f)-(I-P)(T(t)f-f). \end{aligned}$$

Hence, the flight vector is Eberlein weakly almost periodic. Hence, f becomes Eberlein-weakly almost periodic.

To prove (5) it leaves to verify that the flight vector becomes aap. This can be done similarly to the previous for the uniform topology. $\square$

Remark 6.2

In the case where $f\in \textrm{LC}_{\textrm{wrc}}(U,X)$ the assumption that $\{(I-P)(f_s-f):\ s\in U\}$ is pointwise weakly relatively compact in X is surely fulfilled as the translation semigroup is weighted on $\textrm{LC}_{\textrm{wrc}}(U,X)$.

As a direct consequence of Theorem 6.1 we are now able to generalise the result by Farkas and Kreidler, cf. [16, Thm. 3.3 & Rem. 3.4].

Corollary 6.3

Let X be a Banach space which does not contain a copy of $\textrm{c}_0$ and let $T\in \mathscr {L}(X)$ be a power-bounded operator. As $\textrm{AAP}(\mathbf{\mathbb N},X)$ is a weighted semigroup with a unique splitting, let P denote the projection. Let $x \in X$ be such that $\left\{ {k\mapsto (T^{k+l}-T^k)x} \right\} \in \textrm{AAP}(\mathbf{\mathbb N},X),$ for $l\in \mathbf{\mathbb N},$ and $(I-P)(T^n-I)x$ is convergent. Then x is aap.

Proof

Let $f(n):=T^nx,$ and apply Theorem 6.1. $\square$

7 Applications: integration of almost periodic functions

Lemma 7.1

Assume that $f\in \textrm{W}(\mathbf{\mathbb R},X)$ has a bounded integral. Let ${\mathbb {I}}$ denote the function on $\mathbf{\mathbb R}$ constant 1,  and $F(t):=\int _0^tf(s)\,\textrm{d}{s}$, $t\in \mathbf{\mathbb R}$. Then, we have that

$$\begin{aligned} \overline{\mathscr {O}(F)}^{\sigma (\textrm{BUC}(\mathbf{\mathbb R},X^{**}),\ell ^1(\mathbf{\mathbb R},X^*))}\subset \overline{F(\mathbf{\mathbb R})}^{w^*}\times {\mathbb {I}}+ \left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\} . \end{aligned}$$

Moreover, $\left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\}$ is separable.

Proof

Let $\left\{ { {t}_{\gamma } } \right\} _{{\gamma } \in {\Gamma }} \subset \mathbf{\mathbb R}$ be a net. Then

$$\begin{aligned} \int _0^{t+t_{\gamma }}f(s)\,\textrm{d}{s}=\int _0^{t_{\gamma }}f(s)\,\textrm{d}{s}+\int _0^tf_{t_{\gamma }}(s)\,\textrm{d}{s}\in \overline{F(\mathbf{\mathbb R})}^{w^*}+ \left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\} . \end{aligned}$$

Observe that the weak topology on weakly compact sets is an Eberlein topology, cf. [18]. Hence, it leaves to verify that $\left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\}$ is $\sigma (\textrm{BUC}(\mathbf{\mathbb R},X^{**}),\ell ^1(\mathbf{\mathbb R},X^*))$-closed. To do so, let $f_{t_n}\rightarrow g$ weakly and $(t,x^*)\in \mathbf{\mathbb R}\times X^*$. Then

$$\begin{aligned} \int _0^tx^*f_{t_{n}}(s)\,\textrm{d}{s}\rightarrow \int _0^tx^*g(s)\,\textrm{d}{s}, \end{aligned}$$

by making use of the fact the finite integral is a linear functional. It leaves to verify that the set $\left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\}$ is separable. Let

$$\begin{aligned} \left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w^*}} \right\} , \text{ and } \mathscr {O}_{\mathbf{\mathbb Q}}(f):= \left\{ {f_q(s)=f(q+s):\ q\in \mathbf{\mathbb Q}} \right\} . \end{aligned}$$

Then

$$\begin{aligned} \int _0^tf_r(s)\,\textrm{d}{s}-\int _0^tf_q(s)\,\textrm{d}{s}= & {} \int _r^{t+r}f(s)\,\textrm{d}{s}-\int _q^{t+q}f(s)\,\textrm{d}{s}\\= & {} \int _q^rf(s)\,\textrm{d}{s}+\int _{t+q}^{t+r}f(s)\,\textrm{d}{s}. \end{aligned}$$

Let $f=w\text {-} \lim _{ {\alpha }\in {A}} f_{t_{\alpha }}$. For $\varepsilon _{\alpha }\rightarrow 0$ we find $\left\{ { {q}_{\alpha } } \right\} _{{\alpha } \in {A}} \subset \mathbf{\mathbb Q}$ such that $0\le q_{\alpha }-t_{\alpha }\le \varepsilon _{\alpha }.$ Hence

$$\begin{aligned} f-f_{q_{\alpha }}=f-f_{t_{\alpha }} + f_{t_{\alpha }}-f_{q_{\alpha }}. \end{aligned}$$

For $\left\{ {{T}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }$ the translation semigroup, we obtain

$$\begin{aligned} \displaystyle \left\| {f_{t_{\alpha }}-f_{q_{\alpha }}} \right\| = \displaystyle \left\| {T(s_{\alpha })(f-T(q_{\alpha }-s_{\alpha })f)} \right\| \le \displaystyle \left\| {f-T(q_{\alpha }-s_{\alpha })f} \right\| \end{aligned}$$

Consequently, the uniform continuity of f, which we obtain by [36, Prop. 2.1], yields the weak separability, hence the norm-separability. $\square$

Next we going to add a condition to the well known result of Kadets.

Theorem 7.2

Let X be a Banach space with $X\subset Y^*$ and let $f\in \textrm{AP}(\mathbf{\mathbb R},X)$ such that $F(t):=\int _{0}^t f(\tau )\,\textrm{d}{\tau }$ is bounded. Assume that one of the following conditions on the range of F hold:

• (1) $F(\mathbf{\mathbb R})$ is weakly relatively compact

• (2) $\textrm{c}_0\not \subset X$

• (3) $\overline{F(\mathbf{\mathbb R})}^{w^*}$ is norm-separable as a set in $Y^*$

Then $F\in \textrm{AP}(\mathbf{\mathbb R},X)$.

Proof

Without loss of generality we may assume X to be separable, otherwise choose $X=\overline{\textrm{span}}F(\mathbf{\mathbb R})$. By the previous proposition, the weak compactness of the integral range as well as the almost periodicity we have

$$\begin{aligned} \int _0^{t_{\gamma }}f(s)\,\textrm{d}{s}\in X. \end{aligned}$$

In view of Proposition 7.1 it leaves to verify that $\left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w}} \right\}$ is separable for an almost periodic function f. Certainly one has that

$$\begin{aligned} \left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{w^*}} \right\} = \left\{ {\int _0^tg(s)\,\textrm{d}{s}:\ g\in \overline{\mathscr {O}(f)}^{ \displaystyle \left\| { {\cdot } } \right\| _{\infty } }} \right\} . \end{aligned}$$

Consider again $\mathscr {O}_{\mathbf{\mathbb Q}}(f):= \left\{ {f_q(s)=f(q+s):\ q\in \mathbf{\mathbb Q}} \right\}$. Then

$$\begin{aligned} \int _0^tf_r(s)\,\textrm{d}{s}-\int _0^tf_q(s)\,\textrm{d}{s}= & {} \int _r^{t+r}f(s)\,\textrm{d}{s}-\int _q^{t+q}f(s)\,\textrm{d}{s}\\= & {} \int _q^rf(s)\,\textrm{d}{s}+\int _{t+q}^{t+r}f(s)\,\textrm{d}{s}. \end{aligned}$$

which yields the separability. As $x^*\int _0^tf(s)\,\textrm{d}{s}\in \textrm{AP}(\mathbf{\mathbb R})$ by the Bohl–Bochner Theorem we are in the situation of [7, Cor. 5.18]. Hence the reversible part is almost periodic, which coincides with the integral as weakly almost periodic functions are reversible. $\square$

The next corollary is different to [35, Thm. 4.12], as it provides assumptions on range to obtain an almost periodic reversible part, whereby Ruess/Summers provide conditions on the Eberlein weakly almost periodic f,  to obtain an integral in this class.

Corollary 7.3

Let X be a Banach space. Assume that $f\in \textrm{W}(\mathbf{\mathbb R^+},X)$ and that the integral $F(t)=\int _0^tf(\tau )\, \textrm{d}\tau$ is bounded. Consider the corresponding splitting $F=F^a+F^0\in \textrm{LC}(\mathbf{\mathbb R^+},X^{**})=BUC(\mathbf{\mathbb R^+}, X^{**})$.

• (1) If $F(\mathbf{\mathbb R^+})$ is weakly relatively compact, then the splitting is unique up to constant and $F^a$ is almost periodic.

• (2) If in addition $\textrm{c}_0\not \subset X$ and if both $F(\mathbf{\mathbb R^+})$ and $\left\{ {(F^0_{t+s}-F_s^0):\ t\ge 0} \right\}$ are weakly relatively compact, then F is Eberlein weakly almost periodic.

Proof

Let $\left\{ {{T}(t)} \right\} _{t\in {\mathbf{\mathbb R^+}} }$ denote the translation semigroup on $\textrm{BUC}_{\textrm{wrc}}(\mathbf{\mathbb R^+},X)$. As f is uniformly continuous and let P denote the corresponding possibly non-unique norm-1-projection of a splitting in $\textrm{BUC}_{\textrm{wrc}}(\mathbf{\mathbb R^+},X)$. Then, we have $T(t)P=PT(t)$ and

$$\begin{aligned}{} & {} \displaystyle \left\| { {\frac{1}{h}(PF(t+h+\cdot )-PF(t+\cdot ))-Pf(t+\cdot )} } \right\| _{\infty } \\{} & {} \quad \le \displaystyle \left\| { {\frac{1}{h}(F(t+h+\cdot )-F(t+\cdot ))-f(t+\cdot )} } \right\| _{\infty } = \displaystyle \left\| { {\frac{1}{h}\int _0^h \left( {f(t+s+\cdot )-f(t+\cdot )} \right) ds} } \right\| _{\infty } \\{} & {} \quad \le \sup _{r\in [0,h]} \displaystyle \left\| { {f(t+r+\cdot )-f(t+\cdot )} } \right\| _{\infty } \rightarrow 0 \end{aligned}$$

for $h\rightarrow 0$. Surely, $Pf=g$ for a unique almost periodic function g, see [9]. By the previous observation and using $PF=F_a$ we obtain $F^{\prime }_a=g$. Hence, it is a primitive with $F_a\in \textrm{WW}_{\Lambda }(\mathbf{\mathbb R},X)$ for $\Lambda =\ell ^1(\mathbf{\mathbb R},X^*)$. Using $\overline{F^a(\mathbf{\mathbb R})}^w\subset \overline{F(\mathbf{\mathbb R})}^w$ we obtain that $\overline{\mathscr {O}(F^a)}^{w^*}$ is separable. The second part is direct consequence of Theorem 6.1. $\square$

In fact of the above, we obtain the following result.

Corollary 7.4

Let X be a Banach space. Assume that $f\in \textrm{W}(\mathbf{\mathbb R^+},X^*)$ and that the integral $F(t)=\int _0^tf(\tau )\, \textrm{d}\tau$ is bounded. Then the corresponding splitting $F=F^a+F^0$ is unique up to a constant. If in addition $\overline{F(\mathbf{\mathbb R^+})}^{w^*}$ is separable, then $F^a$ is almost periodic.

Remark 7.5

Let $f:\mathbf{\mathbb R}\rightarrow X$ and $Y=X^{**}$. Assume that both X and $K:=\overline{F(\mathbf{\mathbb R})}^{w^*}$ are separable. Then K is a Radon–Nikodym set by an application of [17, Prop. 2.2, Thm. 2.13, & Lem. 2.17]. This condition however which is certainly weaker than being weakly compact. Note that $\overline{\textrm{span}}(K)$ may contain $\textrm{c}_0$ as $\textrm{c}_0$ is separable and therefore weakly compactly generated. Hence the above approach shows that the assumption of separability is necessary and natural.

Corollary 7.6

Let $(\Omega ,\tau )$ be a topological space and $f\in \textrm{AP}(\mathbf{\mathbb R},\mathrm {C_b}(\Omega ))$. Consider the topology of pointwise convergence $\tau _\textrm{p}$ on $\mathrm {C_b}(\Omega )$. If $F(t):=\int _0^tf(s)\, \textrm{d}{s}$ is bounded and $\overline{F(\mathbf{\mathbb R})}^{\tau _p}$ is separable, then $F\in \textrm{AP}(\mathbf{\mathbb R},\mathrm {C_b}(\Omega )).$

Proof

Recall that $\mathrm {C_b}(\Omega )\subset {\mathscr {B}}(\Omega )=\ell ^1(\Omega )^*$. Consider $f\in \textrm{BUC}(\mathbf{\mathbb R},\mathrm {C_b}(\Omega ))\subset B(\mathbf{\mathbb R},\ell ^1(\Omega )^*)=\ell ^1(\mathbf{\mathbb R},\ell ^1(S))^{*}$. $\square$

Corollary 7.7

Let $(\Omega ,\Sigma ,\mu )$ be a measure space with a topology. Moreover, let $f\in \textrm{AP}(\mathbf{\mathbb R},\textrm{L}^1(\Omega ,X))$ and consider the topology $\tau =\sigma (\textrm{L}^1(\Omega ,X),\textrm{C}_\textrm{b}(\Omega ,X^*))$ induced by the bi-linear form $\int _\Omega \left\langle h(s),g(s)\right\rangle \,\textrm{d}{s}$ where $g\in \textrm{C}_\textrm{b}(\Omega ,X^*)$. If $F(t):=\int _0^tf(s)\, \textrm{d}{s}$ is bounded and $\overline{F(\mathbf{\mathbb R})}^{\tau }$ is separable in the space of $\textrm{C}_\textrm{b}(\Omega ,X^*)^*$ then $F\in \textrm{AP}(\mathbf{\mathbb R},\textrm{L}^1(\Omega ,X^*)).$

8 Example: the necessity

To show that the condition $c_0\not \subset X$ is necessary, we enhance the well known example to the missing separability of the $w^*$-closure of the orbit. The proof of the next example was simplified with the help of Manfred Droste [11], who advised us to apply the Chinese Remainder Theorem.

Consider the following mapping

$$\begin{aligned} \begin{array}{ccccc} {f} &{} : &{} {\mathbf{\mathbb R}} &{} \longrightarrow &{} {c} \\ &{} &{} {t} &{} \longmapsto &{} \displaystyle { \left\{ {\exp (it\pi /n)} \right\} _{n\in \mathbf{\mathbb N}}} \end{array} . \end{aligned}$$

By [1, Exa. 4.6.5] the range fails to be norm compact. As it is a primitive in view of Lemma 7.1 we have to verify that $K=\overline{f(\mathbf{\mathbb R})}^{w^*}$ fails to be separable in $\ell ^{\infty }.$ This surely implies that the range fails to be compact.

Indeed, let $P:= \left\{ {p:\ p \text{ prime } \text{ number } } \right\} ,$ and for a real number p let $\left[ p\right] :=\sup _{n\in \mathbf{\mathbb N}} \left\{ {n\le p} \right\} .$ Then ${\mathscr {P}}(P)$ has the cardinality of the continuum. For $A\in {\mathscr {P}}(P)$ we define

$$\begin{aligned} \begin{array}{ccccc} {\chi _A} &{} : &{} {\mathbf{\mathbb N}} &{} \longrightarrow &{} {\mathbf{\mathbb C}} \\ &{} &{} {p} &{} \longmapsto &{} \displaystyle {\left\{ \begin{array}{rcl} 1&{}:&{} p\in A \\ \exp \left( {2\pi i \frac{[p/2]}{p}} \right) &{}:&{} p\in P\setminus A \\ 0 &{}:&{} \text{ otherwise } , \end{array}\right. } \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ccccc} {p} &{} : &{} {{\mathscr {P}}(P)} &{} \longrightarrow &{} {\ell ^{\infty }} \\ &{} &{} {A} &{} \longmapsto &{} \displaystyle { \left\{ {\chi _A(n)} \right\} _{n\in \mathbf{\mathbb N}}.} \end{array} \end{aligned}$$

We observe that $\displaystyle \left\| { {p(A)-p(B)} } \right\| _{\infty } \ge 1$ whenever $A\not =B$. We claim that $p({\mathscr {P}}(P))\subset \overline{ \left\{ {\exp (it/n):\ t\in \mathbf{\mathbb R}} \right\} }^{w*}$. To see this, we make the following estimate.

$$\begin{aligned} \sum _{n=1}^{\infty } \left| {(c_n-\exp (it\pi /n))x_n} \right|= & {} \sum _{n=1}^{N} \left| {(c_n-\exp (it\pi /n))x_n} \right| \\{} & {} +\sum _{n=N}^{\infty } \left| {(c_n-\exp (it\pi /n))x_n} \right| <\varepsilon \end{aligned}$$

For some $A\in {\mathscr {P}}(P)$ let $c_n=\chi _A(n)$, and $x_n=0$ for all n not prime. As the remaining indices $1\le p_1,\ldots ,p_k\le N$ are prime and therefore coprime, by the Chinese Remainder Theorem, [39, 13.9,p.80], for similar congruences, we find for given $a_1,\ldots ,a_k$ an integer $L\in \mathbf{\mathbb N}$ as well as natural numbers $m_1,\ldots ,m_k\in {\mathbb {N}}$ such that

$$\begin{aligned} a_j=m_jp_j+L \text{ or } L=-m_jp_j+a_j \text{ for } j= p_1,\dots ,p_k. \end{aligned}$$

Consequently, on the prime indexes i we have

$$\begin{aligned} \exp (iL2\pi /p_j)=\exp (-2m_j\pi i)\exp (2\pi a_j/p_j)=\exp (2\pi i a_j/p_j) \end{aligned}$$

by chosing $a_j=\left[ p_j/2\right]$ if j is prime but not in A and $a_j=0$ if $j\in A$. Hence,

$$\begin{aligned} \sum _{n=1}^{N} \left| {(c_n-\exp (iL\pi /n))x_n} \right| =0. \end{aligned}$$

Consequently, our claim is verified and we are in the situation of [13, Cor. 2.1.10] which implies the non-separability.