Subordination for sequentially equicontinuous equibounded $C_0$-semigroups

We consider operators $A$ on a sequentially complete Hausdorff locally convex space $X$ such that $-A$ generates a (sequentially) equicontinuous equibounded $C_0$-semigroup. For every Bernstein function $f$ we show that $-f(A)$ generates a semigroup which is of the same `kind' as the one generated by $-A$. As a special case we obtain that fractional powers $-A^{\alpha}$, where $\alpha \in (0,1)$, are generators.


Introduction
In this paper we aim to generalise subordination of bounded C 0 -semigroups from the well-known case of Banach spaces to sequentially complete Hausdorff locally convex spaces.
Subordination (in the sense of Bochner) for bounded C 0 -semigroups on Banach spaces describes a technique to associate a new semigroup to a given one by integrating orbits against a convolution semigroup of measures. It plays an important role in operator theory, functional calculus theory and stochastic processes, see e.g. [2,34,36]. As is well-known (see also Proposition 4.6) these convolution semigroups of measures correspond via Laplace transform to the class of Bernstein functions, cf. the monograph [37]. It turns out that the generator of the subordinated semigroup can be described by means of the Bernstein function and the generator of the original semigroup [34,Theorem 4.3].
Although the framework of C 0 -semigroups on Banach spaces yields a rich theory as described above, for example even the classical heat semigroup on C b (R n ) does not fit in this context, however can be treated in our generalised setting; cf. Example 5.16 below.
Let us outline the content of the paper. In Section 2 we review the Pettisintegral which provides a suitable integral in our context of locally convex spaces, in particular those satifying the so-called metric convex compactness property, see Theorem 2.2 below. The theory of locally sequentially equicontinuous, equibounded C 0 -semigroups on sequentially complete Hausdorff locally convex spaces can be developped analogously to the classical theory of C 0 -semigroups on Banach spaces, apart from the fact that the continuity properties need to be described by sequences. We collect the facts needed in Section 3. We then turn to subordination in this context in Section 4. After introducing Bernstein functions, the right class of functions for this purpose, we develop the theory of subordination in our generalised setting. It turns out that the subordinated semigroup is again a locally sequentially equicontinuous, equibounded C 0 -semigroup (see Proposition 4.9) and that its generator can be related to the one of the original semigroup (see Theorem 4.14 and Corollary 4.17). These are our main abstract results. We will then apply these results to bounded (locally) bi-continuous semigroups (as introduced in [27]) and to transition semigroups for Markov processes in Section 5. In particular, the above-mentioned classical heat semigroup, also called Gauß-Weierstraß semigroup, on C b (R n ) fits into this context.

Integration in locally convex spaces
In this section we review the notion of the Pettis-integral.
In this case x is unique due to the Hausdorff property and we set Recall that a Hausdorff locally convex space X is said to have the metric convex compactness property if for each metrisable compact subset C ⊆ X also the closed convex hull cx C of C is compact. Note that if X is sequentially complete and Hausdorff locally convex, then X has the metric convex compactness property by [41,Remark 4.1.b].
Theorem 2.2 ([41, Theorem 0.1]). Let X be a Hausdorff locally convex space. Then the following are equivalent.
(a) X has the metric convex compactness property.
If Ω is a compact metric space, µ a finite Borel measure on Ω and f ∶ Ω → X continuous, then f is µ-Pettis-integrable.
If X is a Hausdorff locally convex space with metric convex compactness property and µ a positive and finite Borel measure on [0, ∞), then by Theorem 2.2 every In this case we can construct the Pettis-integral explicitly by using Riemann sums. To that end, fix 0 ≤ a < b and for n ∈ N let If f is additionally bounded, by boundedness of µ we can also integrate f with respect to µ over the entire interval [0, ∞). Lemma 2.3. Let X be a sequentially complete Hausdorff locally convex space, µ a finite Borel measure on [0, ∞) and f ∈ C b ([0, ∞), X). Then f is µ-Pettis-integrable.
Proof. By the above we can integrate f over every compact interval. Consider the sequence (x n ) defined by Let n, m ∈ N, m < n. For every seminorm ⋅ p of the family of seminorms generating the topology of X we have

Locally Sequentially Equicontinuous Semigroups
Let (X, τ ) be a Hausdorff locally convex space. The system of seminorms generating the topology τ will be denoted by ( ⋅ p ) p∈P . W.l.o.g. we may assume that ( ⋅ p ) is directed.
(b) locally sequentially equicontinuous if for all sequences (x n ) in X such that x n → 0, t 0 > 0 and p ∈ P we have lim T t x n p = 0, (c) locally equibounded if for all bounded sets B ⊆ X, t 0 > 0 and p ∈ P it holds that sup x∈B t∈ [0,t0] T t x p < ∞.
We drop the word 'locally' if the properties (b) and (c) hold uniformly on [0, ∞).
Let (T t ) be a locally sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ). As in the case of C 0 -semigroups on Banach spaces we define the generator

Remark 3.3.
There is no common agreement whether to use the here presented definition of a generator or its negative. Throughout the entire paper we will stick to the above made definition, i.e. −A is the generator.
Let ρ 0 (−A) ⊆ C be the set of all elements λ ∈ C such that the operator λ + A has a sequentially continuous inverse (note that we allow for complex values here).
Let us collect some basic facts for (T t ) and A. . Let X be a sequentially complete Hausdorff locally convex space, (T t ) a locally sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ) with (a) Note that compared to [15], we only assume local sequential equicontinuity for (T t ). However, this does not affect the results and proofs. (b) Unless we assume (T t ) to be (sequentially) equicontinuous on the whole of [0, ∞), we cannot show that the resolvent family λ(λ + A) −1 λ>0 corresponding to (T t ) is (sequentially) equicontinuous. However, for ε > 0 the rescaled resolvent family λ(λ + A + ε) −1 λ>0 corresponding to the (sequentially) equicontinuous semigroup (e −εt T t ) is (sequentially) equicontinuous.

Subordination for equicontinuous C 0 -Semigroups
We start with the definition of the class of functions in which we will plug in the negative of a generator of a locally sequentially equicontinuous, equibounded C 0 -semigroup in order to get a new generator. Bernstein functions appear in a vast number of fields such as probability theory, harmonic analysis, complex analysis and operator theory under different names, e.g. Laplace exponents, negative definite functions or Pick, Nevanlinna or Herglotz functions (complete Bernstein functions, cf. [37]). They allow for a very useful representation formula in terms of measures. satisfying The representation of f in (1) is called Lévy-Khintchine representation. The function f determines the two numbers a, b and the measure µ uniquely. The triplet (a, b, µ) is called Lévy triplet of f . Every Bernstein function admits a continuous extension to [0, ∞) since by applying dominated convergence to the representation formula (1) one gets f (0+) = a. Hence, Let us now turn to a concept closely related to Bernstein functions.
i.e. µ t [0, ∞) → 1. By [37,Theorem A.4] this implies weak continuity at 0. (b) Let (µ t ) be a convolution semigroup of sub-probability measures which is vaguely continuous at 0 with limit δ 0 . Then (µ t ) is vaguely continuous at every point s ≥ 0 with limit µ s . Indeed, we can define a contractive semigroup via . We claim that (T t ) is strongly continuous. Then vague continuity of (µ t ) follows since this also implies weak continuity and . By (a), one actually sees µ t ([0, δ)) → 1 and consequently µ t ([δ, ∞)) → 0 as t → 0+. Thus, which can be made arbitrarily small, uniformly in x. The estimate shows strong continuity at t = 0 and by standard arguments this holds for all t ≥ 0.
Every Bernstein function is naturally associated to a family (µ t ) of sub-probability measures which form a vaguely (and hence weakly) continuous convolution semigroup and vice versa. . Let (µ t ) be a convolution semigroup of sub-probability measures on [0, ∞) which is vaguely continuous at 0 with limit δ 0 . Then there exists a unique Bernstein function f such that for all t ≥ 0 the Laplace transform of µ t is given by L(µ t ) = e −tf . Conversely, given any Bernstein function f , there exists a unique vaguely continuous convolution semigroup (µ t ) of sub-probability measures on [0, ∞) such that the above equation holds.
By the above proposition we obtain that the sub-probability measures µ t are probability measures if and only if f (0+) = 0, since Then for t > 0 the measure µ t has a density g t w.r.t. the Lebesgue measure given by  Proof. This is just a direct application of Prohorov's theorem [3, Theorem 8.6.2] for which we need to show the existence of a weakly convergent subsequence in a given sequence (µ tn ). But this follows from the fact that the mapping t ↦ µ t is continuous with respect to the weak topology of measures and the compactness of J.
Analogously to the case of bounded C 0 -semigroups on Banach spaces (see [37,Proposition 13.1]) we can construct a new (locally) sequentially equicontinuous, equibounded C 0 -semigroup from an existing one using a vaguely continuous convolution semigroup (µ t ) of sub-probability measures. Proposition 4.9. Let X be a sequentially complete Hausdorff locally convex space, (T t ) be a (locally) sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ) and (µ t ) be a convolution semigroup of sub-probability measures which is vaguely Then (S t ) is a (locally) sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ).
We will call (S t ) the subordinated semigroup to Proof. Let t ≥ 0 and ( ⋅ p ) p∈P the family of seminorms generating the topology τ of X. By Lemma 2.3 and equiboundedness and strong continuity of (T t ) the above integral exists. The linearity of S t is clear. To show equiboundedness of (S t ), for a bounded set B ⊆ X and a seminorm ⋅ p we observe Since (T t ) is equibounded, so is (S t ). The semigroup property of (S t ) is inherited from the semigroup property of (µ t ) and of (T t ). Indeed, let s, t ≥ 0. For x ∈ X, For the strong continuity of (S t ) let x ∈ X. For p ∈ P we estimate with value 0 at 0 and µ t → δ 0 weakly. It remains to show that (S t ) is (locally) sequentially equicontinuous. Let (x n ) in X be such that x n → 0. Let t 0 > 0, p ∈ P and ε > 0. By Remark 4.5 (µ t ) is weakly continuous. Hence, by Lemma 4.8 and equiboundedness of (T t ) we can choose s 0 ≥ 0 such that By virtue of the local sequential equicontinuity of (T s ) there exists n 0 ∈ N such that sup In case (T s ) is even equicontinuous we may directly choose n 0 ∈ N such that which finishes the proof.
Definition 4.10. Let X be a sequentially complete Hausdorff locally convex space, (T t ) a (locally) sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ) with generator −A and f a Bernstein function. Then we will denote the generator of the subordinated semigroup (S t ) by −A f . Our next goal is to represent the generator −A f of a subordinated semigroup (S t ) for a given Bernstein function f as it was performed in [37, Eq. (13.10)] for Banach spaces. We need some preparation. To begin with, we need to show that the function s ↦ T s x − x can be approximated linearly in a neighbourhood of s = 0 and thus is capable to compensate the measure appearing in the Lévy triplet (a, b, µ) which is singular at s = 0.
and thus for every x ′ ∈ X ′ there is a continuous seminorm ⋅ p (remember we assumed the family of seminorms to be directed) and a constant C ≥ 0 such that where r > 0 is a point of continuity of the measure µ, i.e. µ({r}) = 0. As before one shows that both (y n ) and (z n ) are Cauchy sequences in X. Let their limits be denoted by y and z, respectively. Since we thus obtain which finishes the proof.
Lemma 4.12. Let X be a sequentially complete Hausdorff locally convex space, invariant and the operators of both families commute with A (on D(A)) and with the operators of (T t ).
Proof. The argument is the same as for the usual Banach space case. Let (µ t ) be the family of measures associated to f according to Proposition 4.6. Let T s x µ t (ds) by Lemma 2.3. For n ∈ N we can approximate ∫ [0,n] T s x µ t (ds) by (finite) Riemann sums which belong to D(A). Since A is sequentially closed, also ∫ [0,n] T s x µ t (ds) ∈ D(A) and therefore S t x ∈ D(A). Let s, t ≥ 0. We now show that S t commutes with T s . Then S t also commutes with A on D(A) since S t is sequentially continuous by Proposition 4.9. Let x ∈ X. By Lemma 2.3 and sequential continuity of T s we obtain T r x µ t (dr).
Approximating ∫ [0,n] T r x µ t (dr) by (finite) Riemann sums, using the sequential continuity of T s , and taking into account the semigroup law for (T s ), we conclude Thus, the claims for λ(λ + A f ) −1 follow from the claims for (S t ) by approximating the integral by integrals over compact subsets and then by finite Riemann sums and taking into account the sequential closedness of A and of the operators T t (which for them follows from sequential continuity).
Definition 4.13. Let X be a sequentially complete Hausdorff locally convex space, (T t ) a (locally) sequentially equicontinuous, equibounded C 0 -semigroup on (X, τ ) with generator −A, and f a Bernstein function with Lévy-Khintchine representation We define the linear operator A f on X by D(A f ) ∶= D(A) and where the integral exists by Proposition 4.11.
Let x ∈ D(A) and x ′ ∈ X ′ . Let c, C > 0 be such that µ({c, C}) = 0, i.e. c and C are points of continuity. Then nµ 1 n [c,C) → µ [c,C) weakly since vague convergence implies By dominated convergence, we obtain By Lemma 3.4, for c > 0 we have we thus obtain that the convergence is even weakly. Hence, for such c we have and therefore Furthermore, since (T t ) is equibounded, by (6) we obtain Thus, For λ > 0 let x λ ∶= λ(λ + A f ) −1 x. We now apply Lemma 4.12 multiple times. Then, on the one hand, x λ ∈ D(A f ) ∩ D(A), and moreover (approximating the integral by integrals over compacta and then by finite Riemann sums again) Since this holds true for all x ′ ∈ X ′ , we obtain By Lemma 3.4 we have λ(λ+A f ) −1 → I strongly as λ → ∞. Since A f is sequentially closed by Lemma 3.4, we thus obtain x ∈ D(A f ) and A f x = A f x.
(ii) For the general case f (0+) = a ≥ 0 consider h ∶= f − a. Then h is a Bernstein function with h(0+) = 0. Let (ν t ) be the associated family of sub-probability measures. Then (µ t ) given by µ t = e −ta ν t for t ≥ 0 is the family of measures associated to f . Thus, for t ≥ 0 and x ∈ X we have i.e. (S f t ) is a rescaling of (S h t ). Analogously to the case of C 0 -semigroups on Banach spaces one proves that then −A f = −A h − a. Thus the general case follows from (i).
Remark 4.16. In the above proof, we first showed that x ∈ D(A) belongs to the weak generator of (S t ), and then did a regularisation by resolvents to obtain the result. If (S t ) is continuous (and not just sequentially continuous), we can directly conclude that (x, Ax) belongs to the weak closure of A f which coincides with A f since A f is closed. 14] for the case of C 0 -semigroups on Banach spaces). Since A f D(A) = A f by Theorem 4.14, we obtain the assertion.
Analogously to the scalar-valued case we shall write from now on Note that similar to the situation in Banach spaces one could have developed an entire functional calculus in the sense of [19] which enables one to define f (A) with the same outcome.

Applications
We now consider two applications, namely bi-continuous semigroups and transition semigroups of stochastic processes.

5.1.
Bi-continuous semigroups. In this subsection let X be a Banach space with norm-topology τ ⋅ .
First, we study bi-continuous semigroups. In order to do this, we need some preparation.
compact-open topology, i.e. the topology of uniform convergence on compact subsets of R n , and γ the mixed topology determined by τ ⋅ ∞ and τ co . (X, ⋅ ∞ , τ co ) is a Saks space which fulfils condition (i) of Remark 5.3 (c) and γ is also generated by the weighted The topology generated by the seminorms ⋅ g , g ∈ C 0 (R n ), on C b (R n ) was introduced under the name strict topology, denoted by β, in [5,Definition,p. 97] and the example shows that the strict topology is indeed a mixed topology (cf. [8,Proposition 3] and also Remark 5.22 (b), (c) below).
(b) (X, τ ) is sequentially complete on ⋅ -bounded sets if and only if (X, γ) is sequentially complete. Proof.
(a) This is clear since   [40, p. 273]). Obviously, every sequential topological vector space is C-sequential. Further, every bornological topological vector space is C-sequential by [40,Theorem 8]. The bornological spaces D(R n ) of test functions on R n with its inductive limit topology and D(R n ) ′ of distributions with its strong dual topology are examples of C-sequential spaces that are not sequential by [39,Théorème 5] and [12,Proposition 1]. For our next proofs we need a classification of C-sequential Hausdorff locally convex spaces. Let (X, τ ) be a Hausdorff locally convex space and U + be the collection of all absolutely convex subsets U ⊆ X which satisfy the condition that every sequence (x n ) in X converging to 0 is eventually in U . Then U + is a zero neighbourhood basis for a Hausdorff locally convex topology τ + ⊆ τ on X, which is the finest Hausdorff locally convex topology on X with the same convergent sequences as τ by [ Proposition 5.7. Let X be a Banach space with norm-topology τ ⋅ and τ ⊆ τ ⋅ a Hausdorff locally convex topology on X such that τ is metrisable on the ⋅ -unit ball B ⋅ , and γ ∶= γ(τ, τ ⋅ ) the mixed topology. Then (X, γ) is a C-sequential space and γ + = γ.
Proof. Let Y be any Hausdorff locally convex space and f ∶ (X, γ) → Y any linear sequentially continuous map. It follows from [9, Section I.1, Proposition 1.9] that f is even continuous. We conclude that (X, γ) is a C-sequential space and γ + = γ by Proposition 5.6.
In particular, Proposition 5.7 is applicable if (X, τ ) is metrisable, and implies the β + = β part in [25,Theorem 8.1]. Further, Proposition 5.7 gives a sufficient condition for γ + = γ that is simple to check and relevant for the relation between bi-continuous semigroups and SCLE semigroups [25,Section 7]. If (X, γ) is even metrisable or equivalently first countable [46, Proposition 1.1.11 (ii)], then we are in the uninteresting situation that γ = τ ⋅ by [9, Section I.1, Proposition 1.15]. However, X = C b (R n ) with the mixed (=strict) topology from Example 5.4 is a Csequential space by Proposition 5.7 which is not metrisable (not even bornological or barrelled) since the strict topology does not coincide with the norm-topology (cf. [9, Section II.1, Proposition 1.2 5)]).
Proof. Denote γ ∶= γ(τ, τ ⋅ ). From [44, Lemma 2.1.1 (3)] (condition (n) there is satisfied) one gets τ ⊆ γ. The other inclusion will be proved by contradiction. Let us assume that there is U ∈ γ such that U ∉ τ . Due to (X, τ ) being C-sequential we obtain U ∉ τ + by Proposition 5.6. Since U ∉ τ + , there is x 0 ∈ U such that U is not a neighbourhood of x 0 w.r.t. τ + , i.e. for all V ∈ U + it holds that x 0 + V ⊈ U . W.l.o.g. x 0 = 0 because τ + is a locally convex topology. As U ∈ γ and x 0 = 0, there is an absolutely convex zero neighbourhood ) and thus that (x n ) is eventually in V 0 . Therefore, (x n ) is ⋅ -unbounded. W.l.o.g. we assume ∀n ∈ N ∶ x n > n and that all x n are distinct from each other. By the norming property for n ∈ N there exists a τ -continuous functional f n in the unit sphere of the dual space such that ⟨f n , x n ⟩ > n. The sets {⟨f n , x⟩ n ∈ N} are bounded by x for every x ∈ X. Further, the set K ∶= {x n n ∈ N} ∪ {0} is compact w.r.t. τ since the sequence (x n ) is convergent to 0. The topology on K is metrisable as image of the continuous map where the domain is equipped with the standard metric ([4, Chapter IX, § 2.10, Proposition 17]). Using [41, Remark 4.1.b] we know that cx K is compact and convex. By a variant of the uniform boundedness principle [35, Theorem 2.9] we obtain the boundedness of ⋃ n∈N f n (cx K). In particular, (⟨f n , x n ⟩) is bounded, contradicting ⟨f n , x n ⟩ → ∞.
We remark that the condition of sequential completeness of (X, τ ) in the preceding proposition can be weakened to the metric convex compactness property since we only need the compactness of cx K. Corollary 5.9. Let X be a Banach space with norm-topology τ ⋅ , τ ⊆ τ ⋅ a Hausdorff locally convex topology such that (X, τ ) ′ is norming and (X, τ ) is a sequentially complete C-sequential space. Then B ⊆ X is bounded in (X, τ ) if and only if B is bounded in (X, τ ⋅ ).
The next proposition shows that bi-continuity of a semigroup is equivalent to being a C 0 -semigroup with respect to the corresponding mixed topology. Thus, bi-continuous semigroups give rise to examples for subordination.
(a) Let (T t ) be a (locally) bi-continuous semigroup w.r.t. τ . Then (X, γ) is sequentially complete, (X, γ) ′ is norming for X and (T t ) is a (locally) sequentially equicontinuous C 0 -semigroup on (X, γ). Lemma 5.12. Let X be a Banach space with norm-topology τ ⋅ , τ ⊆ τ ⋅ a Hausdorff locally convex topology on X such that (X, τ ) ′ ⊆ (X, τ ⋅ ) ′ is norming for X and (X, τ ) is a sequentially complete C-sequential space. Let (T t ) be a locally bi-continuous semigroup. Then (T t ) is uniformly bounded if and only if (T t ) is equibounded on (X, τ ).
is uniformly bounded if and only if B is bounded in (X, τ ⋅ ). By Corollary 5.9, this is equivalent to boundedness of B in (X, τ ), which in turn is equivalent to equiboundedness of (T t ).
We can now combine Proposition 4.9 and Corollary 4.17 to easily obtain the following.
Theorem 5.13. Let X be a Banach space with norm-topology τ ⋅ , τ ⊆ τ ⋅ a Hausdorff locally convex topology on X such that (X, τ ) ′ ⊆ (X, τ ⋅ ) ′ is norming for X and (X, γ) is a C-sequential space where γ ∶= γ(τ, τ ⋅ ) is the mixed topology. Let (T t ) be a uniformly bounded (locally) bi-continuous semigroup on X w.r.t. τ with generator −A and f a Bernstein function. Then the subordinated semigroup (S t ) to (T t ) w.r.t. f is uniformly bounded and (locally) bi-continuous w.r.t. τ as well and its generator −f (A) is given by the sequential closure of −A f .
Proof. Let us first show that (S t ) is uniformly bounded and (locally) bi-continuous. First, we apply Proposition 5.10 (a) and obtain that (T t ) is a uniformly bounded (locally) sequentially equicontinuous C 0 -semigroup on (X, γ), (X, γ) is sequentially complete and (X, γ) ′ norming for X. Since (X, γ) is a C-sequential space, an application of Proposition 5.10 (b) yields that (T t ) is a uniformly bounded (locally) bi-continuous semigroup w.r.t. γ. From Lemma 5.12 we derive that (T t ) is equibounded on (X, γ). Proposition 4.9 yields that (S t ) is a (locally) sequentially equicontinuous and equibounded C 0 -semigroup on (X, γ). Another application of Proposition 5.10 (b) and then of Lemma 5.12 provides that (S t ) is a uniformly bounded (locally) bi-continuous semigroup w.r.t. γ and thus w.r.t. τ as well.
Let −f (A) be the generator of (S t ). By Corollary 4.17 we have that the sequential closure of A f coincides with f (A).
Remark 5.14. If (T t ) is a bi-continuous semigroup with generator −A, but maybe not uniformly bounded, one may be tempted to first rescale the semigroup, then apply Theorem 5.13 and then rescale the subordinated semigroup again. If f is a Bernstein function, then this procedure ends up with a generator being an extension of − f (A + ω) − ω , where ω is the rescaling parameter.
Analytic semigroups ( [13,30]) provide a basic example for bi-continuous semigroups and the generators coincide.
Lemma 5.15. Let (T t ) be an analytic or C 0 -semigroup on a Banach space (X, τ ⋅ ) with generator −A which is at the same time bi-continuous w.r.t. the topology τ with generator −Ã. Then A =Ã.
Example 5.16. Let X ∶= C b (R n ) with supremum norm ⋅ ∞ , and τ co the compactopen topology, i.e. the topology of uniform convergence on compact subsets of R n . Let k∶ (0, ∞) × R n → R, be the Gauß-Weierstraß kernel. For t ≥ 0 define T t ∈ L(X) by Proof. We only need to prove the part on (S t ) which directly follows from Theorem 5.13 since (X, γ) with the mixed topology γ ∶= γ(τ co , τ ⋅ ∞ ) is a C-sequential space by Proposition 5.7.
Further examples of semigroups (T t ) being strongly continuous for mixed topologies can be found e.g. in [18].
Remark 5.17. The situation of Example 5.16 can be generalised. Namely, let A be a sectorial operator such that −A generates an analytic semigroup which is at the same time bi-continuous. One can consider fractional powers A α , α ∈ (0, 1), either by means of the standard sectorial functional calculus (see [19,Chapter 3]) (for this one does not actually need that −A generates a semigroup) or by using the methods from this paper. Even without the assumption of dealing with a strongly continuous semigroup, one can still use (4). This still works since an analytic semigroup is always strongly continuous on D ∶= D(A). It is essentially equivalent to using the Balakrishnan formula, see [19,  For an operator A as above this means that the here presented approach yields the same fractional powers as its sectorial functional calculus does in the Banach space X.

Transition Semigroups for Markov Processes.
Let (Ω, F , P) be a probability space and (E, E) a measurable space. Let us recall the notions of (normal) Markov processes and their associated transition semigroups.
Definition 5.18. A tuple X ∶= (Ω, F , P, (F t ) t≥0 , (X t ) t≥0 , E, E, (P x ) x∈E ) is called a Markov process if (X t ) is an adapted process on (Ω, F , P) w.r.t. the filtration (F t ) with values in E and (P x ) is a family of probability measures on (Ω, F ) such that E ∋ x ↦ P x (X t ∈ B) is measurable for all B ∈ E and P x (X t+s ∈ B F s ) = P Xt (X s ∈ B) P x -a.s. for all x ∈ E, t, s ≥ 0, and B ∈ E. A Markov process X is called normal if {x} ∈ E for all x ∈ E and P x (X 0 = x) = 1 for all x ∈ E.
We write B b (E) for the bounded measurable (scalar) functions on E.
Definition 5.19. Let X ∶= (Ω, F , P, (F t ), (X t ), E, E, (P x ) x∈E ) be a Markov process. For t ≥ 0 we define where E x is the expectation with respect to P x . We call (T t ) the transition semigroup associated with the Markov process.
Transition semigroups (T t ) for Markov processes satisfy a semigroup law, while normality of the Markov process yields that T 0 = I. We state this well-known fact as a lemma.
Lemma 5.20. Let (Ω, F , P, (F t ), (X t ), E, E, (P x ) x∈E ) be a normal Markov process with transition semigroup (T t ). Then (T t ) is a semigroup. Now, let E be a completely regular Hausdorff space, E ∶= B(E) the Borel σ-field, and X a normal Markov process with transition semigroup (T t ). Let us assume that C b (E) is invariant for (T t ), i.e. T t (C b (E)) ⊆ C b (E) for all t ≥ 0. Sometimes, X is then called a C b -Feller process and (T t ) a C b -Feller semigroup, and we will adopt this notion. We may then try to restrict the transition semigroup (T t ) to C b (E).
We now introduce the strict topology on C b (E) as in [38].
Definition 5.21. Let βE be the Stone-Čech compactification of E. For Q ⊆ βE∖E compact we define C Q (E) ∶= {g E ∈ C b (E) g ∈ C(βE), g Q = 0}. Then C Q (E) induces a topology β Q on C b (E) via the seminorms ⋅ g for g ∈ C Q (E) given by f g ∶= gf ∞ . Then the strict topology β on C b (E) is defined to be the inductive limit topology for (C Q (E), β Q ) Q⊆βE∖E compact . is induced by the seminorms ⋅ g for g ∈ C 0 (E) given by f g ∶= gf ∞ (see Example 5.4).
Let us collect some results on C b -Feller semigroups on (C b (E), β). To this end, we write L 0 ((Ω, F , P); E) for the space of P-equivalence classes of strongly measurable functions from Ω to E equipped with the topology of convergence in measure. Now, let E be a complete metric space.
Proposition 5.23. Let X ∶= (Ω, F , P, (F t ), (X t ), E, E, (P x ) x∈E ) be a C b -Feller process and (T t ) the associated C b -Feller semigroup on (C b (E), β). Proposition 5.23 yields that as soon as the Markov process is continuous in time and initial value, then the corresponding transition semigroup satisfies all the properties needed for subordination.