# Bi-continuous semigroups

### Working Groups: Chair Applied Analysis

### Collaborators (MAT): Dr.Â habil. Christian Seifert

### External Collaborators: PD Dr.Â Karsten Kruse, Jan Meichsner, Felix L. Schwenninger

## Description

Strongly continuous semigroups of operators are a well-established framework in the study of evolution equations on Banach spaces \(X\) [3]. However, in many applications the semigroups are not strongly continuous (\(C_0\)) w.r.t. the norm \(\|\cdot\|\) of the Banach space but strongly continuous with respect to a weaker Hausdorff locally convex topology \(\tau\). Examples of such semigroups are adjoint semigroups of norm-strongly continuous semigroups, implemented semigroups, the left translation semigroup on \(C_{b}(\mathbb{R})\), the GauÃŸ-WeierstraÃŸ semigroup on \(C_{b}(\mathbb{R}^d)\) as well as transition semigroups like the Ornstein-Uhlenbeck semigroup on the space \(C_{b}(\Omega)\) of bounded continuous functions on a Polish space \(\Omega\) [5,7].

These examples belong to the general framework of *bi-continuous semigroups*, where the triple \((X,\|\cdot\|,\tau)\) is a sequentially complete *Saks space* [2] and the semigroups are \(\tau\)-strongly continuous, exponentially bounded and locally bi-equicontinuous, and were first studied by KÃ¼hnemund in [9,10]. Equivalently, such semigroups are strongly continuous and locally sequentially equicontinuous w.r.t. the *mixed topology* \(\gamma:=\gamma(\|\cdot\|,\tau)\) of Wiweger [20] which is the finest Hausdorff locally convex topology that coincides with \(\tau\) on \(\|\cdot\|\)-bounded sets. In particular, strongly continuous, locally equicontinuous semigroups w.r.t. \(\gamma\) on sequentially complete Saks spaces are bi-continuous.

In general, the class of bi-continuous semigroups is larger than the class of strongly continuous, locally equicontinuous semigroups w.r.t. \(\gamma\). The space \((X,\gamma)\) is usually neither barrelled nor bornological [2]. Thus automatic local equicontinuity results for strongly continuous semigroups like in [8] are not applicable. Nevertheless, if \((X,\gamma)\) is a *C-sequential space*, i.e.Â every convex sequentially open set is already open, then both classes of semigroups coincide and such semigroups are even quasi-equicontinuous w.r.t. \(\gamma\) [11,15]. For instance, \((X,\gamma)\) is C-sequential if \(\tau\) is metrisable on the closed \(\|\cdot\|\)-unit ball [12].

In the context of perturbation theory of bi-continuous semigroups the notion of tightness emerged [1,4], which plays a similar role as equicontinuity in perturbation theory of strongly continuous semigroups on Hausdorff locally convex spaces [7]. In [15] we consider the relation between tightness and equicontinuity w.r.t. the mixed topology \(\gamma\) and present sufficient conditions that guarantee their equivalence.

Complementary to the Hille-Yosida generation theorem for bi-continuous semigroups [10], we derive Lumer-Phillips type generation theorems for strongly continuous, equicontinuous semigroups w.r.t. \(\gamma\) in [17]. Turning to a particular class of semigroups, we extensively study weighted composition semigroups induced by semiflows and associated semicocycles on spaces like \(C_{b}(\mathbb{R})\), the Hardy space \(H^{\infty}\) of bounded holomorphic functions on the open complex unit disc or the Bloch type spaces \(\mathcal{B}_{\alpha}\) for \(\alpha>0\). We give necessary and sufficient conditions for their bi-continuity and characterise their generators [12] and also study the topological properties of such spaces equipped with the mixed topology [13].

Another application of bi-continuous semigroups lies in control theory of infinite-dimensional systems. In [18] we consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e.Â for every initial value, estimating the state at a final time \(T>0\) by taking into account the orbit of the initial value under the semigroup for \(t\in [0,T]\), measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the GauÃŸ-WeierstraÃŸ semigroup and the Ornstein-Uhlenbeck semigroup on \(C_{b}(\mathbb{R}^d)\).

In [16] we turn to another question arising in control theory of infinite-dimensional systems. Namely, we aim for a generalisation of a recently proved result for \(\|\cdot\|\)-strongly continuous semigroups to bi-continuous ones:

**Theorem** [6] Let \((X,\|\cdot\|)\) be a Banach space and \((T(t))_{\geq 0}\) a \(\|\cdot\|\)-strongly continuous semigroup on \(X\) with generator \((A,D(A))\). Then the following assertions are equivalent:

1. \(A_{-1}\) is \(L^{\infty}\)-admissible.

2. \(Fav(T) = D(A)\) and \((T(t))_{\geq 0}\) satisfies the \(C\)-maximal regularity property.

3. \(A\) extends to a bounded operator from \(X\) to \(X\).

Here, \(A_{-1}\) is the generator of the extrapolated semigroup \((T_{-1}(t))_{\geq 0}\) on the extrapolation space \(X_{-1}\), and \(Fav(T)\) the Favard space of \((T(t))_{\geq 0}\). For the proof of the non-trivial implications of this theorem the concept of sun dual spaces for \(\|\cdot\|\)-strongly continuous semigroups from [19] is pivotal. As a first step in reaching a generalisation of this theorem we develop a sun dual theory for bi-continuous semigroups and discuss its peculiarities with respect to the properties of the present topologies in [16]. However, the proof of a generalisation of this theorem in the setting of bi-continuous semigroups is still not achieved yet and subject to future work.

## References

[1] C. Budde, Positive Miyadera-Voigt perturbations of bi-continuous semigroups, *Positivity*, 25(3):1107-1129, 2021. doi: 10.1007/s11117-020-00806-1.

[2] J.B. Cooper. Saks spaces and applications to functional analysis. *North-Holland Math. Stud. 28*. North-Holland, Amsterdam, 1978.

[3] K.-J. Engel, R. Nagel. One-parameter semigroups for linear evolution equations. *Grad. Texts in Math. 194*. Springer, New York, 2000. doi: 10.1007/b97696.

[4] A. Es-Sarhir, B. Farkas. Perturbation for a class of transition semigroups on the HÃ¶lder space \(C_{b,loc}^{\theta}(H)\). *J. Math. Anal. Appl.*, 315(2):666-685, 2006. doi: 10.1016/j.jmaa.2005.04.024.

[5] B. Goldys, M. Nendel, M. RÃ¶ckner. Operator semigroups in the mixed topology and the infinitesimal description of Markov processes, 2022. arXiv:2204.07484.

[6] B. Jacob, F.L. Schwenninger, J. Wintermayr. A refinement of BaillonÂ´s theorem on maximal regularity. *Studia Math.*, 263(2):141-158, 2022. doi: 10.4064/sm200731-20-3.

[7] B. Jacob, S.-A. Wegner, J. Wintermayr. Desch-Schappacher perturbation of one-parameter semigroups on locally convex spaces. *Math. Nachr.*, 288(8-9):925-935, 2015. doi: 10.1002/mana.201400116.

[8] T. Komura, Semigroups of operators in locally convex spaces, *J. Funct. Anal.*, 2(3):258-296, 1968. doi: 10.1016/0022-1236(68)90008-6.

[9] F. KÃ¼hnemund. Bi-continuous semigroups on spaces with two topologies: Theory and applications. PhD thesis, Eberhard-Karls-UniversitÃ¤t TÃ¼bingen, 2001. URN: urn:nbn:de:bsz:21-opus-2366.

[10] F. KÃ¼hnemund. A Hille-Yosida theorem for bi-continuous semigroups. *Semigroup Forum*, 67(2):205-225, 2003. doi: 10.1007/s00233-002-5000-3.

[11] R. Kraaij. Strongly continuous and locally equi-continuous semigroups on locally convex spaces. *Semigroup Forum*, 92(1):158-185, 2016. doi: 10.1007/s00233-015-9689-1.

[12] K. Kruse. Linearisation of weak vector-valued functions, 2022. arXiv:2207.04681.

[13] K. Kruse. Weighted composition semigroups on spaces of continuous functions and their subspaces, 2022. arXiv:2207.05384.

[14] K. Kruse, J. Meichsner, C. Seifert. Subordination for sequentially equicontinuous equibounded \(C_0\)-semigroups. *J. Evol. Equ.*, 21(2):2665-2690, 2021. doi: 10.1007/s00028-021-00700-7.

[15] K. Kruse, F.L. Schwenninger. On equicontinuity and tightness of bi-continuous semigroups. *J. Math. Anal. Appl.*, 509(2):1-27, 2022. doi: 10.1016/j.jmaa.2021.125985.

[16] K. Kruse, F.L. Schwenninger. Sun dual theory for bi-continuous semigroups, 2023. arXiv:2203.12765.

[17] K. Kruse, C. Seifert. A note on the Lumer-Phillips theorem for bi-continuous semigroups, 2022. arXiv:2206.00887 (to appear in Z. Anal. Anwend.).

[18] K. Kruse, C. Seifert. Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups, 2022. arXiv:2206.00562.

[19] J. van Neerven. The adjoint of a semigroup of linear operators. *Lecture Notes in Math. 1529*. Springer, Berlin, 1992. doi: 10.1007/BFb0085008.

[20] A. Wiweger. Linear spaces with mixed topology. *Studia Math.*, 20(1):47-68, 1961. doi: 10.4064/sm-20-1-47-68.