Bi-Continuous Semigroups

Collaborators: Jan Meichsner, Felix L. Schwenninger, Christian Seifert Link to heading

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 $\lVert\cdot\rVert$ 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 $(X,\lVert\cdot\rVert,\tau)$ 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 $\gamma:=\gamma(\lVert\cdot\rVert,\tau)$ of Wiweger [20] which is the finest Hausdorff locally convex topology between the $\lVert\cdot\rVert$-topology and $\tau$ that coincides with $\tau$ on $\lVert\cdot\rVert$-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 $\lVert\cdot\rVert$-unit ball [14].

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 [13] and also study the topological properties of such spaces equipped with the mixed topology [12].

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 $\lVert\cdot\rVert$-strongly continuous semigroups to bi-continuous ones:

Theorem [6] Let $(X,\lVert\cdot\rVert)$ be a Banach space and $(T(t))_{\geq 0}$ a $\lVert\cdot\rVert$-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, $A_{-1}$ 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 $\lVert\cdot\rVert$-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 Link to heading

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