Bi-continuous semigroups

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Dr. habil. Christian Seifert

External Collaborators: PD Dr. 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 Gauß-Weierstraß 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 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 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. 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

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[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.
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