Sun dual theory for bi-continuous semigroups

The sun dual space corresponding to a strongly continuous semigroup is a known concept when dealing with dual semigroups, which are in general only weak$^*$-continuous. In this paper we develop a corresponding theory for bi-continuous semigroups under mild assumptions on the involved locally convex topologies. We also discuss sun reflexivity and Favard spaces in this context, extending classical results by van Neerven.


Introduction
Semigroup theory is a well-established tool in the abstract study of evolution equations.Classically, strongly continuous semigroups of bounded linear operators on Banach spaces (also called C 0 -semigroups) are considered, meaning that the semigroup is strongly continuous with respect to the norm topology.This, however, limits the applicability of the theory in spaces such as C b (R n ) or L ∞ , ruling out interesting examples arising from (partial) differential equations.This fact is underlined by Lotz's result [42] asserting that any strongly continuous semigroup on Grothendieck spaces with the Dunford-Pettis property is automatically uniformly continuous.
On the other hand, it has long been known that strong continuity fails to be preserved for the dual semigroup (T ′ (t)) t≥0 ∶= (T (t) ′ ) t≥0 in general, and merely translates into weak * -continuity.Nevertheless, the strong continuity of the "presemigroup" (T (t)) t≥0 encodes enough structure to allow for a rich theory.Following first results in the early days of semigroup theory; by Phillips [48], Hille-Phillips [27], de Leeuw [12], see also Butzer-Berens [5]; intensified research on dual semigroups was conducted in the 1980s centred around a "Dutch school" in a series of papers such as by Clément, Diekmann, Gyllenberg, Heijmans and Thieme [6][7][8][9]14], de Pagter [13].The renewed interest in dual semigroups was partially driven by the interest from applications in e.g.delay equations [15].At a peak of these developments van Neerven [54] finally provided a general comprehensive treatment of the theory, together with many new results clarifying especially the topological aspects, see also [55][56][57].Since then, the interest in dual semigroups which fail to be strongly continuous remained, and we name particularly applications in mathematical neuroscience [52,53].
The key concept to compensate for the lack of strong continuity of dual semigroups is the notion of the sun dual space and the related sun dual semigroup.More precisely, given a strongly continuous semigroup (T (t)) t≥0 on a Banach space X, the sun dual space X ⊙ consists of the elements x ′ in the continuous dual X ′ such that lim t→0+ T ′ (t)x ′ = x ′ .As X ⊙ is closed and T ′ (t)-invariant, the restrictions of T ′ (t) to X ⊙ define a strongly continuous semigroup (T ⊙ (t)) t≥0 on X ⊙ , an object which is in many facets superior to the dual semigroup.
Note that this approach can be viewed as a way to regain symmetry in duality for continuity properties of the semigroup.While this holds trivially for reflexive spaces X -in which case X ⊙ = X ′ -, it is not surprising that sun duality comes with an adapted notion of reflexivity, so-called sun reflexivity (or ⊙-reflexivity), which depends on the semigroup under consideration.In particular, if X is ⊙-reflexive with respect to the semigroup (T (t)) t≥0 , then (T ⊙⊙ (t)) t≥0 can be identified with (T (t)) t≥0 via the canonical isomorphism j∶ X → (X ⊙ ) ′ , x ↦ (x ⊙ ↦ ⟨x ⊙ , x⟩).That this framework indeed leads to a meaningful theory is also reflected by the existence of an Eberlein-Shmulyan type theorem due to van Neerven [56], and de Pagter's characterisation of sun reflexivity [13], which can be seen as a variant of Kakutani's theorem.About ten years after this flourishing period of dual semigroups, Kühnemund [38,39] conceptualized semigroups which only satisfy weaker continuity properties through the notion of bi-continuous semigroups.More precisely, the strong continuity was relaxed to hold with respect to a Hausdorff locally convex topology τ coarser than the norm topology on X.Under the additional conditions that τ is sequentially complete on norm-bounded sets and the dual space of (X, τ ) is norming, an exponentially bounded semigroup (T (t)) t≥0 on X is called τ -bi-continuous if the trajectories T (⋅)x are τ -strong continuous and locally sequentially τ -equicontinuous on norm-bounded sets.Since the weak * -topology shares these properties, dual semigroups naturally fall in this framework.Thus the question becomes how the construction of the sun dual can be seen in this light.With this paper we would like to answer this question and hence generalise existing results for strongly continuous semigroups in the presence of previously missing topological subtleties.
The interest in bi-continuous semigroups goes beyond the above mentioned special case of dual semigroups, as they, for instance, naturally emerge in the study of evolution equations on spaces of bounded continuous functions, most prominently parabolic problems, see e.g.Farkas-Lorenzi [25], Metafune-Pallara-Wacker [45].In the last decades the abstract theory of bi-continuous semigroups has been further developed and variants of the classical case have been established, such as for instance perturbation results; Farkas [22,23], approximation results; Albanese-Mangino [1] and mean ergodic theorems; Albanese-Lorenzi-Manco [2].In [24] Farkas defined a proper concept for a dual bi-continuous semigroup by considering a suitable subspace X ○ of X ′ .In particular, the restriction of the dual semigroup on X ○ is again a σ(X ○ , X)-bi-continuous dual semigroup under some additional topological assumptions.
In this work we develop a sun dual theory for bi-continuous semigroups and discuss its peculiarities with respect to properties of the present topologies.This generalises the classical case, i.e. strongly continuous semigroups with respect to the norm topology; henceforth simply called "strongly continuous".Apart from the abstract interest in developing a sun dual framework for bi-continuous semigroups, one of our main motivations to provide such generalizations are open problems of the following kind: We aim to extend the following theorem for strongly continuous semigroups to bi-countinuous ones.
Here F av(T ) denotes the Favard space of (T (t)) t≥0 given by F av(T ) ∶= {x ∈ X lim sup Note that [28, Theorem 2.9, p. 152] lists another equivalent condition, which relates to control theory, see also [28,Remark 2.4,p. 148].Following this, the question whether Theorem 1.1 can be formulated for bi-continuous semigroups is relevant for studying generalizations of control theoretic notions in non-strongly continuous semigroup settings.The concept of sun dual spaces for strongly continuous semigroups is pivotal in the proof of the non-trivial implication (i) ⇒ (ii) in Theorem 1.1.The argument is based, among other tools, on two characterizations due to van Neerven, [54, Theorems 3.2.8,3.2.9,p. 57]: The first stating that an element x ∈ X belongs to F av(T ) if and only if that there exists a bounded sequence (y n ) n∈N in X such that lim n→∞ R(λ, A)y n = x for some (all) λ in the resolvent set ρ(A) of A where R(λ, A) ∶= (λ id −A) −1 .The second result claims that the property Let us briefly highlight some of our findings in the following.Starting from Farkas' dual space [24] X ○ ∶= {x ′ ∈ X ′ x ′ τ -sequentially continuous on ⋅ -bounded sets}, which is a closed subspace of X ′ and invariant under the dual semigroup, we define the bi-sun dual space X • as the space of strong continuity for the restricted dual semigroup T ○ (t) ∶= T ′ (t) X ○ , t ≥ 0. Under the additional assumptions that ⋅ -bounded sets, we can subsequently show that the norm defined by is equivalent to ⋅ .This result, Theorem 4.3, naturally generalises the corresponding known fact for strongly continuous semigroups (see [54,Theorem 1.3.5,p. 7] and the discussion in the previous paragraph).Further, let us point out that the assumptions (1)-( 3) are fulfilled by Theorem 3.8 if (X, γ) is a sequentially complete c 0 -barrelled Mazur space, e.g. a sequentially complete Mackey-Mazur space, where γ ∶= γ( ⋅ , τ ) denotes the mixed topology of Wiweger [61].We henceforth say that X is •-reflexive with respect to the τ -bi-continuous semigroup (T (t)) t≥0 if the canonical map j∶ X → X •′ given by ⟨j(x), maps the space of strong continuity X cont onto X •• .Given the latter property, we show that j∶ X → X •′ is surjective if and only if the unit ball The article is organized as follows.In the preparatory Section 2 we set the stage by discussing the topological assumptions and recapping some basics on bicontinuous semigroups as well as integral notions in this context.With the level of detail we aim for making the presentation rather self-contained, especially for readers less familiar with bi-continuous semigroups.In Sections 3 and 4 we present our approach to dual semigroups of bi-continuous semigroups and the sun dual space, respectively.The short Section 5 discusses the notion of sun reflexivity in this generalised context and we finish with studying the relation of the obtained results to Favard spaces, Section 6.

Notions and preliminaries
For a vector space X over the field R or C with a Hausdorff locally convex topology τ we denote by (X, τ ) ′ the topological linear dual space and just write X ′ ∶= (X, τ ) ′ if (X, τ ) is a Banach space.For two topologies τ 1 and τ 2 on a space X, we write τ 1 ≤ τ 2 if the topology τ 1 is coarser than τ 2 .Further, we use the symbol L(X; Y ) ∶= L((X, ⋅ X ); (Y, ⋅ Y )) for the space of continuous linear operators from a Banach space (X, ⋅ X ) to a Banach space (Y, ⋅ Y ) and denote by ⋅ L(X;Y ) the operator norm on L(X; Y ).If X = Y , we set L(X) ∶= L(X; X).
2.1.Definition ([35, Definition 2.2, p. 3]).Let (X, ⋅ ) be a Banach space and τ a Hausdorff locally convex topology on X that is coarser than the ⋅ -topology τ ⋅ .Then (a) the mixed topology γ ∶= γ( ⋅ , τ ) is the finest linear topology on X that coincides with τ on ⋅ -bounded sets and such that τ ≤ γ ≤ τ ⋅ , (b) the triple (X, ⋅ , τ ) is called a Saks space if there exists a directed system of seminorms P τ that generates the topology τ such that The mixed topology γ is Hausdorff locally convex and our definition is equivalent to the one from the literature [ In this case x Ω (f ) is unique due to X being Hausdorff and we define the τ -Pettis f (s)dλ(s).
2.4.Definition.Let (X, ⋅ , τ ) be a sequentially complete Saks space and Ω ⊂ R non-empty.We set is the space of continuous functions from Ω to (X, τ ).
in X and the Riemann and the Pettis integral coincide.
Furthermore, the τ -Riemann integrability of f implies that the Riemann sums are τ -convergent.They are even ⋅ -bounded as f is ⋅ -bounded.It follows from [11, I.1.10Proposition, p. 9] that the Riemann sums are γ-convergent and their γ-limit coincides with their τ -limit because γ is stronger than τ .Thus f is γ-Riemann integrable on [a, b] in X and this integral coincides with the τ -Riemann integral.Now, we only need to prove that f ∶ [a, b] → (X, γ) is continuous.Then it follows as above that f is γ-Pettis integrable on [a, b] in X and that the Riemann and the Pettis integral coincide.Let (x n ) n∈N be a sequence in [a, b] . By [11, I.1.10Proposition, p. 9] it follows that (f The proof is analogous to (a).The condition that ⟨x ′ , f ⟩ is improper Riemann integrable on [a, ∞) for all x ′ ∈ (X, τ x, is continuous for all x ∈ X, (iii) (T (t)) t≥0 is locally sequentially γ-equicontinuous, i.e. for every sequence locally uniformly for all t ∈ [0, ∞).
If we want to emphasize the dependence on the Saks space, we say that (T (t)) t≥0 is a bi-continuous semigroup on (X, ⋅ , τ ).[26, Proposition 3.6 (ii), p. 1137] in combination with [61,2.4.1 Corollary,p. 56] gives that a bi-continuous semigroup (T (t)) t≥0 on X is exponentially bounded (of type ω), i.e. there exist M ≥ 1 and ω ∈ R such that T (t) L(X) ≤ M e ωt for all t ≥ 0, and we call its growth bound (see [38, p. 7]).Due to the exponential boundedness of a bicontinuous semigroup and [11, I.1.10Proposition, p. 9] we also may rephrase the definition [21, Definition 1.2.6, p. 7] of the generator of a bi-continuous semigroup in terms of the mixed topology.2.7.Definition.Let (X, ⋅ , τ ) be a sequentially complete Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.The generator (A, D(A)) is defined by We recall that an element λ ∈ C belongs to the resolvent set ρ(A) of the generator Usually, it is required that Y is a ⋅ -closed subspace (or a Banach space normcontinuously embedded in X) which is (T (t)) t≥0 -invariant (see [18, Chap.II, Definition, p. 60]), but this is not needed just for the sake of the definition of A Y .With these definitions at hand, we recall the following properties of the generator of a bi-continuous semigroup given in [39, Definition 9, Propositions 10, 11, Theorem 12, Corollary 13, p. 213-215], which are summarised in [4, Theorems 5.5, 5.6, p. 339-340], and may be rephrased in terms of the mixed topology by [11, I.1.10Proposition, p. 9] and Proposition 2.5 as well.
2.8.Theorem.Let (X, ⋅ , τ ) be a sequentially complete Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).Then the following assertions hold: where the integral is a γ-Pettis integral.(f ) For each ω > ω 0 there exists M ≥ 1 such that for all k ∈ N and Re λ > ω, i.e. the generator (A, D(A)) is a Hille-Yosida operator.(g) Let X cont be the space of ⋅ -strong continuity for (T (t)) t≥0 , i.e.
Then X cont is a ⋅ -closed, sequentially γ-dense, (T (t)) t≥0 -invariant linear subspace of X.Moreover, X cont = D(A) ⋅ and (T (t) Xcont ) t≥0 is the ⋅strongly continuous semigroup on X cont generated by the part A Xcont of A in X cont and We added in part (g) that X cont is sequentially γ-dense in X, which is a consequence of (b).

Dual bi-continuous semigroups
We start this section by recalling the definition of the dual semigroup on X ○ of a bi-continuous semigroup on X given in [24], where for a Saks space (X, ⋅ , τ ) we set 3.1.Remark.Let (X, ⋅ , τ ) be a Saks space.Then X ○ is a closed linear subspace of the norm dual X ′ and hence a Banach space by [24, Proposition 2.1, p. 314].We note that it is assumed in [24, Proposition 2.1, p. 314] that the Saks space (X, ⋅ , τ ) is sequentially complete (see [24, Hypothesis A (ii), p. 310-311]) but an inspection of its proof shows that this assumption is not needed.

3.3.
Proposition ([24, Proposition 2.4, p. 315], [3, Lemma 1, p. 6]).Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space, ⋅ X ○ the restriction of ⋅ X ′ to X ○ and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).Then the following assertions hold: (a) The triple (X ○ , ⋅ X ○ , σ(X ○ , X)) is a sequentially complete Saks space.(b) The operators given by T Next, we take a closer look at the space X ○ and its relation to the dual space (X, γ) ′ where γ is the mixed topology of ⋅ and τ .Both spaces coincide if (X, γ) is a Mazur space.This will be a quite helpful observation in the next sections.3.5.Remark.Let (X, ⋅ , τ ) be a Saks space.Then (X, γ) ′ is a closed linear subspace of X ′ , in particular a Banach space, and seq-γ by [11, I.1.10Proposition, p. 9] and since we always have 3.6.Proposition.Let (X, ⋅ , τ ) be a Saks space.10 Proposition, p. 9] and thus τ -continuous as (X, τ ) is a Mazur space and every τ -convergent sequence ⋅bounded.But this implies that x ′ is γ-continuous since τ is coarser than γ.

3.7.
Definition.We call a Saks space (X, ⋅ , τ ) a Mazur space if (X, γ) is a Mazur space.Now, let us revisit Definition 3.2 and give sufficient conditions in terms of the mixed topology γ when the conditions of this definition are fulfilled.For that purpose we recall that a Hausdorff locally convex space (X, ϑ) is called c 0 -barrelled if every σ((X, ϑ) ′ , X)-null sequence in (X, ϑ) ′ is ϑ-equicontinuous (see [30, p. 249], or [59,Definition,p. 353] where such spaces are called sequentially barrelled ).
(a) Let (X, γ) be a Mazur space.Then condition (i) of Definition 3.2 is fulfilled if and only if Proof.(a) We have X ′ γ = X ○ by Remark 3.5 and so the triple ) is a Saks space by our considerations above Definition 3.2.Our claim follows from [61, 2.3.2Corollary, p. 55] since condition (i) of Definition 3.2 is equivalent to the sequential completeness of (X ○ , ⋅ X ○ , σ(X ○ , X)), and γ ○ = τ c (X ′ γ , (X, ⋅ )) by [35, 3.22   Second, let C b (Ω) be the space of bounded continuous functions on a completely regular Hausdorff space Ω and We denote by τ co the compact-open topology, i.e. the topology of uniform convergence on compact subsets of Ω, which is induced by the directed system of seminorms P τco given by Let V denote the set of all non-negative bounded functions ν on Ω that vanish at infinity, i.e. for every ε > 0 the set {x ∈ Ω ν(x) ≥ ε} is compact.Let β 0 be the Hausdorff locally convex topology on C b (Ω) that is induced by the seminorms for ν ∈ V. Due to [50, Theorem 2.4, p. 316] we have γ( ⋅ ∞ , τ co ) = β 0 .Let M t (Ω) denote the space of bounded Radon measures on a completely regular Hausdorff space Ω and ⋅ Mt(Ω) be the total variation norm (see e.g.[40, p. 439-440] where Furthermore, a Banach space (X, ⋅ ) is called weakly compactly generated (WCG) if there is a σ(X, X ′ )-compact set K ⊂ X such that X = span(K) where span(K) denotes the ⋅ -closure of span(K) (see [20,Definition 13.1, p. 575]).A Banach space (X, ⋅ ) is called strongly weakly compactly generated space (SWCG) if there exists a σ(X, X ′ )-compact set K ⊂ X such that for every σ(X, X ′ )-compact set L ⊂ X and ε > 0 there is n ∈ N with L ⊂ (nK + εB ⋅ ) by [49, p. 387].In particular, every SWCG space is a WCG space by [ .Moreover, we recall that a Banach space (X, ⋅ ) has an almost shrinking basis if it has a Schauder basis such that its associated sequence of coefficient functionals forms a Schauder basis of (X ′ , µ(X ′ , X)) where µ(X ′ , X) is the Mackey topology on X ′ (see [31, p. 75]).
The generator 4.1.Corollary.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space, ⋅ X ○ the restriction of ⋅ X ′ to X ○ and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).We define the bi-sun dual is the ⋅ X ○ -strongly continuous semigroup on X • generated by the part Proof.We only need to prove ω 0 (T • ) ≤ ω 0 (T ).The rest of the corollary is a direct consequence of Theorem 2.8 (g) and Proposition 3.3.We note that for all t ≥ 0, yielding ω 0 (T • ) ≤ ω 0 (T ).
Let us comment on the definition of X •• and its relation to X •○ ∶= (X • ) ○ and (X • ) • .4.5.Remark.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.The triple (X • , ⋅ X • , σ(X • , X •′ )) is a Saks space, where ⋅ X • denotes the restriction of ⋅ X ′ to X • , and we have for all t ≥ 0, we see that Like in [54, Corollary 1.3.6,p. 8] we can consider X as a subspace of X •′ .
4.6.Corollary.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.Then the canonical map j∶ X → X •′ given by ⟨j(x), Proof.j is clearly linear.If j(x) = 0 for some x ∈ X, then ⟨x • , x⟩ = 0 for all x • ∈ X • , which implies that x • = 0 and thus x = 0 by Theorem 4.3.The inclusion j(X cont ) ⊂ (X •• ∩ j(X)) follows directly from the definitions of X cont (see Theorem 2.8 (g)) and X •• .For the converse inclusion let x ∈ X with j(x) ∈ X •• .We note that for any t ≥ 0 and x implying the rest of our statement because In our next theorem we investigate the relation between the resolvent sets ρ(A), ρ(A • ) and ρ(A •′ ) resp. the resolvents R(λ, A), R(λ, A • ) and R(λ, A •′ ).4.7.Theorem.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, ), then we have j(R(λ, A)x) = R(λ, A •′ )j(x) for all X with the canonical map j∶ X → X •′ .
Proof.(a) Let λ ∈ ρ(A).For any x ∈ X and which means that x annihilates the range of λ − A ○ .In particular, x annihilates ).Thus we have x • = 0 and so x = 0 by Theorem 4.3, implying the injectivity of λ − A.
Next, we show that the range of λ − A is ⋅ -dense and ⋅ -closed, which then implies the surjectivity of λ − A. Suppose that the range of λ − A is not ⋅ -dense.Then there is some x ○ ∈ X ○ with x ○ ≠ 0 such that for any x ∈ D(A) we have ⟨(λ − A)x, x ○ ⟩ = 0 since X ○ separates the points of X.It follows that ⟨Ax, x ○ ⟩ = ⟨x, λx ○ ⟩ for all x ∈ D(A) and so x ○ ∈ D(A ○ ) by Proposition 3.3 (b).We deduce that (λ − A ○ )x ○ = 0 as D(A) is sequentially γ-dense by Theorem 2.8 (b), which yields Let us turn to the ⋅ -closedness of the range of λ − A. Let x ∈ D(A).By Theorem 4.3 there is x n = y for some y ∈ X, we derive from the estimate above that (x n ) n∈N is a ⋅ -Cauchy sequence, say with limit z ∈ X, because ⋅ and ⋅ • are equivalent norms on X by Theorem 4.3.Since (A, D(A)) is sequentially γ-closed by Theorem 2.8 (a), in particular ⋅ -closed as γ is coarser than the ⋅ -topology, we get z ∈ D(A) and y = (λ − A)z.Hence λ − A is bijective and (3) yields that (λ − A) −1 ∈ L(X) as well.
Let us turn to sufficient conditions for R(λ, A) • X • ⊂ D(A • ) to hold in Theorem 4.7 (a).4.8.Proposition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).
(a) If Re λ > ω 0 (T ), then we have λ ∈ ρ(A) and R(λ, A) Proof.(a) Let Re λ > ω 0 (T ).Then we have Re λ > ω 0 (T • ) by Corollary 4.1 and thus λ ∈ ρ(A) ∩ ρ(A • ) by Theorem 2.8 (e) as well as and P γ be a directed system of seminorms that generates the mixed topology γ.Since X ○ = (X, γ) ′ by Remark 3.5, there are p γ ∈ P γ and C ≥ 0 such that ⟨R(λ, A) for all x ∈ X. Due to the continuity of R(λ, A)∶ (X, γ) → (X, γ), there are pγ ∈ P γ and C ≥ 0 such that ⟨R(λ, A) For any x ∈ X we note that which means that y ○ ∈ (X, γ) ′ = X ○ .Thus we have ⟨Ax, R(λ, A) . Further, we observe that for any x ∈ D(A) The sequential γ-density of D(A) by Theorem 2.8 (b) and Thus we have for any For any x ∈ X we remark that for all x ∈ X.We deduce that Part (a) shows that the continuity of R(λ, A)∶ (X, γ) → (X, γ) need not be a necessary condition for R(λ, A) • X • ⊂ D(A • ) for all Re λ > ω 0 (T ).This is an open question.Another open question is whether one actually has R(λ, A) The answer is affirmative if τ coincides with the ⋅topology.Because then γ also coincides with the ⋅ -topology, which gives that R(λ, A)∶ (X, γ) → (X, γ) is continuous for all λ ∈ ρ(A).Therefore Proposition 4.8 (b) and Theorem 4.7 imply [54, Theorem 1.4.2,p. 10] (see also [27,Theorem 14.3.3,p. 425]).
where Ω denotes the cardinality of Ω.Let M ⊂ ℓ 1 be σ(ℓ 1 , ℓ ∞ )-compact and absolutely convex.Then we have for all x ∈ ℓ ∞ and λ ∉ σ(A) where Now, we only need to show that M λ is σ(ℓ 1 , ℓ ∞ )-compact and absolutely convex.First, we note that C λ ∶= sup n∈N Due to the characterisation of σ(ℓ 1 , ℓ ∞ )-compactness above it remains to show that M λ is uniformly absolutely summable.Let ε > 0. Since M is uniformly absolutely summable, there is δ > 0 such that for all Ω ⊂ N with Ω < δ and all y ∈ M it holds Further, it is easy to check that M λ is absolutely convex because M is absolutely convex.Hence R(λ, A) is µ(ℓ ∞ , ℓ 1 )-continuous by ( 6) for all λ ∈ ρ(A).Therefore Example 3.9 (b) and Proposition 4.
In particular, if By assumption there is some t 0 > 0 such that x ∉ ⋃ 0≤r≤t0 G r σ(X,X ○ ) .Since the complement of the latter set is σ(X, X ○ )-open, there are some n ∈ N and x ○ i ∈ X ○ , 1 ≤ i ≤ n, and ε > 0 such that the σ(X, X ○ )-neighbourhood V of x given by Since X ○ = (X, γ) ′ by Remark 3.5, for every 1 ≤ i ≤ n there are C i > 0 and p γi ∈ P γ such that ⟨x ○ i , z⟩ ≤ C i p γi (z) for all z ∈ X where P γ is a directed system of seminorms that generates the mixed topology γ.From P γ being directed it follows that there are C ≥ 1 and p γ ∈ P γ such that ⟨x ○ i , z⟩ ≤ Cp γ (z) for all z ∈ X and 1 ≤ i ≤ n.By the proof of Theorem 4.3 we know that γ-lim t→0+ Thus there is some 0 < t 1 ≤ t 0 such that We claim that Ṽ ∩ G = ∅ where Indeed, for g ∈ G there is some which shows that Ṽ ∩ G = ∅ and proves the claim.However, Now, we generalise the definition of (weak) equicontinuity w.r.t. a norm-strongly continuous semigroup from [54, p. 25] and [54, Proposition 2.2.2, p. 26] to the bicontinuous setting.4.11.Definition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.We say that a set G ⊂ X is γ-(T (t)) t≥0 -equicontinuous if the set {t ↦ T (t)g g ∈ G} is γ-equicontinuous at t = 0. We say that G is σ(X, X ○ )-(T (t)) t≥0 -equicontinuous if for each x ○ ∈ X ○ the set {t ↦ ⟨x ○ , T (t)g⟩ g ∈ G} is equicontinuous at t = 0. 4.12.Remark.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space, G ⊂ X and (T (t)) t≥0 a bi-continuous semigroup on X.
Proof.The inclusion ⊂ is clear since G 0 = G.We prove the converse inclusion ⊃ by contraposition.Let x ∉ G σ(X,X ○ ) .We have to show that there is some t 0 > 0 such that x ∉ ⋃ 0≤r≤t0 G r σ(X,X ○ ) .Like in Proposition 4.10 there are some n ∈ N and By the σ(X, X ○ )-(T (t)) t≥0 -equicontinuity there is t 0 > 0 such that for every 0 ≤ r ≤ t 0 , g ∈ G and 1 ≤ i ≤ n we have This yields for every 0 < r ≤ t 0 , g ∈ G and We derive that for every 0 < r ≤ t 0 , g ∈ G and We deduce that Ṽ ∩ G r = ∅ for all 0 < r ≤ t 0 where This finishes the proof because x ∈ Ṽ .

4.16.
Corollary.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.Then a σ(X, X Proof.The implication ⇒ is obvious because σ(X, X ○ ) is a finer topology than σ(X, X • ).Let us turn to the implication ⇐.Let (x n ) n∈N be a σ(X, X ○ )-(T (t)) t≥0equicontinuous sequence in X that is σ(X, X • )-convergent to some x ∈ X.Then the set G ∶= {x n n ∈ N} ∪ {x} is the σ(X, X • )-closure of {x n n ∈ N} and so its σ(X, X ○ )-closure by Corollary 4.14 as well.Hence G is also σ(X, X ○ )-(T (t)) t≥0equicontinuous by Remark 4.12 (c).Let V be a σ(X, X Corollary 4.15.This implies that all but finitely many x n lie in (V ∩ G) ⊂ V , which we had to show.Now, we give a class of sets to which the three preceding corollaries can be applied due to Remark 4.12 (a) if (X, γ) is a Mazur space.4.17.Proposition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, Proof.Let P γ be a directed system of seminorms that generates the mixed topology γ.Due to [37, Lemma 5.5 (a), p. 2680] and [35, Remark 2.3 (c), p. 3] we may choose P γ such that x = sup pγ ∈Pγ p γ (x) for all x ∈ X.We start with noting that the map for all t > 0 and h ∈ H by Theorem 2.8 (c) and (d).For any x ′ ∈ (X, γ) ′ we get ≤ tM e ω t AR(λ, A) L(X) h for any p γ ∈ P γ since (T (t)) t≥0 is exponentially bounded and AR(λ, A) ∈ L(X) because AR(λ, A)x = λR(λ, A)x − x for all x ∈ X.Since H is ⋅ -bounded, there is C > 0 such that h ≤ C for all h ∈ H, which yields for all t > 0 and p γ ∈ P γ .This means that R(λ, A)H is γ-(T (t)) t≥0 -equicontinuous at t = 0. Proposition 4.17 in combination with Remark 4.12 (b) generalises [54, Proposition 2.2.6, p. 27].The next proposition transfers one direction of [54,Corollary 2.2.8,p. 28] to the bi-continuous setting.4.18.Proposition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).Let G ⊂ X be σ(X, X • )-compact.Then the following assertions hold: Proof.(a) Let G ⊂ X be σ(X, X • )-compact.We may regard G as a subset of X •′ via the the canonical map j∶ X → X •′ from Corollary 4.6.Then G is σ(X •′ , X • )compact and thus ⋅ X • ′ -bounded by the uniform boundedness principle, implying the ⋅ -boundedness by Corollary 4.6.
The rest of statement (b) is a consequence of Proposition 4.8 (a) and (b).

Remark.
(a) Let (X, ⋅ ) be a Banach space.For a ⋅ -strongly continuous semigroup (T (t)) t≥0 on X we have X cont = X and X •• = X ⊙⊙ .Thus X is •-reflexive w.r.t.(T (t)) t≥0 if and only if it is ⊙-reflexive w.r.t.(T (t)) t≥0 .(b) One might object to coining the property j(X cont ) = X •• by "•-reflexivity", as it is not symmetric.However, our main point in studying this property lies in its value for describing the Favard space F av(T ) and its relation to the generator (A, D(A)) of (T (t)) t≥0 (and by part (a), it is indeed a reasonable name for this property).
First, we study the relation between a bi-continuous semigroup and its restriction to its space of strong continuity with regard to (bi-)sun reflexivity.5.3.Proposition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.Then the following assertions hold: (a) T •• j(x) = j(T (t)x) for all t ≥ 0 and x ∈ X cont .(b) The maps ι∶ X • → X ⊙ cont , ι(x • ) ∶= x • Xcont , and κ∶ X ⊙⊙ cont → X •• , κ(y) ∶= y ○ ι, are well-defined, linear and continuous, and ι is injective.In particular, we have the continuous embeddings Proof.(a) We note that j(X cont ) ⊂ X •• by Corollary 4.6 and T (t)X cont ⊂ X cont for all t ≥ 0 by Theorem 2.8 (g), which implies for any t ≥ 0, x ∈ X cont and x • ∈ X • .(b) Due Theorem 2.8 (g) and Remark 3.5 X cont is sequentially γ-dense in X and X ○ = X ′ seq-γ .Thus the continuous linear map Xcont , is injective and we note that ι = ι 0 X • .From T ○ (t)x ○ = T ′ (t)x ○ for all t ≥ 0 and x ○ ∈ X ○ it follows ι 0 (X • ) ⊂ X ⊙ cont .Thus we get y ○ ι ∈ X •′ for any y ∈ X ⊙ cont ′ and by Corollary 4.6 where we identified X cont and X with subspaces of X •′ via j.

The Favard space
We begin this section with the definition of the Favard space.6.1.Definition.Let (X, ⋅ , τ ) be a sequentially complete Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.Then the Favard space (class) of (T (t)) t≥0 is defined by 6.2.Remark.Let (X, ⋅ , τ ) be a sequentially complete Saks space and (T (t)) t≥0 a bi-continuous semigroup on X.
(a) It is obvious from the definition of the generator (A, D(A)) that D(A) ⊂ F av(T ).(b) From T (t)x−x = t 1 t T (t)x−x for all t > 0 and x ∈ X, we obtain F av(T ) ⊂ X cont where X cont is the space of ⋅ -strong continuity of (T (t)) t≥0 from Theorem 2.8 (g).
Our goal is to characterise those bi-continuous semigroups on X for which F av(T ) = D(A) holds.A class of bi-continuous semigroups for which this holds are the dual semigroups of norm-strongly continuous semigroups.6.3.Example.Let (X, ⋅ ) be a Banach space and (S(t)) t≥0 a ⋅ -strongly continuous semigroup on X with generator (A, D(A)).Then (S ′ (t)) t≥0 is a bi-continuous semigroup semigroup on (X ′ , ⋅ X ′ , σ(X ′ , X)) by [38,   Next, we present a proposition that extends [54, Theorem 3.2.3,p. 55] to the bi-continuous setting.6.6.Proposition.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).
).The definitions of the Favard space and of T •• yield that where X is identified with its image j(X) in X •′ by Corollary 4.6.Since X cont = X •• ∩ X by Corollary 4.6 again and F av(T ) ⊂ X cont by Remark 6.2 (b), the statement is proved.
Next, we show that the second inclusion is a consequence of the equation for all t > 0 and x ∈ X, which we get from Theorem 2.8 (d).Indeed, take x ∈ R(λ, A •′ )B X •′ ∩ X. Due to Proposition 6.6 we have So, since x ∈ F av(T ), (T (t)) t≥0 is exponentially bounded, ⋅ = sup pγ ∈Pγ p γ on X for some directed system of seminorms P γ that generates γ, and ⋅ • is equivalent to ⋅ by Theorem 4.3, the right-hand side of (8) remains ⋅ • -bounded as t → 0+ whereas the left-hand side γ-converges to x (as a sequence with t = t n for any (t n ) n∈N with t n → 0+) by the proof of Theorem 4.3.Thus there is n ∈ N such that x ∈ nR(λ, A)B (X, ⋅ • ) seq-γ .
Due to the equivalence of ⋅ • and ⋅ there is M ≥ 0 such that B ⋅ ⊂ B (X, ⋅ • ) ⊂ M B ⋅ , which yields that the lemma above is still valid if ⋅ • is replaced by ⋅ .The next theorem is a generalisation of [54, Theorem 3.2.8,p. 57] and describes the space F av(T ) in terms of approximation by elements of D(A).6.9.Theorem.Let (X, ⋅ , τ ) be a sequentially complete d-consistent Mazur-Saks space and (T (t)) t≥0 a bi-continuous semigroup on X with generator (A, D(A)).Then the following assertions are equivalent for x ∈ X: (i) x ∈ F av(T ) (ii) For some (all) λ ∈ ρ(A) such that R(λ, A) • X • ⊂ D(A • ) there exists a ⋅bounded sequence (y n ) n∈N in X with γ-lim n→∞ R(λ, A)y n = x.(iii) For some (all) λ ∈ ρ(A) such that R(λ, A) • X • ⊂ D(A • ) there exist a ⋅bounded sequence (y n ) n∈N in X and k ∈ N 0 with γ-lim n→∞ R(λ, A) k+1 y n = R(λ, A) k x.
(i)⇒(iv) B X •′ is σ(X •′ , X • )-compact by the Banach-Alaoglu theorem.By assumption we may identify X and X •′ as well as B X •′ and B (X, ⋅ • ) via j because j is an isometry as a map from (X, ⋅ • ) to (X •′ , ⋅ X •′ ).

0 T 0 T
(a) The generator (A, D(A)) is sequentially γ-closed, i.e. whenever (x n ) n∈N is a sequence in D(A) such that γlim n→∞ x n = x and γlim n→∞ Ax n = y for some x, y ∈ X, then x ∈ D(A) and Ax = y.(b) The domain D(A) is sequentially γ-dense, i.e. for each x ∈ X there exists a sequence (x n ) n∈N in D(A) such that γlim n→∞ x n = x.(c) For x ∈ D(A) we have T (t)x ∈ D(A) and T (t)Ax = AT (t)x for all t ≥ 0. (d) For t > 0 and x ∈ X we have t (s)xds ∈ D(A) and A t (s)xds = T (t)x − x where the integrals are γ-Pettis integrals.(e) For Re λ > ω 0 we have λ ∈ ρ(A) and
4, the first part of our statement follows from Proposition 6.6.Let us consider the second part.Let (X, γ) be semi-reflexive.Then X is •-reflexive w.r.t.(T (t)) t≥0 by Proposition 5.5 and X = X •′ via the canonical map j.Hence we have F av(T ) = D(A •′ ) by the first part of our statement.As D(A) ⊂ F av(T ) by Remark 6.2 (a), we only need to prove that D(A •′ ) ⊂ D(A).Let Re λ > ω 0 (T ).Then it follows from Theorem 4.7 (c) and Proposition 4.8 (a) that R see Theorem 6.12, implying that F av(T ) = D(A) if one (thus both) of the assertions holds.In analogy to strongly continuous semigroups, we are able to show in Theorem 6.10 that the domain of the semigroup generator equals the Favard space if and only if the set R(λ, A)B (X, ⋅ • ) is closed with respect to τ .The main results are thoroughly laid out by various natural classes of examples.
2.3.Definition.Let (X, τ ) be a Hausdorff locally convex space over the field K ∶= R or C, Ω ⊂ R a measurable set w.r.t. the Lebesgue measure λ and L 1 (Ω) the space of (equivalence classes of) absolutely Lebesgue integrable functions from Ω integrable, τ -Pettis integrable and γ-Pettis integrable on [a, ∞) in X and all four integrals coincide.
Definition.Let (X, ⋅ , τ ) be a sequentially complete Saks space and γ ∶= [19,s a Mackey-Mazur space, then it is c 0 -barrelled by[59, Proposition  4.3, p. 354] because (X, γ) is sequentially complete.Let us come to some examples of sequentially complete d-consistent Mazur-Saks spaces.First, we recall some notions from general topology.A completely regular space Ω is called k R -space if any map f ∶ Ω → R whose restriction to each compact K ⊂ Ω is continuous, is already continuous on Ω (see[46, p. 487]).In particular, locally compact Hausdorff spaces clearly are Hausdorff k R -spaces.In addition Polish spaces, i.e. separably completely metrisable spaces, are Hausdorff k R -spaces by [29, Proposition 11.5, p. 181] and[19, 3.3.20,3.3.21Theorems,p.152].We recall that a Hausdorff space Ω is called hemicompact if there is a sequence (K n ) n∈N of compact sets in Ω such that for every compact set K ⊂ Ω there is N ∈ N such that K ⊂ K N (see[19, Exercises 3.4.E, p. 165]).For instance, σ-compact locally compact Hausdorff spaces are hemicompact Hausdorff k R -spaces by [19, Exercises 3.8.C (b), p. 195].Further, there are hemicompact Hausdorff k R -spaces that are neither locally compact nor metrisible by [58, p. 267].
Theorem 2.8 (d), Proposition 3.3 (b) and Corollary 4.1 and we note that 1 t0 X •′ and thus x ∈ X cont as ⋅ • and ⋅ are equivalent by Theorem 4.3.