Weighted composition semigroups on spaces of continuous functions and their subspaces

This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces $\mathcal{F}(\Omega)$ of continuous functions on a Hausdorff space $\Omega$ such that the norm-topology is stronger than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces, the space of bounded Dirichlet series and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Guti\'errez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces $\mathcal{F}(\mathbb{D})$ of holomorphic functions on the open unit disc $\mathbb{D}$ such that $H^{\infty}\subset\mathcal{F}(\mathbb{D})\subset\mathcal{B}_{1}$ where $H^{\infty}$ is the Hardy space of bounded holomorphic functions on $\mathbb{D}$ and $\mathcal{B}_{1}$ the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity.


Introduction
Let (F (Ω), ⋅ ) be a Banach space of scalar-valued continuous functions on a Hausdorff space Ω such that the ⋅ -topology is stronger than the compact-open topology τ co on F (Ω). Suppose that ϕ ∶= (ϕ t ) t≥0 is a semiflow and m ∶= (m t ) t≥0 an associated semicocycle on Ω such that the induced weighted composition semigroup (C m,ϕ (t)) t≥0 given by C m,ϕ (t)f ∶= m t ⋅ (f ○ ϕ t ) for all t ≥ 0 and f ∈ F (Ω) is a welldefined semigroup of linear maps from F (Ω) to F (Ω).
In the case that Ω = D ⊂ C is the open unit disc, ϕ a jointly continuous holomorphic semiflow and F (D) a space of holomorphic functions on D such semigroups are well-studied in the unweighted case, i.e. m t (z) = 1 for all t ≥ 0 and z ∈ D, on the Hardy spaces H p for 1 ≤ p < ∞ in [10], on the weighted Bergman spaces A p α for α > −1 and 1 ≤ p < ∞ in [76], on the Dirichlet space D in [77] and on more general spaces F (D) in [4,12,27,43,78].In particular, they are always ⋅ -strongly continuous on H p , A p α and D. The weighted case, where m is a jointly continuous holomorphic semicocycle, is more complicated and got more attention recently [5,11,18,19,44,70,84].However, to the best of our knowledge from the spaces mentioned above only for the Hardy spaces H p , 1 ≤ p < ∞, and the weighted Bergman spaces A p α , 1 ≤ p < ∞ and α > −1, sufficient (non-trivial) conditions on general m are known such that the weighted composition semigroup becomes ⋅ -strongly continuous [53,70,75,78,84].In [5] non-trivial sufficient conditions for ⋅ -strong continuity are given for Banach spaces F (D) of holomorphic functions in which the polynomials are dense in the case that m t = ϕ ′ t for all t ≥ 0. Considering the Hardy space for p = ∞, it is shown in [44] that the only ⋅strongly continuous weighted composition semigroups on F (D) such that H ∞ ⊂ F (D) ⊂ B 1 are the trivial ones, i.e. ϕ t = id for all t ≥ 0. Here, B 1 stands for the Bloch space.In the unweighted case this has already been observed in [3,43].Similarly, it is shown in [24,Theorem 7.1,p. 34] that there no non-trivial ⋅ -strongly continuous composition semigroups on the space H ∞ of bounded Dirichlet series on the open right half-plane.Nevertheless, there are non-trivial weighted composition semigroups on such spaces F (D) resp.H ∞ as well and it is said in [43, p. 494] that it would be desirable to substitute the ⋅ -strong continuity by a weaker property so that [43,Main theorem,p. 490] ([44, Theorems 2.1, 3.1, p. [68][69] in the weighted case), which describes the generator of a ⋅ -strongly continuous (weighted) composition semigroup, remains valid.This is one of the problems we solve in the present paper.We substitute the ⋅ -strong continuity by γ-strong continuity where γ ∶= γ( ⋅ , τ co ) is the mixed topology between the ⋅ -topology and τ co , which was introduced in [83] and is a Hausdorff locally convex topology.
Let us outline the content of our paper.In Section 2 we recall the notions of a Saks space (X, ⋅ , τ ), where (X, ⋅ ) is a normed space and τ a coarser norming Hausdorff locally convex topology on X, the mixed topology γ ∶= γ( ⋅ , τ ), some background on semigroups on Hausdorff locally convex spaces and in Theorem 2.10 how γ-strongly continuous, locally γ-equicontinuous semigroups are related to the concept of a τ -bi-continuous semigroup, which was introduced in [60,61].We then give several examples of Saks spaces of the form (F (Ω), ⋅ , τ co ), which include among others the Hardy spaces H p for 1 ≤ p < ∞, the weighted Bergman spaces A p α for 1 ≤ p < ∞ and α > −1, and the Dirichlet space D in Example 2.12, the v-Bloch spaces w.r.t. a continuous weight v in Example 2.13, especially the Bloch type spaces B α for α > 0, as well as weighted spaces of continuous resp.holomorphic functions, especially, the Hardy space H ∞ and the space H ∞ of bounded Dirichlet series in Example 2.14, Example 2.15 and Example 2. 16.
In Section 3 we recap the notions of a semiflow ϕ, a semicocycle m for ϕ and introduce the notion of a co-semiflow (m, ϕ).We give equivalent characterisations of their joint continuity depending on the topological properties of Ω, present several examples and generalise the concept of a generator of a semiflow, which was introduced in [10] for jointly continuous holomorphic semiflows.
In Section 4 we use the concepts and results of the preceding sections to prove one of our main results Proposition 4.5, which generalises [41, Proposition 2.10, p. 5] and [56,Corollary 4.3,p. 20], that the weighted composition semigroup (C m,ϕ (t)) t≥0 on a Saks space (F (Ω), ⋅ , τ co ) is γ-strongly continuous and locally γ-equicontinuous if the semigroup is locally bounded (w.r.t. the operator norm) and the co-semiflow (m, ϕ) is jointly continuous.Then we derive sufficient conditions depending on (m, ϕ) for the local boundedness of the semigroup (C m,ϕ (t)) t≥0 for the Saks spaces mentioned above, in particular in the case F (Ω) = H ∞ in Theorem 4.17.
In Section 5 we turn to the generator (A, D(A)) of locally bounded, γ-strongly continuous weighted composition semigroups and show in Proposition 5.4 that it coincides with the Lie generator, i.e. the pointwise generator, if the Saks space (F (Ω), ⋅ , τ co ) is sequentially complete (w.r.t. the mixed topology γ) and the cosemiflow (m, ϕ) is jointly continuous.This generalises [41,Proposition 2.12,p. 6] and [30,Proposition 2.4,p. 118] where F (Ω) = C b (Ω) is the space of bounded continuous functions on a completely regular Hausdorff k-space resp.Polish space Ω and m trivial.The connection to τ -bi-continuous semigroups from Section 2 also allows us to deduce in Proposition 5.3 that on sequentially complete Saks spaces (F (Ω), ⋅ , τ co ) such semigroups are ⋅ -strongly continuous if (F (Ω), ⋅ ) is reflexive, and to show that the space of ⋅ -strong continuity coincides with the ⋅closure of D(A).In Theorem 5.11 we turn to the special case that Ω ⊂ C is open, the jointly continuous co-semiflow (m, ϕ) continuously differentiable w.r.t.t and F (Ω) for instance a space of holomorphic functions, which results in the representation of the generator, where G ∶= .ϕ 0 and g ∶= .
m 0 denote the derivatives w.r.t.t of ϕ and m in t = 0, respectively.In Theorem 5.10 we also handle the more complicated case where Ω ⊂ R is open and the space F (Ω) need not be a space of continuously differentiable or holomorphic functions, for example F (Ω) = C b (Ω).
Section 6 is dedicated to the converse of (1) in the sense that a γ-strongly continuous semigroup with such a generator for some holomorphic functions G and g on Ω must be a weighted composition semigroup w.r.t.some jointly continuous holomorphic co-semiflow (m, ϕ) at least if Ω = D, (F (D), ⋅ , τ co ) is a sequentially complete Saks space of holomorphic functions and the embedding H(D) ↪ (F (D), ⋅ ) continuous where H(D) denotes the space of holomorphic germs near D with its inductive limit topology (see Theorem 6.1).In Example 6.2 we show that this embedding condition is fulfilled for the spaces like H p for 1 ≤ p ≤ ∞, A p α for α > −1 and 1 ≤ p < ∞, D and B α for α > 0. Theorem 6.1 in combination with Theorem 5.11 (see also Theorem 7.6) is the counterpart of [43,Main theorem,p. 490] and [44, Theorems 2.1, 3.1, p. [68][69] we were searching for.
In the closing Section 7 we generalise in Proposition 7.1, Proposition 7.2 and Proposition 7.3 results from [16,29,30,56] on multiplication semigroups on C b (Ω) for locally compact Hausdorff Ω and unweighted composition semigroups on C b (R) to the more general setting of weighted composition semigroups on weighted spaces of continuous functions.Further, Theorem 7.4 gives us necessary and sufficient conditions for a weighted composition semigroup (C m,ϕ (t)) t≥0 on a sequentially complete Saks space (F (Ω), ⋅ , τ co ) of holomorphic functions on an open connected set Ω ⊂ C to be ⋅ -strongly continuous resp.γ-strongly continuous.In combination with the results from Section 4, see Theorem 4.15, Theorem 4.16 and Theorem 4.18, we obtain sufficient conditions on (m, ϕ) so that (C m,ϕ (t)) t≥0 is ⋅ -strongly continuous on the Hardy spaces H p for 1 < p < ∞ and the weighted Bergman spaces A p α for α > −1 and 1 < p < ∞, where we get back the ones from [70,84] by a different proof which improve the already known ones from [53,75,78], and on the Dirichlet space D as well as γ-strongly continuous on the Hardy space H ∞ and the Bloch type spaces B α for α > 0.

Background on semigroups on Saks spaces
In this section we recall some basic notions and results in the context of semigroups on Hausdorff locally convex spaces, bi-continuous semigroups, the mixed topology and Saks spaces to keep this work practically self-contained.We refer the interested reader for more detailed information to [25,51,52,60,86].For a Hausdorff locally convex space (X, τ ) over the field K = R or C we use the symbol L(X, τ ) for the space of continuous linear operators from (X, τ ) to (X, τ ).If (X, ⋅ ) is a normed space, we just write L(X) ∶= L(X, τ ⋅ ) where τ ⋅ is the ⋅ -topology.First, we recall the notions of strong continuity and equicontinuity.
2.1.Definition.Let (X, τ ) be a Hausdorff locally convex space, I a set and (T (t)) t∈I a family of linear maps X → X.
(a) Let I be a Hausdorff space.(T (t)) t∈I is called τ -strongly continuous if T (t) ∈ L(X, τ ) for every t ∈ I and the map T x ∶ I → (X, τ ), T x (t) ∶= T (t)x, is continuous for every x ∈ X.(b) Let σ be an additional Hausdorff locally convex topology on X.
In the context of semigroups of linear maps there are different degrees of equicontinuity and boundedness.In the following definition we use the symbol id for the identity map on a set X, i.e. the map id∶ X → X, id(x) ∶= x.

2.2.
Definition.Let X be a linear space and (T (t)) t≥0 a family of linear maps X → X.
where T (t) L(X) ∶= sup x∈X, x ≤1 T (t)x .(T (t)) t≥0 is called exponentially bounded if there exist M ≥ 1 and ω ∈ R such that T (t) L(X) ≤ M e ωt for all t ≥ 0.
Since local boundedness of semigroups of linear maps on normed spaces will be an essential condition in our work, we give the following characterisation, which carries over from the case of norm-strongly continuous semigroups on Banach spaces.2.3.Proposition.Let (X, ⋅ ) be a normed space and (T (t)) t≥0 a semigroup of linear maps X → X.Then the following assertions are equivalent.
If (X, ⋅ ) is additionally complete, then each of the preceding assertions is equivalent to: (d) There exists t 0 > 0 such that sup t∈[0,t0] T (t)x < ∞ for all x ∈ X and . Then we get for all t 0 ≤ t ≤ 2t 0 that and thus sup t∈[0,2t0] T (t) L(X) ≤ M + M 2 .By repetition of this procedure we get in finitely many steps that sup t∈[0,1] T (t) L(X) < ∞.
The equivalence of the first three statements to (d) is a consequence of the uniform boundedness principle if (X, ⋅ ) is additionally complete.
Let us recall the definition of the mixed topology, [83, Section 2.1], and the notion of a Saks space, [25, I.3.2Definition, p. 27-28], which will be important for the rest of the paper.
2.4.Definition ([55, 2.1 Definition, p. [3][4]).Let (X, ⋅ ) be a normed 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 continuous seminorms Γ τ that generates the topology τ such that The mixed topology is actually Hausdorff locally convex and the definition given above is equivalent to the one introduced by Wiweger [ We recall from [55, p. 4] that it is often useful to have a characterisation of the mixed topology by generating systems of continuous seminorms, e.g. the definition of dissipativity in Lumer-Phillips generation theorems for bi-continuous semigroups depends on the choice of the generating system of seminorms of the mixed topology (see [58]).For that purpose we recap the following auxiliary topology whose origin is [83, Theorem 3.1.1,p. 62].

By
(a) We have τ ⊂ γ s ⊂ γ and γ s has the same convergent sequences as γ.

(b) If
(i) for every x ∈ X, ε > 0 and p ∈ Γ τ there are y, z ∈ X such that x = y + z, p(z) = 0 and y Further, we will use the following notions.(a) We call (X, ⋅ , τ sequentially open subset of (X, γ) is already open (see [79, p. 273]).
A Saks space is complete if and only if the ⋅ -closed unit ball B ⋅ is τ -complete by [25,I.1.14 Proposition,p. 11].We note that condition (ii) in Remark 2.6 (b) is equivalent to (X, ⋅ , τ ) being a semi-Montel space by [25,I.1.13 Proposition,p. 11].If X is a space of K-valued functions on some set Ω, then the semi-Montel property may be used to derive linearisations of weak vector-valued versions of where E is a Hausdorff locally convex space and E ′ its topological linear dual space (see [55,3.3 Theorem,p. 7]).Since the space (X, γ) is usually not barrelled by [25, I.1.15Proposition, p. 12], one cannot apply automatic local equicontinuity results like [52, Proposition 1.1, p. 259] to γ-strongly continuous semigroups.A way to circumvent this problem is the condition that the space is C-sequential.A sufficient condition that guarantees that (X, γ), and thus (X, ⋅ , τ ), is C-sequential is that τ is metrisable on B ⋅ by [59, Proposition 5.7, p. 2681-2682].
2.8.Remark.If (X, ⋅ , τ ) is a sequentially complete Saks space, then the normed space (X, ⋅ ) is complete because γ is a coarser topology than the norm ⋅ -topology and completeness of a normed space is equivalent to sequential completeness.
We close this section with some examples of Saks spaces and a convention we will use throughout the paper.We denote by C(Ω) the space of K-valued continuous functions on a Hausdorff space Ω and by τ co the compact-open topology on C(Ω), i.e. the topology of uniform convergence on compact subsets of Ω.
Our next example is the Bloch space w.r.t. a weight v. Then (Bv(D), ⋅ Bv(D) , τ co ) is a complete semi-Montel C-sequential Saks space such that γ = γ s .
The case p = ∞ in Example 2.12 (a) can also be covered, namely, we have the following result for weighted H ∞ -spaces.

2.15.
Example.We define the space H ∞ of bounded Dirichlet series as the topological subspace of H ∞ (C + ) consisting of bounded holomorphic functions on C + that can be written as a Dirichlet series on some half-plane contained in C + .Further, we denote by ⋅ H ∞ the restriction of the norm Proof.Due to [7,Lemma 18,p. 227]  For our last example of this section we recall that a completely regular Hausdorff space Ω is called a k R -space if any map f ∶ Ω → R whose restriction to each compact K ⊂ Ω is continuous, is already continuous on Ω (see [65, p. 487] If Ω ⊂ C is open, then Hv(Ω) from Example 2.14 is a subspace of Cv(Ω) and the mixed topologies are compatible, i.e.

Semiflows, semicocycles and semicoboundaries
In this section we recall the notions and properties of semiflows, associated semicoclyes and of semicoboundaries, which form a special class of semicocycles.
where differentiability in t = 0 means right-differentiability in t = 0. Further, we set .
Let us come to semiflows.
The generator is also called the speed of the semiflow (see [6, p. 210] where the symbol λ is used for G).For a separately continuous semiflow ϕ the existence of the generator is equivalent to right-differentiability in t = 0 and continuity of .ϕ t ∈ C(Ω) for all t ≥ 0 if and only if ϕ (⋅) (x) is right-differentiable in t = 0 for all x ∈ Ω and .
ϕ 0 is the generator of ϕ.
Proof.We only need to prove the implication ⇐.Let ϕ (⋅) (x) be right-differentiable in t = 0 for all x ∈ Ω and .
ϕ 0 (ϕ t (x)) for all t ≥ 0. Indeed, we have Further sufficient and necessary conditions for a given continuous function G∶ R → R to be the generator of a jointly continuous flow on R are contained in [6, Lemma 2.2, p. 212].The notion of a generator in the case of a jointly continuous holomorphic semiflow was introduced in [10, p. 103].In this case the generator always exists and is not only continuous but even holomorphic.

Proposition. Let ϕ be a semiflow on an open set
If in addition ϕ has a generator G, then .
for all s, t ≥ 0 and x ∈ Ω, which yields .
ϕ 0 (x) for s = 0.The rest of the statement follows from the definition of a generator and Remark 3.5.
Next, let us recall the notion of a semicocycle for a semiflow.
3.9.Definition.Let ϕ ∶= (ϕ t ) t≥0 be a semiflow on a Hausdorff space We call a semicocycle m trivial and write m = If ϕ is a holomorphic semiflow on an open set Ω ⊂ C, then a simple example of a holomorphic semicocycle m for ϕ is given by the complex derivatives m t ∶= ϕ ′ t for t ≥ 0 by the chain rule.There is an analogon of Proposition 2.3 for semicocycles due to König [53] which will be important later on.The similarity to Proposition 2.3 is not a coincidence because they use the same ideas, which can be found in the proofs of [ Proof.The implications (d)⇒(a)⇒(e)⇒(c)⇒(d) follow from the proof of [53, Lemma 2.1 (a), p. 472] (we note that it is not relevant for the proof that (h t ) t≥0 ∶= m in the cited lemma is assumed to be holomorphic and Ω to be equal to D).Moreover, the implications (a)⇒(b)⇒(c) clearly hold.
Proof.We only need to prove the implication ⇐.Let m (⋅) (x) be right-differentiable in t = 0 for all x ∈ Ω and .
m 0 (ϕ t (x)) for all t ≥ 0. Thus for x ∈ Ω we know that the map t ↦ m t (x) solves the initial value problem . .m 0 (ϕ s (x))ds) for all t ≥ 0.
We have the following construction of a semicocycle given a jointly continuous semiflow and a continuous function on a locally compact metric space.
3.14.Proposition.Let ϕ be a jointly continuous semiflow on a locally compact metric space Ω and g ∈ C(Ω).Then the following assertions hold.
(a) The family m ∶= (m t ) t≥0 given by m t (x) ∶= exp( ∫ t 0 g(ϕ s (x))ds) for all t ≥ 0 and x ∈ Ω is a jointly continuous semicocycle for ϕ.In particular, Proof.(a) For t ≥ 0 we note that g ○ϕ t ∈ C(Ω), the map g(ϕ (⋅) (x)) is continuous and therefore integrable on [0, t] and for every x 0 ∈ Ω there is a compact neighbourhood )ds, we deduce that F t is continuous on the metric space Ω by [36,5.6 Satz,p. 147] and thus m t = exp ○F t as well.From here it is easy to check that m is a C 0 -semicocycle for ϕ and so jointly continuous by Proposition 3.12.The rest of statement (a) follows from the integral form of m t (x) and Proposition 3. On connected proper subsets of C every jointly continuous holomorphic semicocycle of a jointly continuous holomorphic semiflow is actually of the integral form in Proposition 3.14 (a).There is another way to construct semicocycles for a semiflow apart from the one in Proposition 3.14, namely, so-called (semi)coboundaries, see e.g.[49, p. 240], [53, p. 469-470] and [68, p. 513].For that construction we need the notion of a fixed point of a semiflow.

Definition.
Let Ω be a Hausdorff space and ϕ a semiflow on Ω.We call x ∈ Ω a fixed point of ϕ if it is a common fixed point of all ϕ t , i.e. ϕ t (x) = x for all t ≥ 0. We denote the set of all fixed points of ϕ by Fix(ϕ Let Ω be a Hausdorff space and ϕ a semiflow on and note that If N ω ≠ ∅, suppose additionally that m ω t is continuously extendable on Ω for all t ≥ 0 and denote the (unique) extension by m ω t as well.Then (3) also holds for x ∈ N ω by continuity and the density of Ω ∖ N ω in Ω.Thus m ω ∶= (m ω t ) t≥0 is a semicocycle for ϕ under this assumption.
and m ω t ∈ H(Ω) for all t ≥ 0 where ord ω (z) ∈ N is the order of the zero z ∈ N ω of ω.If ϕ is additionally jointly continuous, then m ω is jointly continuous.
Therefore m ω t is continuously extendable on Ω and this extension is holomorphic on Ω by Riemann's removable singularity theorem.Now, if ϕ is additionally jointly continuous, then we have , is continuous for every z ∈ Ω by [49, Lemma 2.1, p. 242] with connected G ∶= Ω. Hence we have lim t→0+ m ω t (z) = (ϕ ′ 0 (z)) ord ω (z) = 1 ord ω (z) = 1 for all z ∈ N ω .We conclude that m ω t is C 0 and thus jointly continuous by Proposition 3.12.For Ω = D the previous example is already contained in [75, p. 361-362].We already observed that (ϕ ′ t ) t≥0 is a simple example of a semicocycle of a holomorphic semiflow ϕ on an open set Ω ⊂ C. If Ω is also connected, then it is even a semicoboundary by Theorem 3.7, Proposition 3.8 and Proposition 3.19 (a), which is jointly continuous by the arguments in the example above.
, is continuously extendable in any z 0 ∈ N ω .By Riemann's removable singularity theorem this extension is holomorphic on Ω and we denote it by g ω,G .

Proposition.
Let Ω ⊊ C be open and simply connected, and ϕ a jointly continuous holomorphic semiflow on Ω with generator G. Then the following assertions hold.
Third, suppose there is ω ∈ H(Ω) such that (4) holds.Then we obtain with ω ∶= ω ○ h that for all t ≥ 0 and z which extends to ( 6) for all z ∈ D by continuity.Due to the first and the second part of (b) this means that Then the first and the second part of (b) imply that there is w ∈ H(D) such that (6) holds.Setting ω ∶= w ○ h −1 , we see that for all t ≥ 0 and z which extends to (4) for all z ∈ Ω by continuity.
If Ω ⊊ C is open and simply connected, and ϕ a jointly continuous holomorphic semiflow on Ω with generator G ≠ 0, then it follows from Proposition 3.22 (a) and Example 3.21 that Corollary 3.23 is already known due [50, Theorem 5, p. 3393] (here Ω = C is also allowed).However, the proof is different.

Semigroups of weighted composition operators
Before introducing weighted composition semigroups induced by a co-semiflow (m, ϕ), we start this section with weighted composition families induced by a tuple (m, ϕ) which need not be a co-semiflow.First, we generalise a part of [47, Proposition 1, p. 307].

Proposition.
Let Ω be a Hausdorff space and (F (Ω), ⋅ , τ co ) a Saks space such that F (Ω) ⊂ C(Ω).Let I be a set, ϕ ∶= (ϕ t ) t∈I and m ∶= (m t ) t∈I be families of functions Proof.First, for compact K ⊂ Ω we note that for all t ∈ I and f ∈ F (Ω), implying that the family (C m,ϕ (t)) t∈I of linear maps F (Ω) → F (Ω) by condition (i) is τ co -equicontinuous on the whole space F (Ω) by condition (ii) and the continuity of the functions in F (Ω). Now, suppose that sup t∈I C m,ϕ (t) L(F (Ω)) < ∞.Since τ co is a coarser topology than γ, we obtain from the first part that the family (C m,ϕ (t)) t∈I is γ-τ coequicontinuous.It follows from [56, 3.16 Proposition, (f)⇔(g), p. 12-13] that (C m,ϕ (t)) t∈I is even γ-equicontinuous.

Remark.
Let Ω be a Hausdorff space, I a compact Hausdorff space, ϕ ∶= (ϕ t ) t∈I and m ∶= (m t ) t∈I families of functions ϕ t ∶ Ω → Ω and m t ∶ Ω → K.If ϕ and m are both jointly continuous, then condition (ii) of Proposition 4.1 is fulfilled since ϕ I (K) and m I (K) are compact for all compact K ⊂ Ω.
Our next goal is to derive necessary and sufficient conditions for the weighted composition family (C m,ϕ (t)) t∈I to be γ-strongly continuous.For the necessary condition we need the following definition.

Definition.
Let Ω and I be Hausdorff spaces, F (Ω) ⊂ C(Ω) a linear space, and ϕ ∶= (ϕ t ) t∈I a family of functions ϕ t ∶ Ω → Ω.We say that the topology of Ω is initial-like w.r.t.(ϕ, F (Ω)) if for every compact set K ⊂ Ω the continuity of the map I × K → K, (t, x) ↦ f (ϕ t (x)), for all f ∈ F (Ω) implies the continuity of the map I × K → Ω, (t, x) ↦ ϕ t (x).

Proposition.
Let Ω be a Hausdorff space and (F (Ω), ⋅ , τ co ) a Saks space such that F (Ω) ⊂ C(Ω).Let I be a metric space, ϕ ∶= (ϕ t ) t∈I and m ∶= (m t ) t∈I be families of continuous functions Then the following assertions hold.
(a) If ϕ and m are jointly continuous, then I is locally compact, m t (x) ≠ 0 for all (t, x) ∈ I × Ω and (C m,ϕ (t)) t∈I is γ-strongly continuous, then ϕ and m are jointly continuous.
Proof.First, we observe that C m,ϕ (t) is linear and γ-continuous, thus C m,ϕ (t) ∈ L(F (Ω), γ), for every t ∈ I due to Proposition 4.1 and Remark 4.2 applied to the singleton I t ∶= {t} and the continuity of ϕ t and m t .Since I is a metric space and F (Ω) ⊂ C(Ω), the family (C m,ϕ (t)) t∈I is γ-strongly continuous if and only if the map is continuous for every compact K ⊂ Ω and f ∈ F (Ω) by [25, I.1.10Proposition, p. 9] and the assumption sup t∈I C m,ϕ (t) L(F (Ω)) < ∞.It follows from [34,Lemma 4.16,p. 56] that this is equivalent to the continuity of the map for every compact K ⊂ Ω and f ∈ F (Ω).
(a) If ϕ and m are jointly continuous, then the map ( 8) is clearly continuous for every compact K ⊂ Ω and f ∈ F (Ω).
(b) Since 1 ∈ F (Ω), the continuity of the map (8) implies the continuity of the map for every compact K ⊂ Ω.The continuity of the maps (8) and (9), that m t (x) ≠ 0 for all (t, x) ∈ I × Ω and that the topology of Ω is initial-like w.r.t.(ϕ, F (Ω)) yield the continuity of the map for every compact K ⊂ Ω.Conversely, the continuity of the maps ( 9) and ( 10) clearly implies the continuity of the map (8) for every compact K ⊂ Ω and f ∈ F (Ω). Hence the continuity of the map (8) for every compact K ⊂ Ω and f ∈ F (Ω) is equivalent to the continuity of the maps ( 9) and ( 10) for every compact K ⊂ Ω.Now, if I is locally compact and Ω a k R -space, then I × Ω is also a k R -space by a comment after the proof of [15, Théorème (2.1), p. [54][55].Thus the γ-strong continuity of (C m,ϕ (t)) t∈I implies the continuity of the maps ( 9) and ( 10) for every compact K ⊂ Ω, which then implies the joint continuity of ϕ and m because I × Ω is a k R -space.
4.6.Remark.Looking at the proof, we see that we can drop the condition that 1 ∈ F (Ω) in Proposition 4.5 (b) if m t = 1 for all t ∈ I.
From now on we restrict to the case that (m, ϕ) is a co-semiflow on a Hausdorff space Ω.

Remark.
Let Ω be a Hausdorff space, F (Ω) ⊂ C(Ω) a linear space and (m, ϕ) a co-semiflow for F (Ω). (a If the semicocycle is actually a semicoboundary, then the weighted composition semigroup may have a quite simple structure.4.9.Remark.Let Ω be a Hausdorff space, F (Ω) ⊂ C(Ω) a linear space and (m, ϕ) a co-semiflow for F (Ω) such that there is ω ∈ C(Ω) with N ω = ∅ and m = m ω .Then a direct calculation shows that . This means that C m,ϕ (t) and C 1,ϕ (t) are similar as linear operators on F (Ω) for all t ≥ 0 if (1, ϕ) is also a co-semiflow for F (Ω) (cf.[44, p. 67] in the case Ω = D and F (D) being a space of holomorphic functions).
Sufficient and necessary conditions for the existence of ω in Remark 4.9 are given in Corollary 3.23 in the case that (m, ϕ) is a jointly continuous holomorphic co-semiflow on a proper open simply connected subset Ω of C.
The question which tuples (m, ϕ) are co-semiflows for a given space F (Ω) is difficult on its own.We recall the following results deduced from characterisations of weighted composition operators.(a) (C m,ϕ (t)) t≥0 is γ-strongly continuous, locally τ co -equicontinuous and locally γ-equicontinuous.
The multiplier space M(D) of the Dirichlet space is more complicated and its elements can be described in terms of the Carleson measure by [80, Theorems 1.1 (c), 2.3, 2.7, p. 115, 122, 125] with α = 1 2 .From Proposition 4.13 and 1 ∈ D it follows that M(D) ⊂ (D ∩ H ∞ ).

4.15.
Theorem.Let (m, ϕ) be a holomorphic co-semiflow on D such that ϕ is jointly continuous.Then (m, ϕ) is a co-semiflow for F (D) and the weighted composition semigroup (C m,ϕ (t)) t≥0 on F (D) is locally bounded in each of the following cases if Proof.The condition lim sup t→0+ m t ∞ < ∞ in (a) and (b) yields that m t ∈ H ∞ for all t ≥ 0 by Proposition 3.10, which is also true in (c) because M(D) ⊂ H ∞ .Hence in all the cases (1, ϕ), (m, id) and (m, ϕ) are co-semiflows for F (D) by Remark 4.12 and Proposition 4.14.
An example in case (c) of the Dirichlet space D is the jointly continuous holomorphic co-semiflow (ϕ ′ , ϕ) on D given by ϕ t ∶ D → D, ϕ t (z) ∶= e −ct z, for all t ≥ 0 for some c ∈ C with Re(c) ≥ 0 since ϕ ′ t (z) = e −ct for all t ≥ 0 and z ∈ D, which implies m t ∶= ϕ ′ t ∈ D for all t ≥ 0. The same is true if we choose ϕ t (z) ∶= e −t z + 1 − e −t for all t ≥ 0 and z ∈ D. Thus (C ϕ ′ ,ϕ (t)) t≥0 is a locally bounded semigroup on D by Theorem 4.15 (c) in both cases.4.16.Theorem.Let (m, ϕ) be a holomorphic co-semiflow on D and α > 0. Then (m, ϕ) is a co-semiflow for B α and the weighted composition semigroup (C m,ϕ (t)) t≥0 on B α is locally bounded if for all t ≥ 0, there exists t 0 > 0 such that sup t∈[0,t0] K α (ϕ t ) < ∞ and Proof.By the proof of [85, Theorem 2.2 (i), p. 115] we have Moreover, the conditions in (a), (b) and (c) guarantee that m t ∈ M(B α ) for all t ≥ 0 by Proposition 3.10 and Proposition 4.14 (b).Hence in all the cases (m, id) and so (m, ϕ) are co-semiflows for B α .
Let us turn to the space of bounded Dirichlet series.We say that a holomorphic function ϕ∶ C + → C + belongs to the class G ∞ if there exist c ϕ ∈ N 0 and a Dirichlet series ρ ϕ which converges on some open half-plane and extends holomorphically to C + such that ϕ(z) = c ϕ z + ρ ϕ (z) for all z ∈ C + (see [24, Definition 2.6, p. 9]).4.17.Theorem.Let (m, ϕ) be a holomorphic co-semiflow on C + .Then (m, ϕ) is a co-semiflow for H ∞ and the weighted composition semigroup (C m,ϕ (t)) t≥0 on H ∞ is locally bounded if ϕ t ∈ G ∞ and m t ∈ H ∞ for all t ≥ 0. The converse is true if inf z∈C+ m t (z) > 0 for all t ≥ 0.
Proof.By [8, Proposition 2, p. 219] (1, ϕ) is a co-semiflow for H ∞ if and only ϕ t ∈ G ∞ for all t ≥ 0. Further, we observe that Now, the first implication follows from our considerations above and Proposition 3.10 using that H ∞ ⊂ H ∞ (C + ).Let us consider the converse implication.The property m t ∈ H ∞ for all t ≥ 0 follows from the definition of the weighted composition semigroup and the observation that 1 ∈ H ∞ since m t = C m,ϕ (t)1 for all t ≥ 0. The property ϕ t ∈ G ∞ for all t ≥ 0 follows from our first observation of the proof, writing C 1,ϕ (t) = (1 m t )C m,ϕ (t) and noting that 1 m t belongs to the Banach algebra H ∞ by [71, Theorem 6.2.1, p. 147] if and only if inf z∈C+ m t (z) > 0.

Generators of weighted composition semigroups
In this section we give several characterisations of the generator of a γ-strongly continuous weighted composition semigroup on a Saks space.Let (X, ⋅ , τ ) be a Saks space and (T (t)) t≥0 a γ-strongly continuous semigroup on X.We define the generator (A, D(A)) of (T (t)) t≥0 according to [52, p. 260] by )) of (T (t)) t≥0 is given by In the context of τ -bi-continuous semigroups their generators are actually defined as bi-generators (see [39, Definition 1.2.6, p. 7]).The notion of the bi-generator was originally introduced in [60,61] (and corrected in [39]).
5.1.Proposition.Let (X, ⋅ , τ ) be a Saks space and (T (t)) t≥0 a γ-strongly continuous, locally bounded semigroup on X.Then we have Proof.The inclusion D(A ⋅ ,τ ) ⊂ D(A) follows from [25, I.1.10Proposition, p. 9], which says that a sequence in X is γ-convergent if and only if it is τ -convergent and ⋅ -bounded.Further, Af = A ⋅ ,τ f for f ∈ D(A) because τ is coarser than γ.Conversely, suppose that there is x ∈ D(A) such that sup t∈(0,1] T (t)x−x t = ∞.Due to the local boundedness of (T (t)) t≥0 this implies that there is a sequence

Corollary.
Let Ω be a Hausdorff space, (F (Ω), ⋅ , τ co ) a Saks space such that F (Ω) ⊂ C(Ω) and (A, D(A)) the generator of a locally bounded weighted composition semigroup (C m,ϕ (t)) t≥0 on F (Ω) w.r.t. a jointly continuous co-semiflow (m, ϕ).Then we have For our next observation we recall the definition of the generator of a normstrongly continuous semigroup on a Banach space.Let (X, ⋅ ) be a Banach space and (T (t)) t≥0 a ⋅ -strongly continuous semigroup on X.We define the normgenerator (A ⋅ , D(A ⋅ )) of (T (t)) t≥0 according to [37, Chap.2, 1.2 Definition, p. 49] by

Proposition.
Let Ω be a Hausdorff space, (F (Ω), ⋅ , τ co ) a sequentially complete Saks space such that F (Ω) ⊂ C(Ω) and (A, D(A)) the generator of a locally bounded weighted composition semigroup (C m,ϕ (t)) t≥0 on F (Ω) w.r.t. a jointly continuous co-semiflow (m, ϕ).Then we have Proof.Since γ is stronger than τ co and thus stronger than the topology of pointwise convergence, we only need to prove the inclusion ) and set g ∶= A m,ϕ f .Since g ∈ F (Ω) and (C m,ϕ (t)) t≥0 is γ-strongly continuous by Theorem 4.11 (a), the γ-Riemann integral for every s ≥ 0 and x ∈ Ω.The right-derivative g x is continuous for every x ∈ Ω as g ∈ F (Ω) and (m, ϕ) is jointly continuous.Hence for every t ≥ 0 and x ∈ Ω by the fundamental theorem of calculus.In combination with the existence of the γ-Riemann integral for every t ≥ 0, implying our statement by [52, Proposition 1.2 (2), p. 260].

Remark.
Let Ω be a Hausdorff space, (F (Ω), ⋅ , τ co ) a sequentially complete Saks space such that F (Ω) ⊂ C(Ω) and (A, D(A)) the generator of a locally bounded weighted composition semigroup (C m,ϕ (t)) t≥0 on F (Ω) w.r.t. a jointly continuous co-semiflow (m, ϕ).We may also define the τ co -generator (A τco , D(A τco )) by and Then it follows from τ co being coarser than γ and Proposition 5.4 that If we have more information on the co-semiflow than just joint continuity, then we may give a simpler characterisation of the generator of a weighted composition semigroup.5.6.Proposition.Let Ω be a Hausdorff space, (F (Ω), ⋅ , τ co ) a sequentially complete Saks space such that F (Ω) ⊂ C(Ω) and (A, D(A)) the generator of a locally bounded weighted composition semigroup (C m,id (t)) t≥0 on F (Ω) w.r.t. a jointly continuous co-semiflow (m, id).If m (⋅) (x) is right-differentiable in t = 0 for all x ∈ Ω, then Proof.For f ∈ F (Ω) we have for all x ∈ Ω, yielding our statement by Proposition 5.4.
(ii) ϕ has a generator G, i.e. there is a function G ∈ C(Ω) such that .
We note that the map , is continuous by condition (i), the continuity of f ′ and the joint continuity of (m, ϕ).Then we have for every 0 < t ≤ t by (i) and the fundamental theorem of calculus.This implies for every 0 < t ≤ t.Using that h x is continuous on the compact interval [0, t ], thus uniformly continuous by the Heine-Cantor theorem, and that for every ε > 0 there is 0 < δ ≤ t such that for all s ≥ 0 with s = s − 0 < δ we have The rest of the statement follows from (17) and Proposition 5.4.(b) Let f ∈ C 1 K (ω) ∩ F (Ω) and x ∈ Ω.First, we consider the case that x ∈ ω.
Since ω is open, ϕ 0 (x) = x ∈ ω, and ϕ (⋅) (x) is continuous, there is δ x > 0 such that ϕ t (x) ∈ ω for all t ∈ [0, δ x ].It follows that the map h x ∶ [0, t ] → K from part (a) is still a well-defined continuous function for the choice t ∶= δ x and the rest of the proof carries over.
Let us turn to the case x ∈ Ω ∖ ω.Now, we need the restriction that .
m 0 f ∈ F (Ω).We set p(x) ∶= inf{t > 0 ϕ t (x) = x}.If p(x) = 0, then x is a fixed point of ϕ, and thus Suppose that p(x) > 0. Setting t(x) ∶= min{p(x), t x }, we observe that t(x) > 0 by condition (iii).Hence the map h x ∶ (0, t ] → K from part (a) is still a well-defined continuous function for the choice t ∶= t(x).Next, we show that h x is continuously extendable in s = 0. We denote by g ∈ F (Ω) the extension of .ϕ 0 f ′ + .
m 0 f and note that g as an element of F (Ω) is continuous on Ω.Then g ∶= g − .m 0 f is a continuous extension of .
since g is continuous in x and (m, ϕ) is a C 0 -co-semiflow.Hence h x is continuously extendable in s = 0 by setting h x (0) ∶= g(x) + .
. From here the rest of the proof of part (a) carries over with .ϕ 0 (x)f ′ (x) + .m 0 (x)f (x) replaced by g(x).
The expression p(x) = inf{t > 0 ϕ t (x) = x} in the proof of part (b) is also called the period of x ∈ Ω w.r.t.ϕ (see [72, p. 660]).For the proof of the converse inclusion in Proposition 5.7 in the case that F (Ω) is not a subspace of C 1 K (Ω) we need to know what happens with ϕ t and m t to the left of t = 0, meaning we consider flows and cocycles instead of just semiflows and semicocycles.

Definition.
Let Ω be a Hausdorff space.A family ϕ ∶= (ϕ t ) t∈R of continuous functions ϕ t ∶ Ω → Ω is called a flow if (i) ϕ 0 (x) = x for all x ∈ Ω, and (ii) ϕ t+s (x) = (ϕ t ○ ϕ s )(x) for all t, s ∈ R and x ∈ Ω.We call a flow trivial and write ϕ = id if ϕ t = id for all t ∈ R. We call a flow ϕ a C 0 -flow if lim t→0 ϕ t (x) = x for all x ∈ Ω.A family m ∶= (m t ) t∈R of continuous functions m t ∶ Ω → K is called a multiplicative cocycle for a flow ϕ if (i) m 0 (x) = 1 for all x ∈ Ω, and (ii) m t+s (x) = m t (x)m s (ϕ t (x)) for all t, s ∈ R and x ∈ Ω.We call a cocycle m trivial and write m = 1 if m t = 1 for all t ∈ R. We call a cocycle m a C 0 -cocycle if lim t→0 m t (x) = 1 for all x ∈ Ω.We call the tuple (m, ϕ) a co-flow on Ω.We call a co-flow (m, ϕ) jointly continuous (separately continuous, C 0 ) if ϕ and m are both jointly continuous (separately continuous, C 0 ).
We have the following characterisation of joint continuity of flows and cocycles on certain Hausdorff spaces Ω. Proof.(a) We only need to prove the implication ⇐.We define ψ ∶= (ψ t ) t≥0 by ψ t (x) ∶= ϕ −t (x) for all t ≥ 0 and x ∈ Ω.Then it is easily checked that ψ is a semiflow.Further, we have (a) If the map V ϕ → Ω, (t, x) ↦ ϕ t (x), is surjective, where and Af = .
Proof.(a) Due to Proposition 5.7 (a) we only need to show that D(A) ⊂ D 0 .Let f ∈ D(A) and x ∈ Ω.By assumption there is (t 0 , x 0 ) ∈ V ϕ such that x = ϕ t0 (x 0 ).The arguments in the proof of Proposition 5.4 in combination with Proposition 5.9 applied to the C 0 -co-flow (m, ϕ) show that By assumption we know that ϕ (⋅) (x 0 ) ∈ C 1 (R) with .ϕ t0 (x 0 ) ≠ 0. By the inverse function theorem there is an open neighbourhood U ∶= U (t 0 ) ⊂ R of t 0 such that ϕ (⋅) (x 0 ) is invertible on U and the inverse is continuously differentiable on the open neighbourhood W ∶= ϕ U (x 0 ) ⊂ Ω of x = ϕ t0 (x 0 ).Noting that for every x ∈ Ω ∖ N G , we obtain that the map V ϕ → Ω ∖ N G , (t, x) ↦ ϕ t (x), is surjective.Hence the proof of part (a) shows that f is continuously differentiable in every x ∈ Ω ∖ N G .The first part of the proof of Proposition 5.7 (b) yields that holds for all x ∈ Ω ∖ N G .The left-hand side g of this equation belongs to Looking at the proof of Theorem 5.10 (b), we see that part (a) is a special case of (b) if there is a generator G such that N G = ∅.If (ϕ t ) t≥0 is the restriction of a jointly continuous holomorphic semiflow ψ on an open set Ω ⊂ C, i.e. ϕ t = ψ t on Ω ∶= Ω ∩ R for all t ≥ 0, then the generator G of (ϕ t ) t≥0 exists by Theorem 3.7, namely, it is the restriction of the generator of ψ to Ω.So condition (i) of Theorem 5.10 (b) is fulfilled in this case.If in addition ψ is non-trivial, Ω simply connected and Ω ≠ C, then N G ≤ Fix(ψ) ≤ 1 by Proposition 3.19 and thus t x = inf ∅ = ∞ for all x ∈ N G , yielding that condition (ii) of Theorem 5.10 (b) is also fulfilled.Now, we will see that the proof of the converse inclusion in Proposition 5.7 is much simpler if F (Ω) is a subspace of C 1 K (Ω).
If (m, ϕ) is a jointly continuous co-semiflow for F (Ω), (C m,ϕ (t)) t≥0 locally bounded and ϕ has a generator G, then m Proof.Let (A, D(A)) be the generator of the γ-strongly continuous weighted composition semigroup (C m,ϕ (t)) t≥0 on F (Ω).We fix x ∈ Ω.By (18) there is F ∈ F (Ω) such that F (x) ≠ 0. Since F is continuous on Ω, there is a compact neighbourhood U ⊂ Ω of x such that F (z) ≠ 0 for all z ∈ U .Due to Theorem 4.11 (a) (C m,ϕ (t)) t≥0 is γ-strongly continuous on the sequentially complete space (X, γ), impying that D(A) is γ-dense in F (Ω) by [52,Proposition 1.3,p. 261].Thus there is f ∈ D(A) such that f (z) ≠ 0 for all z ∈ U since γ is stronger than the topology τ co and U compact.Using that ϕ 0 (x) = x and the joint continuity of ϕ, we deduce that there are t 0 > 0 and a neighbourhood U 0 of x such that ϕ s (ζ) ∈ U for all s ∈ [0, t 0 ] and ζ ∈ U 0 .In particular, f (ϕ s (ζ)) ≠ 0 for all s ∈ [0, t 0 ] and ζ ∈ U 0 .Further, we have m 0 is continuous on Ω because x is arbitrary.If in addition K = C and G ∈ H(Ω), then we also get .
We note that we may replace the condition because we only need it for the function f ∈ D(A) in the proof of Proposition 5.12.Condition (18) is for example fulfilled if 1 ∈ F (Ω).  of more assumptions on (m, ϕ).For example the condition 1 ∈ F (Ω) in Corollary 5.13 implies that m t ∈ F (Ω) for all t ≥ 0 (see Remark 4.8).

Converse of the holomorphic generation theorem
Let Ω ⊂ C be open, (F (Ω), ⋅ , τ co ) a sequentially complete Saks space such that F (Ω) ⊂ H(Ω).Suppose that Ω is connected or that 1 ∈ F (Ω). Due to Theorem 3.7, Theorem 5.11, and Proposition 3.15 or Corollary 5.13 we know that the generator (A, D(A)) of a locally bounded weighted composition semigroup (C m,ϕ (t)) t≥0 on F (Ω) w.r.t. a jointly continuous holomorphic co-semiflow (m, ϕ) is given by where G ∈ H(Ω) is the generator of ϕ and g ∶= .
m 0 ∈ H(Ω).In this section we want to prove the converse statement, namely, if we know that the domain of a γ-strongly continuous semigroup (T (t)) t≥0 on F (Ω) is given by (19), we want to show, under suitable conditions, that this semigroup is a weighted composition semigroup whose semiflow has G as a generator and whose semicocycle m is given by m t (z) ∶= exp( ∫ t 0 g(ϕ s (z))ds) for all t ≥ 0 and z ∈ Ω.Our first result in this direction is an analogon of [43, Main theorem, p. 490] (g = 0) and [44,Theorem 3.1,p. 69] where Ω = D and (T (t)) t≥0 is a ⋅ -strongly continuous semigroup.We define the space of holomorphic germs near the closed unit disc D by the inductive limit If T ∶= (T (t)) t≥0 is a γ-strongly continuous semigroup on F (D) with generator (A, D(A)) of the form for some G, g ∈ H(D), then there is a jointly continuous holomorphic co-semiflow (m, ϕ) for F (D) such that G is the generator of ϕ, m t (z) = exp( ∫ t 0 g(ϕ s (z))ds) for all t ≥ 0, z ∈ D, and T = (C m,ϕ (t)) t≥0 .
Proof.The proof of [44, Theorem 3.1, p. 69] carries over to our setting.We only have to adjust the proof in three instances.First, we have to use that Af Thus for every f ∈ F (D) there is a net (f ι ) ι∈I , I a directed set, which is γ-convergent to f .Since T (t) is γ-continuous for every t ≥ 0, this implies that (T (t)f ι ) ι∈I is γconvergent to T (t)f for every t ≥ 0. This proves the validity of [44, Eq. (3.6), p. 71] because γ is finer than τ co .Third, we note that δ z ∈ (F (D), γ) ′ and δ z ∈ (F (D), ⋅ ) ′ for all z ∈ D, where δ z (f ) ∶= f (z) for all f ∈ F (D), because δ z ∈ (F (D), τ co ) ′ and γ and the ⋅ -topology are finer than τ co .We now get by [44,Eq. (3.10) for all z ∈ D, n ∈ N and 0 ≤ t < t 0 with t 0 > 0 from [44, p. 69].Let Γ γ denote a directed system of continuous seminorms that generates γ.Since δ z ∈ (F (D), γ) ′ , T (t) ∈ L(F (D), γ) and the ⋅ -topology is finer than γ, there are p = p z,t ∈ Γ γ , e n for all z ∈ D, n ∈ N and 0 ≤ t < t 0 .This implies that there is 0 < t 1 ≤ t 0 such that ϕ t (z) < 1 for all z ∈ D and 0 ≤ t < t 1 like in the proof of [43,Claim 2,p. 492].This is all we have to change in the proof of [44,Theorem 3.1,p. 69].Moreover, we note that the joint continuity of ϕ follows from [44, p. 65] and Proposition 3.3 (b), and the joint continuity of m from Proposition 3.14 (a).

3. 12 .
Proposition.Let (m, ϕ) be a co-semiflow on an open subset Ω of a metric space and ϕ jointly continuous.Then m is jointly continuous if and only if m is C 0 .Proof.The implication ⇒ clearly holds.The other implication follows from [35, Definition 3.1, p. 1203], the proof of [35, Theorem 3.1, p. 1204] with D ∶= Ω and A ∶= C, and the observation that the assumption that D is an open connected subset of a Banach space (see [35, p. 1200]) is not needed in the proof of [35, Theorem 3.1, p. 1204].Analogously to Proposition 3.6 we have the following result for semicocycles.3.13.Proposition.Let Ω be a Hausdorff space and (m, ϕ) a separately continuous co-semiflow on Ω.Then m

3. 15 .
Proposition.Let Ω ⊂ C be open and connected, and (m, ϕ) a jointly continuous holomorphic co-semiflow on Ω.Then it holds m

3. 21 .
Example.Let Ω ⊂ C be open and connected, and ϕ a jointly continuous holomorphic semiflow on Ω with generator G ≠ 0. Then m G t (z) = ϕ ′ t (z) for all t ≥ 0 and z ∈ Ω, and m G is jointly continuous.For Ω = D the previous example is also contained in [78, Example 7.4, p. 247-248].We have the following relation between semicoboundaries and the semicocycles from Proposition 3.14 of a jointly continuous holomorphic semiflow on simply connected proper subsets Ω ⊂ C, which generalises [53, Lemma 2.2, p. 472] where Ω = D. Let Ω ⊂ C be open and connected, ϕ a jointly continuous holomorphic semiflow on Ω with generator G (see Theorem 3.7), and ω ∈ H(Ω), ω ≠ 0, such that N ω ⊂ Fix(ϕ).The function z ↦ ω ′ (z) ω(z) has a pole of order one in z 0 ∈ N ω .Due to Proposition 3.19 (a) we have N ω ⊂ Fix(ϕ) = N G and thus the holomorphic function
(a) The γ-strong continuity follows from Proposition 4.5 (a) and the local boundedness with I ∶= [0, t 0 ] for every t 0 ≥ 0. The local τ co -equicontinuity and local γ-equicontinuity are a consequence of Proposition 4.1, Remark 4.2 and the local boundedness with I ∶= [0, t 0 ] for every t 0 ≥ 0. (b)+(c)+(d) The remaining parts follow from Theorem 2.10 and the comments before it.

5. 9 .
Proposition.Let (m, ϕ) be a co-flow on a Hausdorff space Ω.(a) Let Ω be locally compact and σ-compact.Then ϕ is jointly continuous if and only if ϕ is C 0 .(b) Let Ω be an open subset of a metric space and ϕ jointly continuous.Then m is jointly continuous if and only if m is C 0 .

2 and Definition 3 . 4 , 6 . 5 .
and also called a global semiflow.Moreover, we need to recall the evaluation condition[4, p. 168] for a Banach space (F (Ω), ⋅ ) consisting of holomorphic functions on Ω:If (z n ) n∈N is a sequence in Ω that converges to some z ∈ Ω ∪ {∞} and lim n→∞ f (z n ) exists in C for all f ∈ F (Ω), then z ∈ Ω. (E)Here the one-point compactification C ∪ {∞} of C is used, which is only needed if Ω is an unbounded set.For instance, condition (E) is fulfilled for the Hardy spaces H p , 1 ≤ p < ∞, and further examples may be found in [4, Example 3.9, p. 173].In the following theorem we denote by (X, υ) ′ the topological linear dual space of a Hausdorff locally convex space (X, υ).Theorem.Let Ω ⊂ C be open, (F (Ω), ⋅ , τ co ) a sequentially complete Saks space such that F (Ω) ⊂ H(Ω) and suppose that F (Ω) fulfils the evaluation condition (E).Let G, g ∈ H(Ω) where G generates the local semiflow (ϕ t ) 0≤t<τ (z) for each z ∈ Ω.
3.1.Definition.Let I, Ω and Y be Hausdorff spaces and ϕ ∶= (ϕ t ) t∈I a family of functions ϕ t ∶ Ω → Y .(a) We call ϕ separately continuous if ϕ t and ϕ (⋅) (x)∶ I → Y are continuous for all t ∈ I and x ∈ Ω.(b) We call ϕ jointly continuous if the map I × Ω → Y , (t, x) ↦ ϕ t (x), is continuous where I × Ω is equipped with the product topology.
Definition.Let ϕ be a semiflow on a Hausdorff space Ω such that ϕ 4.10.Remark.Let (m, ϕ) be a holomorphic co-semiflow on D.