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topics/Bi-continuous_semigroups.md

+# Bi-continuous semigroups
+
+### Working Groups: aa
+
+### Collaborators (MAT): cseifert, kkruse
+
+### External Collaborators: [Jan Meichsner](https://tu-dresden.de/bu/umwelt/geo/ipg/astro/weltraumastrometrie/mitarbeiter), [Felix L. Schwenninger](https://www.math.uni-hamburg.de/home/schwenninger/hp/index.html)
+
+## Description
+
+Strongly continuous [semigroups](https://en.wikipedia.org/wiki/C0-semigroup) 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 Gauß-Weierstraß semigroup on $C_{b}(\mathbb{R}^d)$ as well as [transition semigroups](https://encyclopediaofmath.org/wiki/Transition-operator_semi-group) like the Ornstein-Uhlenbeck semigroup on the space $C_{b}(\Omega)$ of bounded continuous functions on a [Polish space](https://en.wikipedia.org/wiki/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](https://thehighergeometer.wordpress.com/2018/06/06/saks-spaces-what/)* [2] and the semigroups are $\tau$-strongly continuous, exponentially bounded and locally bi-equicontinuous, and were first studied by Kühnemund 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](https://en.wikipedia.org/wiki/Equicontinuity) 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](https://en.wikipedia.org/wiki/Barrelled_space) nor [bornological](https://en.wikipedia.org/wiki/Bornological_space) [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. 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. 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 Gauß-Weierstraß 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](https://doi.org/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](https://doi.org/10.1007/b97696).\
+[4] A. Es-Sarhir, B. Farkas. Perturbation for a class of transition semigroups on the Hölder space $C_{b,loc}^{\theta}(H)$. *J. Math. Anal. Appl.*, 315(2):666-685, 2006. doi: [10.1016/j.jmaa.2005.04.024](https://doi.org/10.1016/j.jmaa.2005.04.024).\
+[5] B. Goldys, M. Nendel, M. Röckner. Operator semigroups in the mixed topology and the infinitesimal description of Markov processes, 2022. [arXiv:2204.07484](https://arxiv.org/abs/2204.07484).\
+[6] B. Jacob, F.L. Schwenninger, J. Wintermayr. A refinement of Baillon´s theorem on maximal regularity. *Studia Math.*, 263(2):141-158, 2022. doi: [10.4064/sm200731-20-3](https://doi.org/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](https://doi.org/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](https://doi.org/10.1016/0022-1236(68)90008-6).\
+[9] F. Kühnemund. Bi-continuous semigroups on spaces with two topologies: Theory and applications. PhD thesis, Eberhard-Karls-Universität Tübingen, 2001. URN: [urn:nbn:de:bsz:21-opus-2366](http://nbn-resolving.de/urn:nbn:de:bsz:21-opus-2366).\
+[10] F. Kühnemund. A Hille-Yosida theorem for bi-continuous semigroups. *Semigroup Forum*, 67(2):205-225, 2003. doi: [10.1007/s00233-002-5000-3](https://doi.org/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](https://doi.org/10.1007/s00233-015-9689-1).\
+[12] K. Kruse. Linearisation of weak vector-valued functions, 2022. [arXiv:2207.04681](https://arxiv.org/abs/2207.04681).\
+[13] K. Kruse. Weighted composition semigroups on spaces of continuous functions and their subspaces, 2022. [arXiv:2207.05384](https://arxiv.org/abs/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](https://doi.org/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](https://doi.org/10.1016/j.jmaa.2021.125985).\
+[16] K. Kruse, F.L. Schwenninger. Sun dual theory for bi-continuous semigroups, 2023. [arXiv:2203.12765](https://arxiv.org/abs/2203.12765).\
+[17] K. Kruse, C. Seifert. A note on the Lumer-Phillips theorem for bi-continuous semigroups, 2022. [arXiv:2206.00887](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://doi.org/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](https://doi.org/10.4064/sm-20-1-47-68).