A note on the Lumer–Phillips theorem for bi-continuous semigroups

Karsten Kruse 0000-0003-1864-4915 Hamburg University of Technology
Institute of Mathematics
Am Schwarzenberg-Campus 3
21073 Hamburg
Germany karsten.kruse@tuhh.de and Christian Seifert 0000-0001-9182-8687 christian.seifert@tuhh.de

(Date: March 16, 2025)

Abstract.

Given a Banach space $X$ and an additional coarser Hausdorff locally convex topology $\tau$ on $X$ we characterise the generators of $\tau$ -bi-continuous semigroups in the spirit of the Lumer–Phillips theorem, i.e. by means of dissipativity w.r.t. a directed system of seminorms and a range condition.

Key words and phrases:

dissipativity, bi-continuous semigroup, Lumer–Phillips theorem, Saks space, mixed topology

2020 Mathematics Subject Classification:

Primary 47B44 Secondary 47D06, 46A70

K. Kruse acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 “Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

1. Introduction

Characterising generators of strongly continuous semigroups on Banach spaces is a classical topic due to its relation to well-posedness of the corresponding abstract Cauchy problem [8, Chap. II, 6.7 Theorem, p. 150]. The two main generation theorems go back to Hille–Yosida [16, 34] and Feller–Miyadera–Phillips [14, 28, 29] for general semigroups and Lumer–Phillips [25] for contraction semigroups.

However, there are important examples of semigroups which are not strongly continuous for the Banach space norm, e.g. the Gauß–Weierstraß semigroup on $\mathrm{C}_{\operatorname{b}}(\mathbb{R}^{d})$ . To circumvent this issue the concept of bi-continuous semigroups which are strongly continuous only w.r.t. to a weaker Hausdorff locally convex topology $\tau$ has been introduced by Kühnemund [22]. Thus a natural question is to characterise the generators of bi-continuous semigroups in the spirit of the Hille–Yosida theorem and the Lumer–Phillips theorem. While a version of the Hille–Yosida theorem for bi-continuous semigroups was established directly at the beginning of the theory [23], a corresponding version of the Lumer–Phillips theorem for bi-continuous contraction semigroups was missing. Recently, in [5], Budde and Wegner introduced the notion of bi-dissipativity to characterise the generators of bi-continuous contraction semigroups. However, the result in [5] does not cover the key example of the Gauß–Weierstraß semigroup on $\mathrm{C}_{\operatorname{b}}(\mathbb{R}^{d})$ (see [5, Example 3.9, p. 8]), see also Remark 3.25 below.

In this paper we make use of the notion of $\Gamma$ -dissipativity, where $\Gamma$ is a directed set of seminorms generating a topology related to the mixed topology $\gamma\coloneqq\gamma(\|\cdot\|,\tau)$ (see [33] and Definition 2.1 for the mixed topology), introduced in [1] to prove versions of the Lumer–Phillips theorem for bi-continuous (contraction) semigroups in Theorem 3.9 and Theorem 3.10. Note that also [5] used $\Gamma$ -dissipativity to define bi-dissipativity; however, there $\Gamma$ is a fundamental system of seminorms generating the topology $\tau$ . Working instead with the mixed topology yields a more natural concept which can also be applied to the Gauß–Weierstraß semigroup, see Remark 3.25.

Let us mention that strongly continuous semigroups have also been considered in locally convex spaces, in particular including a Lumer–Phillips-type generation theorem, see [1].

Note that with these results we also answer a question on characterising generators of transition semigroups raised by Markus Kunze, see Problem 3 in [15, p. 4].

In Section 2 we review the notion of bi-continuous semigroups and their generators as well as the (sub-)mixed topology. We also recall further properties of locally convex spaces as well as operators which we make use of later on. In Section 3 we recall the notion of $\Gamma$ -dissipativity and then prove the version of the Lumer–Phillips theorem, where we also comment on its relation to [5]. Further, we also provide some examples.

2. Notions and preliminaries

In this short section we recall some basic notions and results in the context of bi-continuous semigroups. For a vector space $X$ over the field $\mathbb{R}$ or $\mathbb{C}$ with a Hausdorff locally convex topology $\tau$ we denote by $(X,\tau)^{\prime}$ the topological linear dual space and just write $X^{\prime}\coloneqq(X,\tau)^{\prime}$ if $(X,\tau)$ is a Banach space. By $\Gamma_{\tau}$ we always denote a directed system of continuous seminorms that generates the Hausdorff locally convex topology $\tau$ on $X$ . Further, for two Hausdorff locally convex spaces $(X,\tau)$ and $(Y,\sigma)$ we use the symbol $\mathcal{L}((X,\tau);(Y,\sigma))$ for the space of continuous linear operators from $(X,\tau)$ to $(Y,\sigma)$ , and abbreviate $\mathcal{L}(X,\tau)\coloneqq\mathcal{L}((X,\tau);(X,\tau))$ . If $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ are Banach spaces, we denote by $\tau_{\|\cdot\|_{X}}$ and $\tau_{\|\cdot\|_{Y}}$ the corresponding topologies induced by the norms and just write $\mathcal{L}(X;Y)\coloneqq\mathcal{L}((X,\tau_{\|\cdot\|_{X}});(Y,\tau_{\|\cdot\|_{Y}}))$ with operator norm $\|\cdot\|_{\mathcal{L}(X;Y)}$ , and $\mathcal{L}(X)\coloneqq\mathcal{L}(X;X)$ .

Let us recall the definition of the mixed topology, [33, Section 2.1], and the notion of a Saks space, [7, I.3.2 Definition, p. 27–28], which will be important for the rest of the paper.

2.1 Definition ([18, Definition 2.2, p. 3], [5, Proposition 3.11 (a), p. 9]).

Let $(X,\|\cdot\|)$ be a Banach space and $\tau$ a Hausdorff locally convex topology on $X$ that is coarser than the $\|\cdot\|$ -topology $\tau_{\|\cdot\|}$ . Then

(a)

the mixed topology $\gamma\coloneqq\gamma(\|\cdot\|,\tau)$ is the finest linear topology on $X$ that coincides with $\tau$ on $\|\cdot\|$ -bounded sets and such that $\tau\subseteq\gamma\subseteq\tau_{\|\cdot\|}$ ;
(b)

a directed system of continuous seminorms $\Gamma_{\tau}$ that generates the topology $\tau$ is called norming if

$\|x\|=\sup_{p\in\Gamma_{\tau}}p(x),\quad x\in X;$ (1)
(c)

the triple $(X,\|\cdot\|,\tau)$ is called a Saks space if there exists a norming directed system of continuous seminorms $\Gamma_{\tau}$ that generates the topology $\tau$ .

The mixed topology is actually Hausdorff locally convex and the definition given above is equivalent to the one introduced by Wiweger [33, Section 2.1] due to [33, Lemmas 2.2.1, 2.2.2, p. 51].

2.2 Definition ([20, Definitions 2.2, 5.4, p. 2, 8]).

Let $(X,\|\cdot\|,\tau)$ be a Saks space.

(a)

We call $(X,\|\cdot\|,\tau)$ (sequentially) complete if $(X,\gamma)$ is (sequentially) complete.
(b)

We call $(X,\|\cdot\|,\tau)$ semi-reflexive if $(X,\gamma)$ is semi-reflexive.
(c)

We call $(X,\|\cdot\|,\tau)$ C-sequential if $(X,\gamma)$ is C-sequential, i.e. every convex sequentially open subset of $(X,\gamma)$ is already open (see [31, p. 273]).

2.3 Remark.

If $(X,\|\cdot\|,\tau)$ is a sequentially complete Saks space, then $(X,\|\cdot\|,\gamma)$ is also a sequentially complete Saks space by [21, Lemma 5.5, p. 2680–2681] and [18, Remark 2.3 (c), p. 3]. In particular, there exists a norming directed system of continuous seminorms $\Gamma_{\gamma}$ that generates $\gamma$ .

There is another kind of mixed topology (see [7, p. 41]) which becomes quite handy if one has to deal with the mixed topology because it is generated by a quite simple directed system of continuous seminorms and often coincides with the mixed topology.

2.4 Definition ([18, Definition 3.9, p. 9]).

Let $(X,\|\cdot\|,\tau)$ be a Saks space and $\Gamma_{\tau}$ a norming directed system of continuous seminorms that generates the topology $\tau$ . We set

\mathcal{N}\coloneqq\{(p_{n},a_{n})_{n\in\mathbb{N}}\;|\;(p_{n})_{n\in\mathbb{N}}\subseteq\Gamma_{\tau},\,(a_{n})_{n\in\mathbb{N}}\in c_{0},\,a_{n}\geq 0\;\text{for all }n\in\mathbb{N}\}

where $c_{0}$ is the space of real null-sequences. For $(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}$ we define the seminorm

{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}\coloneqq\sup_{n\in\mathbb{N}}p_{n}(x)a_{n},\quad x\in X.

We denote by $\gamma_{s}\coloneqq\gamma_{s}(\|\cdot\|,\tau)$ the Hausdorff locally convex topology that is generated by the system of seminorms $({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}})_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}}$ and call it the submixed topology.

Due to [7, I.1.10 Proposition, p. 9], [7, I.4.5 Proposition, p. 41–42] and [11, Lemma A.1.2, p. 72] we have the following observation.

2.5 Remark ([18, Remark 3.10, p. 9]).

Let $(X,\|\cdot\|,\tau)$ be a Saks space, $\Gamma_{\tau}$ a norming directed system of continuous seminorms that generates the topology $\tau$ , $\gamma\coloneqq\gamma(\|\cdot\|,\tau)$ the mixed and $\gamma_{s}\coloneqq\gamma_{s}(\|\cdot\|,\tau)$ the submixed topology.

(a)

We have $\tau\subseteq\gamma_{s}\subseteq\gamma$ and $\gamma_{s}$ has the same convergent sequences as $\gamma$ .
(b)
If
1. (i)
  
  for every $x\in X$ , $\varepsilon>0$ and $p\in\Gamma_{\tau}$ there are $y,z\in X$ such that $x=y+z$ , $p(z)=0$ and $\|y\|\leq p(x)+\varepsilon$ , or
2. (ii)
  
  the $\|\cdot\|$ -unit ball $B_{\|\cdot\|}=\{x\in X\;|\;\|x\|\leq 1\}$ is $\tau$ -compact,
then $\gamma=\gamma_{s}$ holds.

The submixed topology $\gamma_{s}$ was originally introduced in [33, Theorem 3.1.1, p. 62] where a proof of Remark 2.5 (b) can be found as well.

2.6 Remark.

Let $(X,\|\cdot\|,\tau)$ be a Saks space and $\Gamma_{\tau}$ a norming directed system of continuous seminorms that generates the topology $\tau$ . Then there is a norming directed system of continuous seminorms $\Gamma_{\gamma_{s}}$ that generates the submixed topology $\gamma_{s}=\gamma_{s}(\|\cdot\|,\tau)$ . Indeed, we set

\mathcal{N}_{1}\coloneqq\{(p_{n},a_{n})_{n\in\mathbb{N}}\;|\;(p_{n})_{n\in\mathbb{N}}\subseteq\Gamma_{\tau},\,(a_{n})_{n\in\mathbb{N}}\in c_{0},\,0\leq a_{n}\leq 1\;\text{for all }n\in\mathbb{N}\}

and $\Gamma_{\gamma_{s}}\coloneqq\{{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}\;|\;(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}_{1}\}$ . Let $(p_{n})_{n\in\mathbb{N}}\subseteq\Gamma_{\tau}$ and $(a_{n})_{n\in\mathbb{N}}\in c_{0}$ with $a_{n}\geq 0$ for all $n\in\mathbb{N}$ . Then $C\coloneqq\sup_{n\in\mathbb{N}}a_{n}<\infty$ and w.l.o.g. $C>0$ . We have

{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}=\sup_{n\in\mathbb{N}}p_{n}(x)a_{n}\leq C\sup_{n\in\mathbb{N}}p_{n}(x)\frac{a_{n}}{C}=C{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\left(p_{n},\tfrac{a_{n}}{C}\right)_{n\in\mathbb{N}}}

for all $x\in X$ . In combination with $\mathcal{N}_{1}\subseteq\mathcal{N}$ this shows that $\Gamma_{\gamma_{s}}$ generates $\gamma_{s}$ . Furthermore, for every $p\in\Gamma_{\tau}$ we have with $p_{n}\coloneqq p$ for all $n\in\mathbb{N}$ that $p(x)\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},1/n)_{n\in\mathbb{N}}}$ for all $x\in X$ . Together with the norming property of $\Gamma_{\tau}$ this implies

\|x\|=\sup_{p\in\Gamma_{\tau}}p(x)\leq\sup_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}_{1}}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}\leq\sup_{(a_{n})_{n\in\mathbb{N}}\in c_{0},\,0\leq a_{n}\leq 1}\|x\|a_{n}\leq\|x\|

for all $x\in X$ . Hence $\Gamma_{\gamma_{s}}$ is norming.

Recall that for a Banach space $(X,\|\cdot\|)$ and a Hausdorff locally convex topology $\tau$ on $X$ the triple $(X,\|\cdot\|,\tau)$ is called bi-admissible space if $\tau$ is coarser than $\tau_{\|\cdot\|}$ , $\tau$ is sequentially complete on the $\|\cdot\|$ -closed unit ball (or equivalently on $\|\cdot\|$ -bounded sets), and $(X,\tau)^{\prime}$ is norming for $X$ ; cf. [5, Assumption 2.1, p. 3].

2.7 Lemma.

Let $(X,\|\cdot\|)$ be a Banach space and $\tau$ a Hausdorff locally convex topology on $X$ that is coarser than the $\|\cdot\|$ -topology $\tau_{\|\cdot\|}$ . Then the following are equivalent:

(i)

$(X,\|\cdot\|,\tau)$ is a sequentially complete Saks space.
(ii)

$(X,\|\cdot\|,\tau)$ is a bi-admissible space.

Proof.

By [33, Corollary 2.3.2, p. 55] a Saks space $(X,\|\cdot\|,\tau)$ is sequentially complete if and only if $(X,\tau)$ is sequentially complete on $\|\cdot\|$ -bounded sets, meaning that every $\|\cdot\|$ -bounded $\tau$ -Cauchy sequence converges in $X$ . Combined with [18, Remark 2.3 (c), p. 3] it follows that a triple $(X,\|\cdot\|,\tau)$ fulfils [23, Assumptions 1, p. 206], which provides a bi-admissible space, if and only if it is a sequentially complete Saks space. ∎

Let us recall the notion of a bi-continuous semigroup.

2.8 Definition ([23, Definition 3, p. 207]).

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space. A family $(T(t))_{t\geq 0}$ in $\mathcal{L}(X)$ is called $\tau$ -bi-continuous semigroup if

(i)

$(T(t))_{t\geq 0}$ is a semigroup, i.e. $T(t+s)=T(t)T(s)$ and $T(0)=\operatorname{id}$ for all $t,s\geq 0$ ,
(ii)

$(T(t))_{t\geq 0}$ is $\tau$ -strongly continuous, i.e. the map $T_{x}\colon[0,\infty)\to(X,\tau)$ , $T_{x}(t)\coloneqq T(t)x$ , is continuous for all $x\in X$ ,
(iii)

$(T(t))_{t\geq 0}$ is exponentially bounded (of type $\omega$ ), i.e. there exists $M\geq 1$ , $\omega\in\mathbb{R}$ such that $\|T(t)\|_{\mathcal{L}(X)}\leq M\mathrm{e}^{\omega t}$ for all $t\geq 0$ ,
(iv)

$(T(t))_{t\geq 0}$ is locally bi-equicontinuous, i.e. for every sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ , $x\in X$ with $\sup\limits_{n\in\mathbb{N}}\|x_{n}\|<\infty$ and $\tau\text{-}\lim\limits_{n\to\infty}x_{n}=x$ it holds that

$\tau\text{-}\lim_{n\to\infty}T(t)(x_{n}-x)=0$

locally uniformly for all $t\in[0,\infty)$ .

2.9 Remark.

Let $(X,\|\cdot\|,\tau)$ be a Saks space.

(a)

A sequence in $X$ is $\gamma$ -convergent if and only if it is $\|\cdot\|$ -bounded and $\tau$ -convergent by [7, I.1.10 Proposition, p. 9].
(b)

Let $(X,\|\cdot\|,\tau)$ be sequentially complete. A semigroup of linear operators $(T(t))_{t\geq 0}$ from $X$ to $X$ is $\gamma$ -strongly continuous and locally sequentially $\gamma$ -equicontinuous (i.e. for all $\gamma$ -null sequences $(x_{n})_{n\in\mathbb{N}}$ in $X$ , $t_{0}>0$ and $p\in\Gamma_{\gamma}$ we have $\lim_{n\to\infty}\sup_{t\in[0,t_{0}]}p(T(t)x_{n})=0$ ) if and only if it is a $\tau$ -bi-continuous semigroup on $X$ . This follows directly fom part (a) and [18, Remark 2.6 (b), p. 5], and remains true if $\gamma$ is replaced by any other Hausdorff locally convex topology on $X$ that has the same convergent sequences as $\gamma$ (cf. [11, Proposition A.1.3, p. 73] for $\gamma$ replaced by $\gamma_{s}$ ).

We already observed in Remark 2.9 (b) that $\tau$ -bi-continuous semigroups are locally sequentially $\gamma$ -equicontinuous. Under some mild conditions on the Saks space $(X,\|\cdot\|,\tau)$ they are even quasi- $\gamma$ -equicontinuous. Let us recall what that means.

2.10 Definition.

Let $(X,\upsilon)$ be a Hausdorff locally convex space and $\Gamma_{\upsilon}$ a directed system of continuous seminorms that generates $\upsilon$ . A family $(T(t))_{t\in I}$ of linear maps from $X$ to $X$ is called $\upsilon$ -equicontinuous if

\forall\;p\in\Gamma_{\upsilon}\;\exists\;\widetilde{p}\in\Gamma_{\upsilon},\;C\geq 0\;\forall\;t\in I,\,x\in X:\;p(T(t)x)\leq C\widetilde{p}(x).

The family $(T(t))_{t\geq 0}$ is called locally $\upsilon$ -equicontinuous if $(T(t))_{t\in[0,t_{0}]}$ is $\upsilon$ -equicontinuous for all $t_{0}\geq 0$ . The family $(T(t))_{t\geq 0}$ is called quasi- $\upsilon$ -equicontinuous if there is $\alpha\in\mathbb{R}$ such that $(\mathrm{e}^{-\alpha t}T(t))_{t\geq 0}$ is $\upsilon$ -equicontinuous. Note that one often drops the $\upsilon$ if the topology is clear.

2.11 Remark.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space. Due to Remark 2.9 (b) a $\gamma$ -strongly continuous, locally $\gamma$ -equicontinuous semigroup of linear operators $(T(t))_{t\geq 0}$ from $X$ to $X$ is a $\tau$ -bi-continuous semigroup on $X$ . The converse is not true in general by [13, Example 4.1, p. 320]. However, if $(X,\|\cdot\|,\tau)$ is C-sequential, then the converse also holds by [18, Theorem 3.17 (a), p. 13]. Even more is true, namely, that every $\tau$ -bi-continuous semigroup on $X$ is quasi- $\gamma$ -equicontinuous if $(X,\|\cdot\|,\tau)$ is C-sequential.

There is another related notion to equicontinuity on Saks spaces.

2.12 Definition ([18, Definitions 3.4, 3.5, p. 6, 7]).

Let $(X,\|\cdot\|,\tau)$ be a Saks space and $\Gamma_{\tau}$ a directed system of continuous seminorms generating the topology $\tau$ . A family of linear maps $(T(t))_{t\in I}$ from $X$ to $X$ is called $(\|\cdot\|,\tau)$ -equitight if

\forall\;\varepsilon>0,\,p\in\Gamma_{\tau}\;\exists\;\widetilde{p}\in\Gamma_{\tau},\,C\geq 0\;\forall\;t\in I,\,x\in X:\;p(T(t)x)\leq C\widetilde{p}(x)+\varepsilon\|x\|.

The family $(T(t))_{t\geq 0}$ is called locally $(\|\cdot\|,\tau)$ -equitight if $(T(t))_{t\in[0,t_{0}]}$ is $(\|\cdot\|,\tau)$ -equitight for all $t_{0}\geq 0$ . The family $(T(t))_{t\geq 0}$ is called quasi- $(\|\cdot\|,\tau)$ -equitight if there is $\alpha\in\mathbb{R}$ such that $(\mathrm{e}^{-\alpha t}T(t))_{t\geq 0}$ is $(\|\cdot\|,\tau)$ -equitight.

At first, tight operators $T\in\mathcal{L}(X)$ as well as families of equitight operators $(T(t))_{t\in[0,t_{0}]}$ in $\mathcal{L}(X)$ for $t_{0}\geq 0$ appeared in [11, Definitions 1.2.20, 1.2.21, p. 12] under the name local. In the setting of $\tau$ -bi-continuous semigroups $(T(t))_{t\geq 0}$ the notion of tightness is used in [9, Definition 1.1, p. 668], meaning that $(T(t))_{t\in[0,t_{0}]}$ is equitight (or local) for all $t_{0}\geq 0$ . Local equitightness plays an important role in perturbation results for bi-continuous semigroups, see e.g. [9, Theorem 1.2, p. 669], [12, Theorems 2.4, 3.2, p. 92, 94–95], [12, Remark 4.1, p. 101], [2, Theorem 5, p. 8], [3, Theorem 3.3, p. 582], and the corrections regarding [2] in [18, Remark 3.8, p. 8–9].

Due to [18, Proposition 3.16, p. 12–13] $(\|\cdot\|,\tau)$ -equitightness of a family of linear maps $(T(t))_{t\in I}$ from $X$ to $X$ implies $\gamma$ -equicontinuity. If $(X,\|\cdot\|,\tau)$ is a sequentially complete C-sequential Saks space and $\gamma=\gamma_{s}$ , then any $\tau$ -bi-continuous semigroup $(T(t))_{t\geq 0}$ on $X$ is quasi- $\gamma$ -equicontinuous and quasi- $(\|\cdot\|,\tau)$ -equitight by [18, Theorem 3.17, p. 13], and both properties are equivalent by [18, Proposition 3.16, p. 12–13].

We close this section by recalling the definition of the generator of a $\tau$ -bi-continuous semigroup and two of its properties which we will need.

2.13 Definition ([11, Definition 1.2.6, p. 7]).

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space and $(T(t))_{t\geq 0}$ a $\tau$ -bi-continuous semigroup on $X$ . The generator $(A,D(A))$ is defined by

	$\displaystyle D(A)\coloneqq$	$\displaystyle\Bigl{\{}x\in X\;\|\;\tau\text{-}\lim_{t\to 0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{T(t)x-x}{t}\;\text{exists in }X\;\text{and }\sup_{t\in(0,1]}\frac{\\|T(t)x-x\\|}{t}<\infty\Bigr{\}},$
	$\displaystyle Ax\coloneqq$	$\displaystyle\tau\text{-}\lim_{t\to 0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{T(t)x-x}{t},\quad x\in D(A).$

2.14 Proposition ([23, Corollary 13, p. 215]).

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space and $(T(t))_{t\geq 0}$ a $\tau$ -bi-continuous semigroup on $X$ with generator $(A,D(A))$ . Then the following assertions hold:

(a)

The generator $(A,D(A))$ is bi-closed, i.e. whenever $(x_{n})_{n\in\mathbb{N}}$ is a sequence in $D(A)$ such that $\tau\text{-}\lim_{n\to\infty}x_{n}=x$ and $\tau\text{-}\lim_{n\to\infty}Ax_{n}=y$ for some $x,y\in X$ and both sequences are $\|\cdot\|$ -bounded, then $x\in D(A)$ and $Ax=y$ .
(b)

The generator $(A,D(A))$ is bi-densely defined, i.e. for each $x\in X$ there exists a $\|\cdot\|$ -bounded sequence $(x_{n})_{n\in\mathbb{N}}$ in $D(A)$ such that $\tau\text{-}\lim_{n\to\infty}x_{n}=x$ .

3. Lumer–Phillips for bi-continuous semigroups

First, we recall the relevant notions from [1] concerning dissipative linear operators on Hausdorff locally convex spaces. We write in short that $(A,D(A))$ is a linear operator on a Hausdorff locally convex space $X$ if $A\colon D(A)\subseteq X\to X$ is a linear operator.

3.1 Definition ([1, Definitions 3.1, 3.5, p. 923]).

Let $(X,\upsilon)$ be a Hausdorff locally convex space and $(A,D(A))$ a linear operator on $X$ .

(a)

$(A,D(A))$ is called $\upsilon$ -closed if for each net $(x_{i})_{i\in I}\subseteq D(A)$ satisfying $x_{i}\to x$ and $Ax_{i}\to y$ w.r.t. $\upsilon$ for some $x,y\in X$ , we have $x\in D(A)$ and $Ax=y$ .
(b)

A linear operator $(B,D(B))$ on $X$ is called an extension of $(A,D(A))$ if $D(A)\subseteq D(B)$ and $B_{\mid D(A)}=A$ . The operator $(A,D(A))$ is called $\upsilon$ -closable if it admits a $\upsilon$ -closed extension. The smallest $\upsilon$ -closed extension of an $\upsilon$ -closable operator $(A,D(A))$ is called the $\upsilon$ -closure of $(A,D(A))$ and denoted by $(\overline{A},D(\overline{A}))$ .
(c)

$(A,D(A))$ is called (sequentially) $\upsilon$ -densely defined if $D(A)$ is (sequentially) $\upsilon$ -dense in $X$ .

(d)

Let $(A,D(A))$ be $\upsilon$ -densely defined. The $\upsilon$ -dual operator $(A^{\prime},D(A^{\prime}))$ of $(A,D(A))$ on $(X,\upsilon)^{\prime}$ is defined by setting

D(A^{\prime})\coloneqq\{x^{\prime}\in(X,\upsilon)^{\prime}\;|\;\exists\;y^{\prime}\in(X,\upsilon)^{\prime}\;\forall\;x\in D(A):\;\langle Ax,x^{\prime}\rangle=\langle x,y^{\prime}\rangle\}

and $A^{\prime}x^{\prime}\coloneqq y^{\prime}$ for $x^{\prime}\in D(A^{\prime})$ .

We have the following relation to the notions of a bi-closed resp. bi-densely defined operator.

3.2 Remark.

Let $(X,\|\cdot\|,\tau)$ be a Saks space and $(A,D(A))$ a linear operator. Due to Remark 2.9 (a) $(A,D(A))$ is bi-closed (see Proposition 2.14 (a)) if and only if it is sequentially $\gamma$ -closed (which is defined analogously to $\gamma$ -closedness but with nets replaced by sequences). Again by Remark 2.9 (a) $(A,D(A))$ is bi-densely defined (see Proposition 2.14 (b)) if and only if it is sequentially $\gamma$ -densely defined. Moreover, if $(A,D(A))$ is sequentially $\gamma$ -densely defined, then it is obviously $\gamma$ -densely defined, too (as every sequence is a net).

3.3 Definition ([1, p. 922]).

Let $(X,\upsilon)$ be a Hausdorff locally convex space and $(A,D(A))$ a linear operator on $X$ . If $\lambda\in\mathbb{C}$ is such that $\lambda-A\coloneqq\lambda\operatorname{id}-A\colon D(A)\to X$ is injective, then the linear operator $(\lambda-A)^{-1}$ exists and is defined on the domain $\operatorname{Ran}(\lambda-A)\coloneqq\{(\lambda-A)x\;|\;x\in D(A)\}$ , i.e. the range of $\lambda-A$ . The resolvent set of $A$ is defined by

\rho_{\upsilon}(A)\coloneqq\{\lambda\in\mathbb{C}\;|\;\lambda-A\text{ is bijective and }(\lambda-A)^{-1}\in\mathcal{L}(X,\upsilon)\}.

If $\upsilon=\tau_{\|\cdot\|}$ for a Banach space $(X,\|\cdot\|)$ , we just write $R(\lambda,A)\coloneqq(\lambda-A)^{-1}$ and $\rho(A)\coloneqq\rho_{\upsilon}(A)$ .

3.4 Definition ([1, Definition 3.9, p. 925]).

Let $(X,\upsilon)$ be a Hausdorff locally convex space and $\Gamma_{\upsilon}$ a directed system of continuous seminorms that generates $\upsilon$ . A linear operator $(A,D(A))$ on $X$ is called $\Gamma_{\upsilon}$ -dissipative if

\forall\;\lambda>0,\,x\in D(A),\,p\in\Gamma_{\upsilon}:\;p((\lambda-A)x)\geq\lambda p(x).

It is important to note that in contrast to equicontinuity or equitightness the notion of dissipativity depends on the selection of the directed system of continuous seminorms that generates the topology $\upsilon$ by [1, Remark 3.10, p. 925–926].

3.5 Remark.

Let $(X,\|\cdot\|,\tau)$ be a Saks space, $\upsilon$ a Hausdorff locally convex topology on $X$ and $(A,D(A))$ a $\Gamma_{\upsilon}$ -dissipative operator on $X$ . If $\Gamma_{\upsilon}$ is norming, then it follows from the $\Gamma_{\upsilon}$ -dissipativity and (1) that

\forall\;\lambda>0,\,x\in D(A):\;\|(\lambda-A)x\|\geq\lambda\|x\|.

Thus $(A,D(A))$ is also a dissipative operator on the Banach space $(X,\|\cdot\|)$ in the sense of [8, Chap. II, 3.13 Definition, p. 82] (cf. [5, Remark 3.3 (i), p. 5] for $\upsilon=\tau$ ). We also denote such kind of dissipativity on a Banach space $(X,\|\cdot\|)$ by $\|\cdot\|$ -dissipativity.

In [5] another notion of dissipativity on Saks spaces was introduced.

3.6 Remark.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space. In [5, Definition 3.2, p. 5] a linear operator $(A,D(A))$ on $X$ is called bi-dissipative if there exists a norming directed system of continuous seminorms $\Gamma_{\tau}$ that generates $\tau$ such that $(A,D(A))$ is $\Gamma_{\tau}$ -dissipative. It is then shown in the proof of [5, Theorem 3.15, p. 11] that a bi-dissipative operator $(A,D(A))$ is also $({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}})_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}}$ -dissipative since

{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|(\lambda-A)x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}=\sup_{n\in\mathbb{N}}p_{n}((\lambda-A)x)a_{n}\geq\sup_{n\in\mathbb{N}}\lambda p_{n}(x)a_{n}=\lambda{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}}

for all $\lambda>0$ , $x\in D(A)$ and $(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}$ , where $({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}})_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}}$ is the system of seminorms that generates the submixed topology $\gamma_{s}$ from Definition 2.4. We observe that this also implies that $(A,D(A))$ is $\Gamma_{\gamma_{s}}$ -dissipative w.r.t the norming directed system of continuous seminorms $\Gamma_{\gamma_{s}}$ that generates the submixed topology $\gamma_{s}$ from Remark 2.6.

3.7 Proposition.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space, $\upsilon$ a Hausdorff locally convex topology on $X$ , $(A,D(A))$ a $\Gamma_{\upsilon}$ -dissipative operator on $X$ . Then the following assertions hold:

(a)

$\lambda-A$ is injective for all $\lambda>0$ . Moreover, we have

\forall\;\lambda>0,\,x\in\operatorname{Ran}(\lambda-A),\,p\in\Gamma_{\upsilon}:\;p((\lambda-A)^{-1}x)\leq\frac{1}{\lambda}p(x).

(2)

(b)

If $\operatorname{Ran}(\lambda-A)$ is (sequentially) $\upsilon$ -closed for some $\lambda>0$ , then $(A,D(A))$ is (sequentially) $\upsilon$ -closed. If $\upsilon=\gamma$ , then the converse even holds for all $\lambda>0$ .
(c)

Let $\Gamma_{\upsilon}$ be norming. Then $\lambda-A$ is surjective for some $\lambda>0$ if and only if it is surjective for all $\lambda>0$ . In such a case, $(0,\infty)\subseteq\rho(A)$ .
(d)

Let $\upsilon=\gamma$ . Then $\lambda-A$ is surjective for some $\lambda>0$ if and only if it is surjective for all $\lambda>0$ . In such a case, $(0,\infty)\subseteq\rho_{\gamma}(A)$ .

Proof.

Parts (a), (b) and (d) are just [1, Proposition 3.11, p. 927] in combination with the sequential completeness of $(X,\gamma)$ . Part (c) is a consequence of [8, Chap. II, 3.14 Proposition (ii), p. 82] and Remark 3.5. ∎

In the case $\upsilon=\tau$ parts (a) and (c) of Proposition 3.7 are [5, Proposition 3.4, p. 6].

3.8 Definition.

Let $(X,\|\cdot\|)$ be a Banach space. We call a semigroup of linear operators $(T(t))_{t\geq 0}$ from $X$ to $X$ a contraction semigroup if $\|T(t)\|_{\mathcal{L}(X)}\leq 1$ for all $t\geq 0$ .

3.9 Theorem.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space, $\upsilon$ a Hausdorff locally convex topology on $X$ with $\tau\subseteq\upsilon\subseteq\tau_{\|\cdot\|}$ such that $\gamma$ -convergent sequences are $\upsilon$ -convergent, $(A,D(A))$ a bi-densely defined, $\Gamma_{\upsilon}$ -dissipative operator on $X$ and $\Gamma_{\upsilon}$ norming. Then the following assertions are equivalent:

(a)

$(A,D(A))$ generates a $\tau$ -bi-continuous contraction semigroup on $X$ .
(b)

$\lambda-A$ is surjective for some $\lambda>0$ .

Proof.

We use the Hille–Yosida theorem for bi-continuous semigroups to prove both implications (see [23, Theorem 16, p. 217] and [4, Theorem 5.6, p. 340]).

(a) $\Rightarrow$ (b): Let $(A,D(A))$ generate a $\tau$ -bi-continuous contraction semigroup on $X$ . Due to [23, Theorem 16, p. 217] with $\omega=0$ we obtain that $(0,\infty)\subseteq\rho(A)$ , in particular, that $\lambda-A$ is surjective for all $\lambda>0$ .

(b) $\Rightarrow$ (a): Let $\lambda-A$ be surjective for some $\lambda>0$ . Due to [4, Theorem 5.6, p. 340] we only need to prove that

(i)

$(0,\infty)\subseteq\rho(A)$ ,
(ii)

$\|R(\lambda,A)^{n}\|_{\mathcal{L}(X)}\leq\frac{1}{\lambda^{n}}$ for all $n\in\mathbb{N}$ and all $\lambda>0$ , and
(iii)

$\{(\lambda-\alpha)^{n}R(\lambda,A)^{n}\;|\;n\in\mathbb{N},\,\lambda\geq\alpha\}$ is bi-equicontinuous for each $\alpha>0$ , i.e. for each $\alpha>0$ and each $\|\cdot\|$ -bounded $\tau$ -null sequence $(x_{m})_{m\in\mathbb{N}}$ in $X$ one has that $\tau$ - $\lim_{m\to\infty}(\lambda-\alpha)^{n}R(\lambda,A)^{n}x_{m}=0$ uniformly for all $n\in\mathbb{N}$ and all $\lambda\geq\alpha$ .

Since $(A,D(A))$ is $\Gamma_{\upsilon}$ -dissipative, we get that $\lambda-A$ is bijective for all $\lambda>0$ and $\rho(A)\subseteq(0,\infty)$ by Proposition 3.7 (a) and (c). From Remark 3.5 and $\Gamma_{\upsilon}$ being norming we deduce that $\|R(\lambda,A)x\|\leq\tfrac{1}{\lambda}\|x\|$ for all $\lambda>0$ and $x\in\operatorname{Ran}(\lambda-A)=X$ , yielding $\|R(\lambda,A)^{n}\|_{\mathcal{L}(X)}\leq\tfrac{1}{\lambda^{n}}$ for all $n\in\mathbb{N}$ and $\lambda>0$ . Let $\Gamma_{\tau}$ be a directed system of continuous seminorms that generates the topology $\tau$ and $q\in\Gamma_{\tau}$ . Thanks to (2) we know that $p((\lambda-A)^{-1}x)\leq\frac{1}{\lambda}p(x)$ for all $p\in\Gamma_{\upsilon}$ , $\lambda>0$ and $x\in\operatorname{Ran}(\lambda-A)=X$ . As $\tau\subseteq\upsilon$ , there are $p\in\Gamma_{\upsilon}$ and $C\geq 0$ such that for each $\alpha>0$ we have

	$\displaystyle q((\lambda-\alpha)^{n}R(\lambda,A)^{n}x)$	$\displaystyle\leq C(\lambda-\alpha)^{n}p((\lambda-A)^{-n}x)\leq C\frac{(\lambda-\alpha)^{n}}{\lambda^{n}}p(x)$
		$\displaystyle\leq C\left(1-\frac{\alpha}{\lambda}\right)^{n}p(x)\leq Cp(x)$

for all $x\in X$ , $n\in\mathbb{N}$ and $\lambda\geq\alpha$ . Since $\|\cdot\|$ -bounded $\tau$ -null sequences are exactly the $\gamma$ -null-sequences by Remark 2.9 (a) and $\gamma$ -convergent sequences are assumed to be $\upsilon$ -convergent, this inequality implies that $\{(\lambda-\alpha)^{n}R(\lambda,A)^{n}\;|\;n\in\mathbb{N},\,\lambda\geq\alpha\}$ is bi-equicontinuous for all $\alpha>0$ . This finishes the proof. ∎

In the case $\upsilon=\tau$ we know that $\gamma$ -convergent sequences are $\tau$ -convergent and thus Theorem 3.9 is [5, Theorem 3.6, p. 6] (without the superfluous assumption that $A$ should be norm-closed) in this case. Another possible choice is $\upsilon=\gamma_{s}$ since the submixed topology $\gamma_{s}$ has the same convergent sequences as $\gamma$ by Remark 2.5 (a). However, we are mostly interested in the choice $\upsilon=\gamma$ . Our second generation result involves complete Saks spaces.

3.10 Theorem.

Let $(X,\|\cdot\|,\tau)$ be a complete Saks space and $(A,D(A))$ a $\gamma$ -densely defined, $\Gamma_{\gamma}$ -dissipative operator. Assume that $\operatorname{Ran}(\lambda-A)$ is $\gamma$ -dense in $X$ for some $\lambda>0$ . Then the following assertions hold:

(a)

The $\gamma$ -closure $(\overline{A},D(\overline{A}))$ generates a $\gamma$ -strongly continuous, $\gamma$ -equicontinuous semigroup $(T(t))_{t\geq 0}$ on $X$ .
(b)

If $\Gamma_{\gamma}$ is norming, then $(T(t))_{t\geq 0}$ is a contraction semigroup.
(c)

If $\Gamma_{\gamma}$ is norming and $\gamma=\gamma_{s}$ , then $(T(t))_{t\geq 0}$ is $(\|\cdot\|,\tau)$ -equitight.

Proof.

(a) Due to [1, Theorem 3.14, p. 929] $(\overline{A},D(\overline{A}))$ generates a $\gamma$ -equicontinuous, $\gamma$ -strongly continuous semigroup $(T(t))_{t\geq 0}$ on $X$ .

(b) By [1, Proposition 3.13, p. 929] the operator $(\overline{A},D(\overline{A}))$ is also $\Gamma_{\gamma}$ -dissipative and $\lambda-\overline{A}$ is surjective for all $\lambda>0$ . As a consequence of part (a) and Remark 2.9 $(T(t))_{t\geq 0}$ is also a $\tau$ -bi-continuous semigroup on $X$ . By [19, p. 5] the generator $(\overline{A},D(\overline{A}))$ of $(T(t))_{t\geq 0}$ as a $\gamma$ -strongly continuous, $\gamma$ -equicontinuous semigroup (see [1, p. 922]) and the generator of $(T(t))_{t\geq 0}$ as a $\tau$ -bi-continuous semigroup (see Definition 2.13) coincide. Thus $(\overline{A},D(\overline{A}))$ is bi-densely defined by Proposition 2.14 (b). Hence we get that $(T(t))_{t\geq 0}$ is a contraction semigroup by Theorem 3.9 with $\upsilon=\gamma$ and the norming property of $\Gamma_{\gamma}$ .

(c) It follows from part (b) that $\|T(t)\|_{\mathcal{L}(X)}\leq 1$ for all $t\geq 0$ . Further, $(T(t))_{t\geq 0}$ is $\gamma$ -equicontinuous by part (a). In combination with $\gamma=\gamma_{s}$ we derive that $(T(t))_{t\geq 0}$ is $(\|\cdot\|,\tau)$ -equitight by [18, Proposition 3.16, p. 12–13]. ∎

Let us compare Theorem 3.10 with one of the main theorems of [5], namely, [5, Theorem 3.15, p. 11]. We note that the topology that is called mixed topology (and denoted by $\gamma$ there) in [5, p. 10] is actually the submixed topology $\gamma_{s}$ . With this observation at hand let us phrase [5, Theorem 3.15, p. 11] in our terminology.

3.11 Theorem ([5, Theorem 3.15, p. 11]).

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space such that $(X,\gamma_{s})$ is complete, and $(A,D(A))$ a bi-densely defined, bi-dissipative operator. Assume that $\operatorname{Ran}(\lambda-A)$ is bi-dense, i.e. sequentially $\gamma$ -dense, in $X$ for some $\lambda>0$ . Then the $\gamma_{s}$ -closure $(\overline{A}^{\gamma_{s}},D(\overline{A}^{\gamma_{s}}))$ generates a $\tau$ -bi-continuous contraction semigroup on $X$ .

3.12 Remark.

(a)

First, it is actually shown in the proof of Theorem 3.11 that $(\overline{A}^{\gamma_{s}},D(\overline{A}^{\gamma_{s}}))$ generates a $\gamma_{s}$ -strongly continuous, $\gamma_{s}$ -equicontinuous semigroup on $X$ , in particular, a $\tau$ -bi-continuous semigroup by Remark 2.9 and Remark 2.5 (a). However, the proof that the generated semigroup is contractive is missing. In this proof it is used that a bi-dissipative operator is $({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}})_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}}$ -dissipative as well (see Remark 3.6). In order to prove that the generated semigroup is also contractive, the only available tool in [5] is [5, Theorem 3.6, p. 6]. However, to apply the latter theorem one has to show that $(\overline{A}^{\gamma_{s}},D(\overline{A}^{\gamma_{s}}))$ is also bi-dissipative. Due to [1, Proposition 3.13, p. 929] we only know that $(\overline{A}^{\gamma_{s}},D(\overline{A}^{\gamma_{s}}))$ is $({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(p_{n},a_{n})_{n\in\mathbb{N}}})_{(p_{n},a_{n})_{n\in\mathbb{N}}\in\mathcal{N}}$ -dissipative (and $\lambda-\overline{A}^{\gamma_{s}}$ is surjective for all $\lambda>0$ ). To circumvent this obstacle, we relaxed [5, Theorem 3.6, p. 6] to Theorem 3.9 where one has several possible choices for the topology $\upsilon$ , not only $\upsilon=\tau$ as in [5, Theorem 3.6, p. 6]. Using Remark 3.6 and [1, Proposition 3.13, p. 929], we see that $(\overline{A}^{\gamma_{s}},D(\overline{A}^{\gamma_{s}}))$ is $\Gamma_{\gamma_{s}}$ -dissipative w.r.t the norming directed system of continuous seminorms $\Gamma_{\gamma_{s}}$ from Remark 2.6. Now, it is possible to apply Theorem 3.9 with $\upsilon=\gamma_{s}$ to conclude that the generated semigroup is contractive. This closes the gap in the proof of Theorem 3.11.
(b)

There is no nice characterisation (known to us) of the completeness of $(X,\gamma_{s})$ , that is assumed in Theorem 3.11. However, there is a nice characterisation of the completeness of the Saks space $(X,\|\cdot\|,\tau)$ . By definition the Saks space is complete if and only if $(X,\gamma)$ is complete. The space $(X,\gamma)$ is complete if and only if $B_{\|\cdot\|}=\{x\in X\;|\;\|x\|\leq 1\}$ is $\tau$ -complete by [7, I.1.14 Proposition, p. 11]. But, since $\gamma_{s}$ is in general a weaker topology than $\gamma$ by Remark 2.5 (a), the completeness of $(X,\gamma)$ does in general not imply the completeness of $(X,\gamma_{s})$ .
(c)

Let us suppose that $\gamma=\gamma_{s}$ . Then Theorem 3.11 is covered by Theorem 3.10 (a) and (b). Further, we point out that in comparison to Theorem 3.11 we weakened the assumptions from $(A,D(A))$ being a bi-densely defined, bi-dissipative operator and $\operatorname{Ran}(\lambda-A)$ being bi-dense for some $\lambda>0$ to $(A,D(A))$ being a $\gamma$ -densely defined, $\Gamma_{\gamma}$ -dissipative operator and $\operatorname{Ran}(\lambda-A)$ being $\gamma$ -dense for some $\lambda>0$ (see Remark 3.2) in Theorem 3.10.

Let us take a closer look at the completeness assumption on the Saks space $(X,\|\cdot\|,\tau)$ in Theorem 3.10, which is actually fulfilled for many important examples, and its characterisation in Remark 3.12 (b). Especially, $(X,\gamma)$ is complete, thus $(X,\|\cdot\|,\tau)$ as well, if $B_{\|\cdot\|}$ is $\tau$ -compact, which is condition (ii) of Remark 2.5 (b) and also a sufficient condition for $\gamma=\gamma_{s}$ . We recall the following observations from [18, Examples 2.4, 3.11, p. 4–5, 10], [18, Remark 3.20 (a), p. 15], [18, Example 4.12, p. 24–25] and [18, Corollary 3.23, p. 17], and add a proof of the completeness of the Saks spaces considered in Remark 3.13 (c), (d) and (f) below.

3.13 Remark.

(a)

Let $\Omega$ be a Hausdorff $k_{\mathbb{R}}$ -space and recall that a completely regular space $\Omega$ is called $k_{\mathbb{R}}$ -space if any map $f\colon\Omega\to\mathbb{R}$ whose restriction to each compact $K\subset\Omega$ is continuous, is already continuous on $\Omega$ (see [27, p. 487]). Further, let $\mathrm{C}_{\operatorname{b}}(\Omega)$ be the space of bounded continuous functions on $\Omega$ , and $\|\cdot\|_{\infty}$ the sup-norm as well as $\tau_{\operatorname{co}}$ the compact-open topology, i.e. the topology of uniform convergence on compact subsets of $\Omega$ . Then $(\mathrm{C}_{\operatorname{b}}(\Omega),\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ is a complete Saks space and $\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})=\gamma_{s}(\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ .

Let $\mathcal{V}$ denote the set of all non-negative bounded functions $\nu$ on $\Omega$ that vanish at infinity, i.e. for every $\varepsilon>0$ the set $\{x\in\Omega\;|\;\nu(x)\geq\varepsilon\}$ is compact. Let $\beta_{0}$ be the Hausdorff locally convex topology on $\mathrm{C}_{\operatorname{b}}(\Omega)$ that is induced by the seminorms

$|f|_{\nu}\coloneqq\sup_{x\in\Omega}|f(x)|\nu(x),\quad f\in\mathrm{C}_{\operatorname{b}}(\Omega),$

for $\nu\in\mathcal{V}$ . Then we have $\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})=\beta_{0}$ . If $\Omega$ is locally compact, then $\mathcal{V}$ may be replaced by the functions in $\mathrm{C}_{0}(\Omega)$ that are non-negative where $\mathrm{C}_{0}(\Omega)$ is the space of real-valued continuous functions on $\Omega$ that vanish at infinity.

If $\Omega$ is a hemicompact Hausdorff $k_{\mathbb{R}}$ -space or a Polish space, then we even have

$\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})=\beta_{0}=\mu(\mathrm{C}_{\operatorname{b}}(\Omega),\mathrm{M}_{\operatorname{t}}(\Omega))$

where $\mathrm{M}_{\operatorname{t}}(\Omega)=(\mathrm{C}_{\operatorname{b}}(\Omega),\beta_{0})^{\prime}$ is the space of bounded Radon measures and $\mu(\mathrm{C}_{\operatorname{b}}(\Omega),\mathrm{M}_{\operatorname{t}}(\Omega))$ the Mackey-topology of the dual pair $(\mathrm{C}_{\operatorname{b}}(\Omega),\mathrm{M}_{\operatorname{t}}(\Omega))$ .
(b)

Let $(X,\|\cdot\|)$ be a Banach space and $\sigma^{\ast}\coloneqq\sigma(X^{\prime},X)$ the weak^∗-topology. Then condition (ii) of Remark 2.5 (b) is fulfilled, $(X^{\prime},\|\cdot\|_{X^{\prime}},\sigma^{\ast})$ is a complete Saks space and $\gamma(\|\cdot\|_{X^{\prime}},\sigma^{\ast})=\gamma_{s}(\|\cdot\|_{\infty},\sigma^{\ast})=\tau_{\operatorname{c}}(X^{\prime},X)$ where $\tau_{\operatorname{c}}(X^{\prime},X)$ is the topology of uniform convergence on compact subsets of $X$ .
(c)

Let $(X,\|\cdot\|)$ be a Banach space and $\mu^{\ast}\coloneqq\mu(X^{\prime},X)$ the dual Mackey-topology. Then $(X^{\prime},\|\cdot\|_{X^{\prime}},\mu^{\ast})$ is a complete Saks space, where the completeness follows from [17, p. 74], and $\gamma(\|\cdot\|_{X^{\prime}},\mu^{\ast})=\mu^{\ast}$ . If $X$ is a Schur space, i.e. every $\sigma(X,X^{\prime})$ -convergent sequence is $\|\cdot\|$ -convergent (see [10, p. 253]), then condition (ii) of Remark 2.5 (b) is fulfilled and $\gamma(\|\cdot\|_{X^{\prime}},\mu^{\ast})=\gamma_{s}(\|\cdot\|_{X^{\prime}},\mu^{\ast})$ .
(d)

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be Banach spaces, and $\tau_{\operatorname{sot}}$ the strong operator topology on $\mathcal{L}(X;Y)$ . Then $(\mathcal{L}(X;Y),\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})$ is a Saks space. Let $(T_{i})_{i\in I}$ be a $\tau_{\operatorname{sot}}$ -Cauchy net in $B_{\|\cdot\|_{\mathcal{L}(X;Y)}}=\{T\in\mathcal{L}(X;Y)\;|\;\|T\|_{\mathcal{L}(X;Y)}\leq 1\}$ . Then for each $x\in X$ the net $(T_{i}x)_{i\in I}$ is $\|\cdot\|_{Y}$ -convergent to some $Tx\in Y$ with $\|Tx\|_{Y}\leq\|x\|_{X}$ in the Banach space $(Y,\|\cdot\|_{Y})$ . Thus the map $T\colon x\mapsto Tx$ belongs to $\mathcal{L}(X;Y)$ with $\|T\|_{\mathcal{L}(X;Y)}\leq 1$ and $(T_{i})_{i\in I}$ is $\tau_{\operatorname{sot}}$ -convergent to $T$ . Hence $B_{\|\cdot\|_{\mathcal{L}(X;Y)}}$ is $\tau_{\operatorname{sot}}$ -complete and so $(\mathcal{L}(X;Y),\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})$ is complete. If $Y$ is in addition finite-dimensional, then condition (ii) of Remark 2.5 (b) is fulfilled and $\gamma(\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})=\gamma_{s}(\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})$ .

(e)

Let $H$ be a separable Hilbert space and $\mathcal{N}(H)$ the space of trace class operators in $\mathcal{L}(H)$ and note that $\mathcal{L}(H)=\mathcal{N}(H)^{\prime}$ . Let $\tau_{\operatorname{sot}^{\ast}}$ be the symmetric strong operator topology, i.e. the Hausdorff locally convex topology on $\mathcal{L}(H)$ generated by the directed system of seminorms

p_{N}(R)\coloneqq\max\bigl{(}\sup_{x\in N}\|Rx\|_{H},\sup_{x\in N}\|R^{\ast}x\|_{H}\bigr{)},\quad R\in\mathcal{L}(H),

for finite $N\subset H$ where $R^{\ast}$ is the adjoint of $R$ . We denote by $\beta_{\operatorname{sot}^{\ast}}$ the mixed topology $\gamma(\|\cdot\|_{\mathcal{L}(H)},\tau_{\operatorname{sot}^{\ast}})$ . Then the triple $(\mathcal{L}(H),\|\cdot\|_{\mathcal{L}(H)},\tau_{\operatorname{sot}^{\ast}})$ is a complete Saks space and $\beta_{\operatorname{sot}^{\ast}}=\mu(\mathcal{L}(H),\mathcal{N}(H))$ .

(f)

Let $\Omega$ be a completely regular Hausdorff space, $\mathrm{M}_{\operatorname{t}}(\Omega)$ the space of bounded Radon measures on $\Omega$ , and $\|\cdot\|_{\mathrm{M}_{\operatorname{t}}(\Omega)}$ the total variation norm on $\mathrm{M}_{\operatorname{t}}(\Omega)$ . Then $(\mathrm{M}_{\operatorname{t}}(\Omega),\|\cdot\|_{\mathrm{M}_{\operatorname{t}}(\Omega)},\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega)))$ is a complete Saks space where the completeness follows from $B_{\|\cdot\|_{\mathrm{M}_{\operatorname{t}}(\Omega)}}$ being $\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega))$ -compact by [18, Corollary 3.23 (a), p. 17], i.e. condition (ii) of Remark 2.5 (b) is fulfilled. Furthermore, we have

	$\displaystyle\beta_{0}^{\prime}\coloneqq$	$\displaystyle\gamma(\\|\cdot\\|_{\mathrm{M}_{\operatorname{t}}(\Omega)},\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega)))=\gamma_{s}(\\|\cdot\\|_{\mathrm{M}_{\operatorname{t}}(\Omega)},\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega)))$
	$\displaystyle=$	$\displaystyle\tau_{\operatorname{c}}(\mathrm{M}_{\operatorname{t}}(\Omega),(\mathrm{C}_{\operatorname{b}}(\Omega),\\|\cdot\\|_{\infty})).$

Let us consider a toy example for an application of Theorem 3.10, namely, the multiplication operator on $\mathrm{C}_{\operatorname{b}}(\Omega)$ , which we will revisit for other generation results.

3.14 Example.

Let $\Omega$ be a Hausdorff $k_{\mathbb{R}}$ -space and $q\colon\Omega\to\mathbb{C}$ be continuous with $C\coloneqq\sup_{x\in\Omega}\operatorname{Re}q(x)<\infty$ . We define the multiplication operator $(M_{q},D(M_{q}))$ by setting

D(M_{q})\coloneqq\{f\in\mathrm{C}_{\operatorname{b}}(\Omega)\;|\;qf\in\mathrm{C}_{\operatorname{b}}(\Omega)\}

and $M_{q}\coloneqq qf$ for $f\in D(M_{q})$ . By solving the equation $(\lambda-q)f=g$ we can compute the resolvent $R(\lambda,M_{q})$ of $M_{q}$ explicitly by

R(\lambda,M_{q})f=\frac{1}{\lambda-q}f,\quad f\in\mathrm{C}_{\operatorname{b}}(\Omega),

for all $\lambda\in(\mathbb{C}\setminus\overline{q(\Omega)})=\rho(M_{q})$ , which shows that $\lambda-M_{q}$ is surjective, i.e. $\operatorname{Ran}(\lambda-M_{q})=\mathrm{C}_{\operatorname{b}}(\Omega)$ , for all $\lambda\in\mathbb{C}\setminus\overline{q(\Omega)}$ . Suppose that $C\leq 0$ . Then $(0,\infty)\in\rho(M_{q})$ and $\operatorname{Ran}(\lambda-M_{q})=\mathrm{C}_{\operatorname{b}}(\Omega)$ for all $\lambda>0$ . Furthermore, we have for all $\lambda>0$ , $f\in\mathrm{C}_{\operatorname{b}}(\Omega)$ and $\nu\in\mathcal{V}$ from Remark 3.13 (a) that

	$\displaystyle\|R(\lambda,M_{q})f\|_{\nu}$	$\displaystyle=\sup_{x\in\Omega}\frac{1}{\|\lambda-q(x)\|}\|f(x)\|\nu(x)\underset{C\leq 0}{\leq}\sup_{x\in\Omega}\frac{1}{\lambda-\operatorname{Re}q(x)}\|f(x)\|\nu(x)$
		$\displaystyle\leq\frac{1}{\lambda}\sup_{x\in\Omega}\|f(x)\|\nu(x)=\frac{1}{\lambda}\|f\|_{\nu}.$

Therefore $(M_{q},D(M_{q}))$ is $\Gamma_{\beta_{0}}$ -dissipative for the directed system of seminorms $\Gamma_{\beta_{0}}\coloneqq(|\cdot|_{\nu})_{\nu\in\mathcal{V}}$ that generates the mixed topology $\beta_{0}=\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ . Moreover, due to Proposition 3.7 (b) and $\operatorname{Ran}(\lambda-M_{q})=\mathrm{C}_{\operatorname{b}}(\Omega)$ for all $\lambda>0$ the operator $(M_{q},D(M_{q}))$ is $\beta_{0}$ -closed and thus generates a $\beta_{0}$ -strongly continuous, $\beta_{0}$ -equicontinuous semigroup $(T(t))_{t\geq 0}$ on $\mathrm{C}_{\operatorname{b}}(\Omega)$ by Theorem 3.10 (a) and Remark 3.13 (a). Choosing $\mathcal{V}_{1}\coloneqq\{\nu\in\mathcal{V}\;|\;\forall\;x\in\Omega:\;\nu(x)\leq 1\}$ instead of $\mathcal{V}$ , we get a norming directed system of continuous seminorms that generates $\beta_{0}$ for which $(M_{q},D(M_{q}))$ is dissipative, too. Hence $(T(t))_{t\geq 0}$ is also a $(\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ -equitight contraction semigroup by Theorem 3.10 (b) and (c) since $\beta_{0}=\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})=\gamma_{s}(\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ by Remark 3.13 (a).

Our next generation result involves the $\gamma$ -dual operator. Let us recall some observations from [19, Remark 4.5, p. 9]. Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space. Then $X_{\gamma}^{\prime}\coloneqq(X,\gamma)^{\prime}$ is a closed linear subspace of $X^{\prime}$ , in particular a Banach space, by [7, I.1.18 Proposition, p. 15], and we denote by $\|\cdot\|_{X_{\gamma}^{\prime}}$ the restriction of $\|\cdot\|_{X^{\prime}}$ to $X_{\gamma}^{\prime}$ . We note that $(X,\gamma)$ is a Mazur space, i.e. $X_{\gamma}^{\prime}$ coincides with the space of linear $\gamma$ -sequentially continuous functionals on $X$ (see [32, p. 50]), if and only if

X_{\gamma}^{\prime}=\{x^{\prime}\in X^{\prime}\;|\;x^{\prime}\;\tau\text{-sequentially continuous on }\|\cdot\|\text{-bounded sets}\}\eqqcolon X^{\circ}

by [7, I.1.10 Proposition, p. 9]. The space $X^{\circ}$ was introduced in [13, Proposition 2.1, p. 314] in the context of dual semigroups of bi-continuous semigroups.

3.15 Corollary.

Let $(X,\|\cdot\|,\tau)$ be a complete Saks space. Let both $(A,D(A))$ and its $\gamma$ -dual operator $(A^{\prime},D(A^{\prime}))$ be $\Gamma_{\gamma}$ -dissipative and $\|\cdot\|_{X_{\gamma}^{\prime}}$ -dissipative operators, respectively. Then the following assertions hold:

(a)

The $\gamma$ -closure $(\overline{A},D(\overline{A}))$ generates a $\gamma$ -equicontinuous, $\gamma$ -strongly continuous semigroup $(T(t))_{t\geq 0}$ on $X$ .
(b)

If $\Gamma_{\gamma}$ is norming, then $(T(t))_{t\geq 0}$ is a contraction semigroup.
(c)

If $\Gamma_{\gamma}$ is norming and $\gamma=\gamma_{s}$ , then $(T(t))_{t\geq 0}$ is $(\|\cdot\|,\tau)$ -equitight.

Proof.

By [7, I.1.18 Proposition (i), p. 15] we have $(X_{\gamma}^{\prime},\tau_{b})=(X_{\gamma}^{\prime},\|\cdot\|_{X_{\gamma}^{\prime}})$ where $\tau_{b}$ denotes the topology of uniform convergence on $\gamma$ -bounded sets. Due to [1, Corollary 3.17, p. 931] we get that $(\overline{A},D(\overline{A}))$ generates a $\gamma$ -strongly continuous, $\gamma$ -equicontinuous semigroup on $X$ . Parts (b) and (c) follow as in Theorem 3.10. ∎

3.16 Example.

Let $\Omega\coloneqq\mathbb{N}$ be equipped with the metric induced by the absolute value. Then $\Omega$ is a Polish space, in particular, a Hausdorff $k_{\mathbb{R}}$ -space. Moreover, $\mathrm{C}_{\operatorname{b}}(\mathbb{N})=\ell^{\infty}$ and $\mathrm{M}_{\operatorname{t}}(\mathbb{N})=\ell^{1}$ (see e.g. [6, p. 477]). It follows from Remark 3.13 (a) that $\beta_{0}=\mu(\ell^{\infty},\ell^{1})$ and so

(\ell^{\infty},\beta_{0})^{\prime}=(\ell^{\infty},\mu(\ell^{\infty},\ell^{1}))^{\prime}=\ell^{1}.

Let $q\colon\mathbb{N}\to\mathbb{C}$ be a function with $C\coloneqq\sup_{n\in\mathbb{N}}\operatorname{Re}q(n)\leq 0$ . Again, we consider the multiplication operator $M_{q}$ from Example 3.14, i.e.

D(M_{q})=\{f\in\ell^{\infty}\;|\;qf\in\ell^{\infty}\}

and $M_{q}=qf$ for $f\in D(M_{q})$ . We already know that $(M_{q},D(M_{q}))$ is $\Gamma_{\beta_{0}}$ -dissipative with $\Gamma_{\beta_{0}}$ from Example 3.14. Furthermore, we have for the $\beta_{0}$ -dual operator $(M_{q}^{\prime},D(M_{q}^{\prime}))$ that

D(M_{q}^{\prime})=\{f\in\ell^{1}\;|\;qf\in\ell^{1}\}

and $M_{q}^{\prime}=qf$ for $f\in D(M_{q}^{\prime})$ . For all $\lambda>0$ and $f\in D(M_{q}^{\prime})$ we get

\|(\lambda-M_{q}^{\prime})f\|_{\ell^{1}}=\sum_{n=1}^{\infty}|(\lambda-q(n))f_{n}|\underset{C\leq 0}{\geq}\sum_{n=1}^{\infty}(\lambda-\operatorname{Re}q(n))|f_{n}|\geq\lambda\sum_{n=1}^{\infty}|f_{n}|=\lambda\|f\|_{\ell^{1}},

meaning that $(M_{q}^{\prime},D(M_{q}^{\prime}))$ is $\|\cdot\|_{\ell^{1}}$ -dissipative. Thus we may apply Corollary 3.15 (a) to deduce that $(M_{q},D(M_{q}))$ generates a $\mu(\ell^{\infty},\ell^{1})$ -strongly continuous, $\mu(\ell^{\infty},\ell^{1})$ -equicontinuous semigroup on $\ell^{\infty}$ .

Instead of Remark 3.13 (a) we may also use Remark 3.13 (c) in Example 3.16 since $\ell^{1}$ is a Schur space by [10, Theorem 5.36, p. 252].

Next, we would like to transfer [1, Theorem 3.18, p. 931] to the setting of Saks spaces $(X,\|\cdot\|,\tau)$ . However, looking at the assumptions of [1, Theorem 3.18, p. 931], we see that this requires $(X,\gamma)$ to be reflexive. Since reflexive spaces are barrelled, this requirement implies that $\tau=\gamma=\tau_{\|\cdot\|}$ by [7, I.1.15 Proposition, p. 12] and so we are in an uninteresting situation from the perspective of $\tau$ -bi-continuous semigroups. But if we could relax the assumption to $(X,\gamma)$ being semi-reflexive, then there are non-trivial (i.e. not Banach) Saks spaces. This is actually possible by the following observation.

3.17 Remark.

[1, Theorem 3.18, p. 931] is stated for reflexive Hausdorff locally convex spaces $(X,\upsilon)$ . However, a closer look at its proof reveals that it is actually valid for semi-reflexive $(X,\upsilon)$ because the only part where reflexivity comes into play is that it implies that a $\upsilon$ -bounded set $B\subset X$ is relatively $\sigma(X,(X,\upsilon)^{\prime})$ -compact; see [1, p. 931, l. 9–10 from below]. But the latter assertion is equivalent to semi-reflexivity by [26, Proposition 23.18, p. 270].

3.18 Theorem.

Let $(X,\|\cdot\|,\tau)$ be a complete, semi-reflexive Saks space, $(A,D(A))$ a $\gamma$ -densely defined, $\Gamma_{\gamma}$ -dissipative operator and $\operatorname{Ran}(\lambda-A)=X$ for some $\lambda>0$ . Then the following assertions hold:

(a)

$(A,D(A))$ generates a $\gamma$ -equicontinuous, $\gamma$ -strongly continuous semigroup $(T(t))_{t\geq 0}$ on $X$ .
(b)

If $\Gamma_{\gamma}$ is norming, then $(T(t))_{t\geq 0}$ is a contraction semigroup.
(c)

If $\Gamma_{\gamma}$ is norming and $\gamma=\gamma_{s}$ , then $(T(t))_{t\geq 0}$ is $(\|\cdot\|,\tau)$ -equitight.

Proof.

Part (a) follows from [1, Theorem 3.18, p. 931] (noting that $1-A$ is surjective by Proposition 3.7 (d)) and Remark 3.17. Parts (b) and (c) follow as in Theorem 3.10. ∎

Let $(X,\|\cdot\|,\tau)$ be a Saks space. By definition the Saks space is semi-reflexive if and only if $(X,\gamma)$ is semi-reflexive. The space $(X,\gamma)$ is semi-reflexive if and only if $B_{\|\cdot\|}=\{x\in X\;|\;\|x\|\leq 1\}$ is $\sigma(X,(X,\tau)^{\prime})$ -compact by [7, I.1.21 Corollary, p. 16]. Due to [7, I.1.20 Proposition, p. 16], $B_{\|\cdot\|}$ is $\sigma(X,(X,\tau)^{\prime})$ -compact if and only if it is $\sigma(X,(X,\gamma)^{\prime})$ -compact. Further, $(X,\gamma)$ is a semi-Montel space, thus semi-reflexive, if and only if $B_{\|\cdot\|}$ is $\tau$ -compact by [7, I.1.13 Proposition, p. 11] which is condition (ii) in Remark 2.5 (b) again and also a sufficient condition for $\gamma=\gamma_{s}$ . Therefore we have by Remark 3.13 the following observations where we only have to add an additional argument in parts (a), (c) and (e) of Remark 3.19 below.

3.19 Remark.

(a)

Let $\Omega$ be a discrete space. Then $(\mathrm{C}_{\operatorname{b}}(\Omega),\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ is a complete, semi-reflexive Saks space by [7, II.1.24 Remark 4), p. 88–89].
(b)

Let $(X,\|\cdot\|)$ be a Banach space. Then $(X^{\prime},\|\cdot\|_{X^{\prime}},\sigma^{\ast})$ is a complete, semi-reflexive Saks space.
(c)

Let $(X,\|\cdot\|)$ be a Banach space. Then $(X^{\prime},\|\cdot\|_{X^{\prime}},\mu^{\ast})$ is a complete, semi-reflexive Saks space where the semi-reflexivity follows from $(X^{\prime},\mu^{\ast})^{\prime\prime}=X^{\prime}$ by the Mackey–Arens theorem.
(d)

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be Banach spaces and $Y$ finite-dimensional. Then $(\mathcal{L}(X;Y),\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})$ is a complete, semi-reflexive Saks space.
(e)

Let $H$ be a separable Hilbert space. Then $(\mathcal{L}(H),\|\cdot\|_{\mathcal{L}(H)},\tau_{\operatorname{sot}^{\ast}})$ is a complete, semi-reflexive Saks space where the semi-reflexivity follows from $(\mathcal{L}(H),\beta_{\operatorname{sot}^{\ast}})^{\prime\prime}=\mathcal{N}(H)^{\prime}=\mathcal{L}(H)$ .
(f)

Let $\Omega$ be a completely regular Hausdorff space. Then we have that the triple $(\mathrm{M}_{\operatorname{t}}(\Omega),\|\cdot\|_{\mathrm{M}_{\operatorname{t}}(\Omega)},\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega)))$ is a complete, semi-reflexive Saks space.

3.20 Example.

Due to Example 3.16 and Remark 3.19 (a) $(\ell^{\infty},\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ is a complete, semi-reflexive Saks space. Therefore we may also apply Theorem 3.18 (a) to prove that the multiplication operator $(M_{q},D(M_{q}))$ with $\sup_{n\in\mathbb{N}}\operatorname{Re}q(n)\leq 0$ generates a $\mu(\ell^{\infty},\ell^{1})$ -strongly continuous, $\mu(\ell^{\infty},\ell^{1})$ -equicontinuous semigroup on $\ell^{\infty}$ (we already checked in Example 3.14 that the other assumptions of Theorem 3.18 are satisfied).

We close this section with a characterisation of the bi-continuous semigroups with dissipative generators. First, we start with a refinement of Theorem 3.9. In the case $\upsilon=\tau$ this was already done in [5, Proposition 3.11, p. 9] whose prove needs some adaptations in the case of more general Hausdorff locally convex topologies $\upsilon$ with $\tau\subseteq\upsilon\subseteq\tau_{\|\cdot\|}$ for sequentially complete Saks spaces $(X,\|\cdot\|,\tau)$ .

3.21 Proposition.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space, $\upsilon$ a Hausdorff locally convex topology on $X$ with $\tau\subseteq\upsilon\subseteq\tau_{\|\cdot\|}$ such that $\gamma$ -convergent sequences are $\upsilon$ -convergent, and $(A,D(A))$ bi-densely defined. Then the following assertions are equivalent:

(a)

$(A,D(A))$ generates a $\tau$ -bi-continuous contraction semigroup $(T(t))_{t\geq 0}$ on $X$ and there exists a norming directed system of continuous seminorms $\Gamma_{\upsilon}$ that generates $\upsilon$ such that $p(T(t)x)\leq p(x)$ for all $t\geq 0$ , $p\in\Gamma_{\upsilon}$ and $x\in X$ .
(b)

$\lambda-A$ is surjective for some $\lambda>0$ and $(A,D(A))$ is a $\Gamma_{\upsilon}$ -dissipative operator on $X$ for some norming directed system of continuous seminorms $\Gamma_{\upsilon}$ that generates $\upsilon$ .

Proof.

(a) $\Rightarrow$ (b): First, we show that $(A,D(A))$ is $\Gamma_{\upsilon}$ -dissipative. We note that $(0,\infty)\subseteq\rho(A)$ and

R(\lambda,A)x=\int_{0}^{\infty}\mathrm{e}^{-\lambda t}T(t)x\mathrm{d}t

for all $\lambda>0$ and $x\in X$ by [23, Theorem 12, p. 215] and [23, Definition 9, p. 213] where the integral is an improper $\tau$ -Riemann integral. The sequence of Riemann sums that approximate the integral on the right-hand side w.r.t. $\tau$ are $\|\cdot\|$ -bounded for each $\lambda>0$ and $x\in X$ . Due to Remark 2.9 (a) this means that this sequence of Riemann sums is actually $\gamma$ -convergent and thus $\upsilon$ -convergent by assumption. Therefore we have for all $\lambda>0$ , $p\in\Gamma_{\upsilon}$ and $x\in X$ that

p\left(R(\lambda,A)x\right)\leq\int_{0}^{\infty}\mathrm{e}^{-\lambda t}p(T(t)x)\mathrm{d}t\leq\int_{0}^{\infty}\mathrm{e}^{-\lambda t}p(x)\mathrm{d}t=\frac{1}{\lambda}p(x)

where we used that $p$ is $\upsilon$ -continuous for the first inequality. Hence $(A,D(A))$ is $\Gamma_{\upsilon}$ -dissipative. In combination with Theorem 3.9 this yields that $\lambda-A$ is surjective for some $\lambda>0$ .

(b) $\Rightarrow$ (a): Due to Theorem 3.9, $(A,D(A))$ generates a $\tau$ -bi-continuous contraction semigroup on $X$ , and $(0,\infty)\subseteq\rho(A)$ by Proposition 3.7 (c). Furthermore, we have by the Post–Widder inversion formula [22, Corollary 2.10, p. 47] that

T(t)x=\tau\text{-}\lim_{n\to\infty}\left(\frac{n}{t}R\left(\frac{n}{t},A\right)\right)^{n}x

for all $t>0$ and $x\in X$ . As a consequence of Remark 3.5 the $\tau$ -convergent sequence $((\tfrac{n}{t}R(\tfrac{n}{t},A))^{n}x)_{n\in\mathbb{N}}$ is $\|\cdot\|$ -bounded for each $t>0$ and $x\in X$ , thus $\gamma$ -convergent by Remark 2.9 (a) and so $\upsilon$ -convergent by assumption. We deduce from (2) that for all $t>0$ , $p\in\Gamma_{\upsilon}$ and $x\in X$ it holds that

p(T(t)x)=\lim_{n\to\infty}\left(\frac{n}{t}\right)^{n}p\left(R\left(\frac{n}{t},A\right)^{n}x\right)\underset{\eqref{eq:res_cont}}{\leq}p(x)

where we used that $p$ is $\upsilon$ -continuous for the first equality. Further, for $t=0$ we have $p(T(t)x)=p(x)$ . We conclude that statement (a) holds. ∎

The assumptions of Proposition 3.21 (a) are up to rescaling fulfilled for any $\tau$ -bi-continuous semigroup on $X$ if it is $\upsilon$ -equicontinuous for $\upsilon=\tau$ or $\gamma_{s}$ or $\gamma$ (the assumption on $\upsilon$ -equicontinuity may fail if $\upsilon=\tau$ by Remark 3.25).

3.22 Remark.

Let $(X,\|\cdot\|,\tau)$ be a sequentially complete Saks space and $\upsilon=\tau$ , $\gamma_{s}$ or $\gamma$ . Then there exists a norming directed system of continuous seminorms that generates the topology $\upsilon$ by Definition 2.1 (c) for $\upsilon=\tau$ , by Remark 2.6 for $\upsilon=\gamma_{s}$ and by Remark 2.3 for $\upsilon=\gamma$ . Further, let $(T(t))_{t\geq 0}$ be a $\tau$ -bi-continuous, $\upsilon$ -equicontinuous semigroup on $X$ and $\omega\in\mathbb{R}$ be its type (see Definition 2.8 (iii)). By modifying the proof of [5, Remark 3.12, p. 9–10] one can show that for the $\tau$ -bi-continuous, $\upsilon$ -equicontinuous contraction semigroup $(e^{-\omega t}T(t))_{t\geq 0}$ on $X$ there exists a norming directed system of continuous seminorms $\Gamma_{\upsilon}$ that generates $\upsilon$ such that $p(e^{-\omega t}T(t)x)\leq p(x)$ for all $t\geq 0$ , $p\in\Gamma_{\upsilon}$ and $x\in X$ .

In addition, we have the following characterisation in the case of complete, C-sequential Saks spaces $(X,\|\cdot\|,\tau)$ and $\upsilon=\gamma$ .

3.23 Proposition.

Let $(X,\|\cdot\|,\tau)$ be a complete, C-sequential Saks space and $(A,D(A))$ the generator of a $\tau$ -bi-continuous semigroup $(T(t))_{t\geq 0}$ on $X$ . Then the following assertions are eqivalent:

(a)

$(T(t))_{t\geq 0}$ is $\gamma$ -equicontinuous.
(b)

There is a directed system of continuous seminorms $\Gamma_{\gamma}$ that generates the mixed topology $\gamma$ such that $(A,D(A))$ is $\Gamma_{\gamma}$ -dissipative.

Proof.

By Remark 2.11 we know that $(T(t))_{t\geq 0}$ is quasi- $\gamma$ -equicontinuous if $(X,\|\cdot\|,\tau)$ is a sequentially complete C-sequential Saks space. Hence the equivalence of the assertions (a) and (b) follows from [1, Propositions 4.2, 4.4, p. 933, 935]. ∎

The condition that $(X,\|\cdot\|,\tau)$ is a complete, C-sequential Saks space is quite often fulfilled, e.g. for the examples from Remark 3.13 under some minor constraints by [18, Remark 3.19, p. 14], [18, Remark 3.20 (c), p. 15], [18, Example 4.12, p. 24–25] and [18, Corollary 3.23 (b), p. 17].

3.24 Remark.

(a)

Let $\Omega$ be a hemicompact Hausdorff $k_{\mathbb{R}}$ -space or a Polish space. Then $(\mathrm{C}_{\operatorname{b}}(\Omega),\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ is a complete, C-sequential Saks space.
(b)

Let $(X,\|\cdot\|)$ be a separable Banach space. Then $(X^{\prime},\|\cdot\|_{X^{\prime}},\sigma^{\ast})$ is a complete, C-sequential Saks space.
(c)

Let $(X,\|\cdot\|)$ be an SWCG space (see [30, p. 387]), or a sequentially $\sigma(X,X^{\prime})$ -complete space with an almost shrinking basis (see [17, p. 75]). Then the triple $(X^{\prime},\|\cdot\|_{X^{\prime}},\mu^{\ast})$ is a complete, C-sequential Saks space.
(d)

Let $(X,\|\cdot\|_{X})$ be a separable Banach space and $(Y,\|\cdot\|_{Y})$ a Banach space. Then $(\mathcal{L}(X;Y),\|\cdot\|_{\mathcal{L}(X;Y)},\tau_{\operatorname{sot}})$ is a complete, C-sequential Saks space.
(e)

Let $H$ be a separable Hilbert space. Then $(\mathcal{L}(H),\|\cdot\|_{\mathcal{L}(H)},\tau_{\operatorname{sot}^{\ast}})$ is a complete, C-sequential Saks space.
(f)

Let $\Omega$ be a Polish space. Then $(\mathrm{M}_{\operatorname{t}}(\Omega),\|\cdot\|_{\mathrm{M}_{\operatorname{t}}(\Omega)},\sigma(\mathrm{M}_{\operatorname{t}}(\Omega),\mathrm{C}_{\operatorname{b}}(\Omega)))$ is a complete, C-sequential Saks space.

3.25 Remark.

Let $(X,\|\cdot\|,\tau)$ be a complete, C-sequential Saks space. Due to Remark 2.11 assertion (a) of Proposition 3.23 always holds up to rescaling, and thus for any $\tau$ -bi-continuous semigroup $(T(t))_{t\geq 0}$ on $X$ there is a rescaling such that the generator of the rescaled semigroup is $\Gamma_{\gamma}$ -dissipative for some system of continuous seminorms $\Gamma_{\gamma}$ that generates the mixed topology $\gamma$ .

On the other hand, there are important examples of $\tau$ -bi-continuous semigroups that are not quasi- $\tau$ -equicontinuous. For instance, the Gauß–Weierstraß semigroup on the complete, C-sequential Saks space $(\mathrm{C}_{\operatorname{b}}(\mathbb{R}^{d}),\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ is $\tau_{\operatorname{co}}$ -bi-continuous but not locally $\tau_{\operatorname{co}}$ -equicontinuous by [23, Examples 6 (a), p. 209–210]. Since there is some $\lambda>0$ such that $\lambda-A$ is surjective for the generator $(A,D(A))$ of the Gauß–Weierstraß semigroup by [23, Lemma 7, Proposition 8, Theorem 12, p. 211–212, 215], it follows from Proposition 3.21 with $\upsilon=\tau_{\operatorname{co}}$ that its generator $(A,D(A))$ is not bi-dissipative (even after rescaling, cf. [5, Example 3.9, p. 8]). Another example of a $\tau_{\operatorname{co}}$ -bi-continuous semigroup which has no bi-dissipative generator (even after rescaling) is the left translation semigroup on the complete, C-sequential Saks space $(\mathrm{C}_{\operatorname{b}}(\mathbb{R}),\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ which is $\tau_{\operatorname{co}}$ -bi-continuous, even locally $\tau_{\operatorname{co}}$ -equicontinuous but not quasi- $\tau_{\operatorname{co}}$ -equicontinuous by [23, Examples 6 (b), p. 209–210] and [24, Example 3.2, p. 549].

Nevertheless, the Gauß–Weierstraß semigroup and the left translation semigroup are quasi- $\beta_{0}$ -equicontinuous by Remark 2.11 and Remark 3.24 (a), and thus both have (after rescaling) a $\Gamma_{\beta_{0}}$ -dissipative generator for some system of continuous seminorms $\Gamma_{\beta_{0}}$ that generates the mixed topology $\beta_{0}=\gamma(\|\cdot\|_{\infty},\tau_{\operatorname{co}})$ . This underlines that in the framework of $\tau$ -bi-continuous semigroups the concept of a bi-dissipative operator resp. generator is not the correct choice whereas the concept of a $\Gamma_{\gamma}$ -dissipative operator resp. generator is the more reasonable one.

Acknowledgement

We would like to thank David Jornet for his kind explanations regarding the proof of [1, Proposition 3.11 (iii), p. 927].

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