Relatively uniformly continuous semigroups on vector lattices
In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology . We introduce the notions of strong continuity with respect to and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as for and spaces such as and . We provide the general framework and standard constructions for such semigroups. We also introduce a version of a Banach-Steinhaus result for relative uniform convergence which holds true for important classes of vector lattices. By using this we prove relative uniform continuity of the (left) translation semigroup on .
Key words and phrases:Vector lattices, relative uniform convergence, relative uniform topology, relative uniform continuity, strongly continuous semigroups
2010 Mathematics Subject Classification:46A40, 47D06, 46B42.
1. Introduction and preliminaries
In the 1940s, E. Hille [Hil42, Hil48] and K. Yosida [Yos48] introduced the theory of strongly continuous semigroups on Banach spaces in order to treat evolution equations. By now, their theory is well established, and its applications reach well beyond the classical field of partial differential equations. However, from the very beginning many situations occurred in which the underlying space is not a Banach space. In order to deal with such phenomena, already I. Miyadera [Miy59], H. Komatsu [Kom64], K. Yosida [Yos65], K. Singbal-Vedak [SV65], T. Komura [K6̄8], S. Ouchi [Ō73] and others generalized the theory to strongly continuous semigroups on locally convex spaces. Also, strongly continuous semigroups of positive operators on Banach lattices have been discussed in [BR84], [AGG86], [BKR17] and others.
Our purpose is to provide a general framework for the theory of strongly continuous semigroups on vector lattices. Although vector lattices themselves are initially order and algebraic theoretical construct, they admit topologies which arise purely from order. The natural question that appears is whether one can study dynamical systems on general vector lattices. Since we want that our notion of strong continuity of semigroups on general vector lattices agrees with strong continuity for semigroups on Banach lattices, relative uniform topology seems to be the correct choice. This allows us to consider semigroups on non-Banach spaces, such as , , , and even on non locally convex spaces, such as for . We discuss two types of continuity of semigroups on vector lattices: the strong continuity with respect to and the relative uniform continuity. The former notion is defined by -convergence and the latter by relative uniform convergence.
This paper is structured as follows. In Section 2 we consider some general properties of and we provide examples of vector lattices together with corresponding relative uniform topologies. In Section 3 we introduce notions of -strongly continuous semigroups and relatively uniformly continuous semigroups. While relative uniform continuity implies -strong continuity, in general, these notions do not coincide as it is shown by the (left) translation semigroup on . In the rest of Section 3 we present how one can lift strong continuity with respect to from a -dense set to the whole space. Parallel theory for relatively uniformly continuous semigroups is considered in Section 5. A comparison between vector lattice case and Banach space case reveals that general vector lattices lack some property related to the principle of uniform boundedness. We introduce such property and call it “relative uniform Banach-Steinhaus property”. We prove that many important vector lattices posses it. This property enables us to provide the extension theorem for relatively uniformly continuous semigroups. In Section 4 we study standard constructions of new semigroups from a given one.
Let us now recall some preliminary facts and notation that are needed throughout the text. A family of linear operators on a vector space is a semigroup if it satisfies the functional equation
for all and . If is a Banach space, then we call a semigroup a -semigroup or strongly continuous on when the operator is bounded for each and for each the orbit map
is continuous with respect to the Euclidean topology on and the norm topology on . If is a vector lattice and is a positive operator on for each , then the semigroup is called a positive semigroup.
If is a linear topology on , then a net of linear operators is -equicontinuous when for each -neighborhood of zero there exists another -neighborhood of zero such that for all . If for each the family of operators is -equicontinuous, then is locally -equicontinuous. If for , and a topology on we have as and as , then we write “ as for ”.
A net is relatively uniformly convergent to if there exists some such that for each there exists such that holds for all . We call such an element a regulator of and we write . It is well-known that limits of relatively uniformly convergent sequences in are unique if and only if is Archimedean. Throughout this paper, stands for an Archimedean vector lattice unless specified otherwise.
2. Relative uniform topology
A subset of is called relatively uniformly closed whenever and imply . By [LM67, Section 3], the relatively uniformly closed sets are exactly the closed sets of a certain topology in , the relative uniform topology which we denote by . This topology has been first studied by W.A.J. Luxemburg and L.C. Moore in [LM67]; see also [Moo68]. When a net converges to in , we write . Since is Archimedean, the topological space satisfies the -separation axiom.
The following proposition yields that if one starts by defining closed sets through nets, one ends up with the same topology.
A subset of is relatively uniformly closed if and only if for each net and with we have .
It suffices to prove the “only if” statement. Fix a relatively uniformly closed set , and a net satisfying with respect to some regulator . We show that . For each pick any index such that . Then , and since is relatively uniformly closed, we conclude . ∎
We proceed with various examples of important vector lattices together with their relative uniform topologies and convergences which will be needed throughout the paper.
On a vector lattice with an order unit the relative uniform topology is generated by the norm
since if and only if . Such vector lattices are -complete if and only if they are uniformly complete.
It is well-known that in a completely metrizable locally solid vector lattice every convergent sequence has a subsequence which converges relatively uniformly to the same limit, see [AT07, Lemma 2.30]. This immediately yields that a subset of is relatively uniformly closed if and only if it is -closed, so that topologies and agree. In particular, if is a Banach lattice, then agrees with norm topology.
For the vector lattice equipped with the topology induced by the metric
is a completely metrizable locally solid vector lattice which is not locally convex.
The vector lattice equipped with the topology of uniform convergence on compact sets is a completely metrizable locally convex solid vector lattice.
In the following proposition we characterize relative uniform convergence in .
A net converges relatively uniformly to if and only if and there exists a compact set and such that for all .
() Fix . There exist , independent of , and such that
for all . This immediately implies that
for all and hence, and for all where is the compact support of the function .
() To construct a regulator , pick compact sets and such that and hold for all and set . Then and since we also have for all . By assumption, for each there exists such that holds for all . Hence, for any we have
Now it is easy to see that any positive function with for all regulates the convergence . ∎
By creftypecap 2.3, a set is relatively uniformly closed if and only if for and the existence of a compact set such that for all and imply .
If a vector lattice has an order unit , creftypecap 2.2(a) yields that on agrees with the norm topology induced by the norm . The following proposition shows that vector lattices of Lipschitz continuous functions and uniformly continuous functions on the real line posses order units.
The function is an order unit of vector lattices and .
Since , it suffices to prove that is an order unit of . To prove this, fix and find such that whenever .
Pick any . There exist and such that . Then
where . The case when can be treated similarly. ∎
It is well-known that implies , see [LM67, Section 3]. While the backward implication is not true in general, for sequences convergence is equivalent to the following. A sequence is relatively uniformly -convergent to if every subsequence of contains a further subsequence that is relatively uniformly convergent to . Similarly, a net is relatively uniformly -convergent to if every subnet of contains a further subnet that is relatively uniformly convergent to . We write if a net or a sequence relatively uniformly -converges to .
It is clear that relative uniform convergence implies relative uniform -convergence in case of sequences and nets. Moreover, relative uniform -convergence is tightly connected to -convergence. By [LM67, Theorem 3.5], is equivalent to . The natural question that appears here is what happens when one replaces sequences by nets. The following proposition shows that relative uniform -convergence always implies -convergence. On the other hand, creftypecap 2.6 will show that the converse implication, in general, is not true.
If , then
We first consider the special case when Fix an open -neighborhood for and with with respect to a regulator .
We claim that there exists such that implies . Assume otherwise. Then for each there exists such that and . From we conclude which is a contradiction to for all . Hence, there exists such that implies . Since , there exists such that holds for all and hence, we have for all
For the general case, assume that while . Then there exists an open -neighborhood of such that for each there exists with . We claim that is a subnet of . For each and find such that and take . Hence is a net and by construction of it is a subnet of . By assumption, there exists a subnet of which converges relatively uniformly to . This subnet necessarily -converges to . This is a contradiction to for all . ∎
Consider the first uncountable ordinal . It is well-known that is an uncountable well-ordered set and all countable subsets of have suprema. This immediately yields that no cofinal subset of is countable.
Let be the vector lattice of all real functions on with countable support. By [Moo68, Example 2.2], the relative uniform topology on is the topology of pointwise convergence. Consider the net in where is the characteristic function of . It is clear that converges pointwise to .
Assume that there exists a subnet of such that . Then there exists and such that for all . Hence, for all we have . Since has no countable cofinal subsets, the set is uncountable, so that the support of is uncountable. This is absurd.
3. Semigroups on
In this section we introduce two notions of continuity for semigroups on general vector lattices. By providing an example we show that these notions truly differ. We show that these notions expand semigroup theory to spaces which are not locally convex or complete. Furthermore, we provide conditions under which it is enough to check continuity of a semigroup on a -dense set to obtain continuity on the whole vector lattice.
A semigroup on is strongly continuous with respect to or -strongly continuous if for each the orbit map is continuous, i.e.
for each as . If, in addition, we have
for each and as , then is relatively uniformly continuous. Since relative uniform convergence implies -convergence, every relatively uniformly continuous semigroup is -strongly continuous. In the special case when is a Banach lattice, a positive semigroup is -strongly continuous on if and only if it is a positive -semigroup on .
For a function and , we consider the (left) translation operator
of by . It is evident that by fixing a translation invariant space of functions on one obtains a semigroup on which we call the (left) translation semigroup on .
The (left) translation semigroup is relatively uniformly continuous on , and .
We first prove that the translation semigroup is relatively uniformly continuous on by applying creftypecap 2.3. Fix , and . Since is uniformly continuous on , there exists such that for all This proves in as
Since , there exists such that Choose any . If , then a direct computation shows that .
We consider the remaining two cases simultaneously. Fix , and . Since is uniformly continuous, there exists such that whenever . This implies that for each and with we have . To finish the proof note that constant functions are Lipschitz continuous. ∎
In the remaining part of this section we will weaken the hypothesis under which the semigroup is still -strongly continuous or relatively uniformly continuous. The motivation comes from the general theory of -semigroups on Banach spaces. As it is shown in [EN00, Proposition I.5.3], a semigroup on a Banach space is a -semigroup if and only if it is norm bounded on bounded time intervals and the orbit maps are continuous on a norm dense set of elements of . This result heavily relies on the principle of uniform boundedness which is unavailable in general vector lattices. The following theorem is a vector lattice version of the above result for strong continuity with respect to in the case when is a linear topology. By [Moo68, Theorem 2.1], the topology is linear whenever it is first countable.
If is a linear topology on , then a semigroup on is -strongly continuous if and only if for each the following two assertions hold.
There exists a -dense subset of such that as for each .
For each net with and each open -neighborhood of zero there exists and such that
holds for all when and all when .
Since the forward implication is clear we only prove the backward implication.
Fix an open -neighborhood of zero and take any open -neighborhood of zero such that . Fix and . By (i) there exists a net such that and as for each . Hence, by (ii) there exist and such that
hold for all when and all when . Therefore,
holds for all when and all when . This proves that on is -strongly continuous. ∎
In Section 5 we will establish an analogous version of creftypecap 3.2 for relatively uniformly continuous semigroups on a particular class of vector lattices which allow a version of the principle of uniform boundedness.
The importance of the following corollary lies in its applicability. For locally -equicontinuous semigroups, -strong continuity is equivalent to -strong continuity at zero.
Let be a linear topology on and a locally -equicontinuous semigroup on . Then is -strongly continuous if and only if there exists a -dense subset of such that as for each .
It suffices to prove the “only if” statement. We will check (i) and (ii) from creftypecap 3.2. Fix and an open -neighborhood of zero . There exists an open -neighborhood of zero such that . Since is a locally -equicontinuous semigroup on , there exists a symmetric open -neighborhood of zero such that for all .
(i) Assume that there exists a -dense subset of such that for each we have as . Fix . There exists such that for all . Then
holds for all for all and
holds for all when . This proves (i).
(ii) Pick a net with and find such that . If , then . If , then again . Hence, if satisfies , we have
which completes the proof. ∎
The following proposition shows that the notion of relative uniform continuity is, in general, stronger than the notion of strong continuity with respect to . Furthermore, it also provides an example of a completely metrizable locally solid vector lattice that is not locally convex and a -strongly continuous semigroup on which is not relatively uniformly continuous.
Having in mind that on Banach lattices agrees with norm topology, in the case the following proposition recovers [EN00, Example I.5.4].
For each the (left) translation semigroup on is -strongly continuous but not relatively uniformly continuous.
Pick and denote by the topology induced by the metric on . Since is a completely metrizable locally solid vector lattice, and agree on by creftypecap 2.2(c). The same arguments as in the classical sense show that is dense in .
Since is relatively uniformly continuous on by creftypecap 3.1, it is also -strongly continuous. Furthermore, for each the operator preserves every open -ball with center at zero, from where it follows that the semigroup is locally -equicontinuous on . By creftypecap 3.3, we conclude that is -strongly continuous on .
To show that is not relatively uniformly continuous on , consider the function
in . Assume that there exist a function and such that holds in for all , i.e.,
holds for all and almost every . Hence, the family is bounded above and since is Dedekind complete, there exists in . This is impossible since attains infinity on a set of positive measure. ∎
The remaining part of this section is devoted to relatively uniformly continuous semigroups of positive operators. Our goal is to obtain a version (see creftypecap 3.6) of creftypecap 3.3 for relatively uniformly continuous semigroups. In the case of -strongly continuous semigroups, we were able to provide such a result only for locally -equicontinuous semigroups on . The reason behind is not so surprising. Consider a semigroup of bounded operators on a Banach lattice. Since agrees with norm topology, local -equicontinuity agrees with local equicontinuity which is equivalent to uniform boundedness of the semigroup on bounded time intervals. Furthermore, applying the principle of uniform boundedness, the latter is equivalent to the fact that the semigroup is pointwise bounded on bounded time intervals. For more details see [EN00, Proposition I.5.3]. In the case of relatively uniformly continuous semigroups we conclude that relatively uniformly continuous positive semigroups are pointwise order bounded on bounded time intervals.
Suppose is a positive semigroup on such that for each we have as . Then for each and the set
is order bounded in .
Fix and . There exist and such that for all we have Pick and find and such that Then
Let be the smallest positive integer such that If we define
we have . ∎
As was already announced, the following result is a version of [EN00, Proposition I.5.3] for relatively uniformly continuous semigroups of positive operators. It says that a semigroup is relatively uniformly continuous if and only if it is relatively uniformly continuous at and positive .
Let be a positive semigroup on . Then is relatively uniformly continuous on if and only if as for positive vectors .
Only the “if statement” requires a proof. Fix and . Let be one of the regulators for and as . Pick and find such that for all we have and By creftypecap 3.5 we can also find such that for all . Then
hold for all . This proves that is relatively uniformly continuous on . ∎
4. Standard constructions
In this section we construct different relatively uniformly continuous semigroups from a given one. All the constructions are motivated by [EN00, Chapter I.5.b]. To prove that a given semigroup is relatively uniformly continuous we will tacitly use creftypecap 3.6. For the sake of clarity, in this section always denotes a given relatively uniformly continuous positive semigroup on a vector lattice .
Let be a lattice isomorphism between vector lattices and . Then is a relatively uniformly continuous positive semigroup on .
It is easy to see that that where is a positive semigroup on . To prove that is relatively uniformly continuous on , pick and . Due to relative uniform continuity of there exist and such that
holds for all . Since is a lattice homomorphism, we obtain
for all . ∎
Next we consider semigroups on quotient vector lattices. Let be an ideal in and let be the quotient projection between vector lattices and . In order to guarantee that is Archimedean, we require our ideal to be relatively uniformly closed (see [LM67, Theorem 5.1]).
Suppose is a relatively uniformly closed ideal which is invariant under operator for each . Then the family of operators defined for each and by
is a relatively uniformly continuous positive semigroup on .
It is easy to check that is a positive semigroup on . To prove that is relatively uniformly continuous on , pick and which regulates as . Choose . and find such that for all we have Since is a lattice homomorphism, we obtain
for all . ∎
The next standard construction on our list are rescaled semigroups.
For any numbers and , the rescaled semigroup defined by
is relatively uniformly continuous.
A direct computation shows that is a positive semigroup. To prove that is relatively uniformly continuous on , pick and find which regulates as . Given any , there exists such that for all we have Since the function is continuous, there exists such that for all we have If , then we have
for each ∎
Next we deal with product semigroups. It is worth pointing out that the proof in our case is more complicated than the proof in the case of -semigroups on Banach spaces.
Let and be relatively uniformly continuous positive semigroups such that
holds for each . Then is a relatively uniformly continuous positive semigroup.
We prove first that is a semigroup. As in [EN00, I.5.15] one can show that holds for all . Fix and . Find which regulates , , and as and . Pick and find