# On the Adjoint of a Strongly Continuous Semigroup

Authors: Diómedes Bárcenas; Luis Gerardo Mármol
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

Using some techniques from vector integration,
we prove the weak measurability of the adjoint of strongly
continuous semigroups which factor through Banach spaces
without isomorphic copy ofl1;
we also prove the strong continuity away from zero of the
adjoint if the semigroup factors through Grothendieck spaces.
These results are used, in particular, to characterize the
space of strong continuity of
{T**(t)}t≥0,
which, in addition, is also characterized for abstract
L- and M-spaces. As a
corollary, it is proven that abstract L-spaces with no copy of
l1
are finite-dimensional.


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## Body

## 1. Introduction

The study of
the adjoint of a strongly continuous semigroups was initiated by Phillips   [1], who proved, among other things, that for the
general case, the domain of continuity of the adjoint semigroup on a Banach
space X is different
from X∗. A systematic
study of the adjoint semigroup {T∗(t)}t≥0 can  be found in [2], where, among other things, the basic properties of  {T∗(t)}t≥0 and of the
canonical spaces X⊙ and X⊙⊙ associated with
it are studied. Also, in [2], Neerven introduces the space X⊙⊙ and studies the
relationship between X⊙⊙ and X⊙⊙, giving some
conditions guaranteeing that X⊙⊙ must be equal
to kX⊙⊙, where k:X⊙⊙→X∗∗ is a natural
embedding of X⊙⊙ into X∗∗, defined by the
formula 〈kx⊙⊙,x∗〉:=limλ→∞〈x⊙⊙,λR(λ,A∗)x∗∗〉,x⊙⊙∈X⊙⊙.Some  of the principal tools used by van Neerven come
from vector measures,  like Gelfand and Pettis
integrals, in order to provide the reader with some examples where the above-mentioned equality holds.In this paper, we also employ certain techniques from
vector  measures  to show that semigroups factoring through Banach spaces without
isomorphic copy ofl1 have weakly
measurable adjoint. On the other hand, we prove the strong continuity away from
zero of the adjoint when the semigroup factors through Grothendieck spaces.
These results have several applications; in particular, there is one related to
our other main topic of interest: to find a characterization of the space X⊙⊙ of strong
continuity of a given second dual semigroup {T∗∗(t)}t≥0. This can be done in the case of spaces satisfying
the above-mentioned hypotheses. This is also done when X is
sun-reflexive and X⊙ is isomorphic
to a dual Banach space. Additionally, we give a characterization of the strong
continuity of {T∗∗(t)}t≥0 for  abstract L-
and M-spaces, and prove that abstract L-spaces with no copy of l1 are
finite-dimensional.


## 2. Preliminaries

## 2.1. The Semigroup Dual

LetX be a complex
Banach space and let {T(t)}t≥0 be a C0-semigroup of
bounded linear operators on X. The adjoint
semigroup {T∗(t)}t≥0 fails in
general to be strongly continuous. The semigroup dual ofXwith respect to{T(t)}t≥0, notation X⊙ (pronounced X-sun) is
defined as the linear subspace of X∗ on which {T∗(t)}t≥0 acts in a
strongly continuous way (2.1)X⊙={x∗∈X∗:limt↓0∥T∗(t)x∗−x∗∥=0}. From this definition, we have that X⊙ is  T∗(t)-invariant, that is, T∗X⊙⊆X⊙, for all t≥0.It can also be shown (see, e.g., [2]) that X⊙ is a closed,
weak∗-dense linear subspace of X∗.


## 2.2. The SpaceX⊙⊙

Let{T⊙(t)}t≥0 denote the
restriction of {T∗(t)}t≥0 to the T∗(t)-invariant
subspace X⊙. Since X⊙ is closed, X⊙ is a Banach
space. Also, it is clear from the definition of X⊙ that {T⊙(t)}t≥0 is a strongly
continuous semigroup on X⊙.We can also define{T⊙∗(t)}t≥0 to be the
adjoint of {T⊙(t)}t≥0 and write X⊙⊙ for (X⊙)⊙ (read as X-sun-sun or X-bosom).Ifj:X→X⊙∗  is defined by 〈jx,x⊙〉:=〈x⊙,x〉, then ∥j∥≤1 and j(X)⊆X⊙⊙. Moreover, j is an embedding
and we can identify X isomorphically
with the closed subspace jX of X⊙⊙. If jX=X⊙⊙, then X is said to be sun-reflexive or ⊙-reflexive with respect to   {T(t)}t≥0.The following results can be found in [2].Theorem 2.1.
LetX
be a complex
Banach space,   {T(t)}t≥0   a   C0-semigroup on   X
with
infinitesimal generator  A,   satisfying  ∥T(t)∥≤Mewt, and let A∗ be the adjoint
of A.The formula(2.2)〈kx⊙⊙,x∗〉:=limλ→∞〈x⊙⊙,λR(λ,A∗)x∗〉,x⊙⊙∈X⊙⊙, defines a natural embedding  k:X⊙⊙→X∗∗. Moreover, 1≤∥k∥≤M and kx⊙⊙|X⊙=x⊙⊙. If i:X→X∗∗ is the natural
map, then we have i=kj.We also have the following characterization ofkX⊙⊙.Theorem 2.2.
It holds that(2.3)kX⊙⊙={x∗∗∈X∗∗:limλ→∞λR(λ,A)∗∗x∗∗=x∗∗}.The  former theorem suggests to define
the following.Definition 2.3.
(2.4)X⊙⊙:={x∗∗∈X∗∗:limt↓0T∗∗(t)x∗∗=x∗∗}.A natural question is:do the spacesX⊙⊙andkX⊙⊙coincide?  Trivially, this is true if X∗=X⊙, since then we
have the coincidence of both definitions. It is also shown in [2] that k maps X⊙⊙ into X⊙⊙, and so kX⊙⊙⊆X⊙⊙ always holds,
but this inclusion can be proper. More specifically, if {T(t)}t≥0 is the rotation
group on X=C(T),T the unit
circle, then X⊙⊙≠kX⊙⊙ (see [2, page 99]).In the following, we will be interested in giving
conditions guaranteeing thatX⊙⊙ is equal to kX⊙⊙.


## 2.3. The Gelfand and Pettis Integrals

Let(Ω,Σ,μ) be finite
measure space and let X be a Banach
space. Suppose that f:Ω→X∗ is weakly∗-μ-measurable and
suppose further that for each x∈X the function 〈f(⋅),x〉 belongs to L1(μ). For each E∈Σ define a map TE:X→L1(μ),  TEx=〈fχE(⋅),x〉. It can be easily
shown that TE is bounded.
This implies that the linear map xE∗ defined by (2.5)〈xE∗,x〉:=∫E〈f(⋅),x〉dμ is bounded. The element xE∗∈X∗ is called the weak∗-integral or Gelfand integral offoverE with respect to   μ, notation xE∗=weak∗∫Efdμ.We have from the definition that the weak∗-integral satisfies (2.6)〈weak∗∫Efdμ,x〉=∫E〈f(⋅),x〉dμ for all E∈Σ and x∈X.Now, iff:Ω→X is weakly μ-measurable and
for each x∗∈X∗, the function 〈x∗,f(⋅)〉 belongs to L1(μ), then using the same
argument as above, each E∈Σ defines an
element x∗∗∈X∗∗ such that (2.7)〈xE∗∗,x∗〉=∫E〈x∗,f(⋅)〉dμ for all x∗∈X∗. If for all E∈Σ the element xE∗∗ belongs to X, then f is said to be Pettis integrable with respect toμ.For a detailed study of the Gelfand and Pettis
integrals, see [3–5].Theorem 2.4.
An elementx⊙⊙∈X⊙⊙   belongs  to   kX⊙⊙   if and only if
for all   t>0   and   x∗∈X∗    one has  (2.8)〈x⊙⊙,weak∗∫0tT∗(σ)x∗dσ〉=∫0t〈x⊙⊙,T∗(σ)x∗〉dσ.For  a proof
of   this, see  [2].Corollary 2.5.
X⊙⊙=kX⊙⊙   if and only if
for all  t>0, x∗∈X∗,   and   x⊙⊙∈X⊙⊙   one has  (2.9)〈x⊙⊙,weak∗∫0tT∗(σ)x∗dσ〉=∫0t〈x⊙⊙,T∗(σ)x∗〉dσ.Hence, if{T∗(t)}t≥0 is Pettis
integrable, that is, for all x∗∈X∗ and τ>0, the map t→T∗(t)x∗ is Pettis
integrable on [0,τ], then X⊙⊙=kX⊙⊙. In particular,
this holds if {T∗(t)}t≥0 is strongly
continuous away from zero (notation: C>0).Definition 2.6.
Letf,g:Ω→X   be two weakly  μ-measurable
functions. One says that   f,g    are  weaklyμ-equivalent if 〈x∗,f(⋅)〉=〈x∗,g(⋅)〉μ- almost
everywhere, for each  x∗∈X∗.   If   f,g:Ω→X∗ are weakly∗μ-measurable, one says that  f, g   are  weakly∗μ-equivalent   if   〈f(⋅),x〉=〈g(⋅),x〉   for each  x∈X.Following [5], we will denote by 𝒫(μ,X) the space of classes of weakly μ-equivalent
Pettis μ-integrable X-valued
functions. It is a linear space with ordinary algebraic operations.Definition 2.7.
X   has the  μ-weak
Radon-Nikodym property (μ-WRNP)
if for each  X-valued   μ-continuous
measure of   σ-finite
variation   γ, there exists  f∈𝒫(μ,X)   such that  (2.10)〈x∗,γ(E)〉=∫E〈x∗,f〉dμ for each x∗∈X∗ and E∈Σ.X has the WRNP if it has the μ-WRNP for every μ.For more details about the WRNP, see [4, 5].One of the principal tools used here is the following
theorem.Theorem 2.8.
X∗   has the weak
Radon-Nikodym property if and only if  X
contains no
isomorphic copy of  l1.


## 3. Strong Continuity of the Adjoint Semigroup

It is well
known that a semigroup is strongly continuous if and only if it is weakly
continuous (see [6]). In [7], Bárcenas and Diestel  have proved that if a C0-semigroup {T(t)}t≥0 is such that,
for each t>0,T(t) factors through
an Asplund space, then the adjoint semigroup {T∗(t)}t≥0 is C>0. (A Banach space X is an Asplund
space if and only if X∗ has the
Radon-Nikodym property.) Since weakly compact operators factor through
reflexive Banach spaces (which are Asplund spaces), we see that the adjoint
semigroup of a semigroup of weakly compact operators is strongly continuous
away  from  O (see [8]). Bárcenas and Diestel used those results to get
some applications in optimal control theory.In this section, we get the same conclusion of
Bárcenas-Diestel if aC0-semigroup {T(t)}t≥0 is such that,
for each t>0, T(t) factors through
a Grothendieck space.We recall that a Banach space is aGrothendieck
space if every weakly∗-convergent sequence in X∗ is also weakly
convergent. Equivalently, X is a
Grothendieck space if every linear bounded operator from X to any
separable space, Banach space is weakly compact. Among Grothendieck spaces, we
list all reflexive Banach spaces and L∞(Ω,Σ,μ), where (Ω,Σ,μ) is a positive
measure space. A Banach space isomorphic to a complemented subspace of a
Grothendieck space is also a Grothendieck space. Several characterizations of
Grothendieck spaces are found in [9].A Banach space is said to have theDunford-Pettis property if every weakly
compact operator in L(X) applies
relatively weakly compact sets onto relatively norm compact sets. The most
common examples of Banach spaces with the Dunford-Pettis property are L1(μ) and C(K). Complemented
subspaces of a space with the Dunford-Pettis property have also the
Dunford-Pettis property. For more details about this, see [10].IfX is a
Grothendieck space with the Dunford-Pettis property, Lotz (see [11])
has shown that every strongly continuous semigroup is uniformly continuous and
so the adjoint semigroup is uniformly continuous.We also recall that a bounded linear operatorT:X→Y  (where X and Y are Banach
spaces) factors through a Banach spaceZ if there are
bounded linear operators u:X→Z and v:Z→Y such that the
following diagram: (3.1) commutes.Theorem 3.1.
LetX
be a Banach
space and  {T(t)}t≥0   a  C0-semigroup
defined on   X.
Suppose that
for every   a>0
there exists a
Grothendieck space  Ya   such that  T(a)   factors through  Ya.   Then   {T∗(t)}t≥0   is  C>0.Proof.
More generally,
we will prove that, givena>0, the adjoint semigroup is strongly continuous for t≥a if the operator T(a) factors through
a Grothendieck space Ya.
Leta be a positive
number. There exist a Grothendieck space Ya and bounded
linear operators u:X→Ya and v:Ya→X such that the
following diagram is commutative:(3.2) For every t>a, the following diagram also commutes, due to
semigroup properties: (3.3) Hence v∗T∗(t−a)=(T(t−a)v)∗ is w∗-continuous. If (tn)∈(a,+∞) and tn→t, then v∗T∗(tn−a)→w∗v∗T∗(t−a). But Ya is a
Grothendieck space, and consequently v∗T∗(tn−a)→wv∗T∗(t−a). From this, we
can deduce that v∗T∗(t−a) is weakly
continuous for t≥a. Finally, u∗(T(t−a)v)∗=(T(t−a)vu)∗=T(t)∗ is w-continuous
for t≥a, which implies that {T∗(t)}t≥a is strongly
continuous.Remark 3.2.
Our proof that
semigroups factorizing through Grothendieck spaces have adjoint semigroupC>0   can be adapted
to prove that the adjoint of a   C0-semigroup on a
Grothendieck space is also a   C0-semigroup.
This result can also be obtained by combining the results of 2.3.2 and 2.3.3 from
[12].Remark 3.3.
Since reflexive Banach
spaces are Grothendieck spaces, this gives an alternative proof thatC0-semigroups of
weakly compact operators have adjoint semigroups which are   C>0,
thanks to the
Davis, Figiel, Johnson, and Pelczynski factorization scheme (see [8]).Remark 3.4.
Theorem3.1 also has applications in optimal control theory. See
[7] for details.The following theorem shows, in the spirit of Remark3.3, new examples of semigroups satisfying the hypotheses of Theorem 3.1.Theorem 3.5.
Let{T(t)}t≥0   be a strongly
continuous semigroup on an AL-space. If, for   t>0,T(t)X
does not
contain any isomorphic copy of   l1, then   {T(t)}t≥0
is a compact
semigroup.Proof.
LetX be an AL-space
and T(t):X→X  a strongly
continuous semigroup such that for t>0,T(t)X does not
contain any isomorphic copy of l1. Then T(t) applies bounded
sets onto sets which contain a weakly Cauchy sequence. Since AL-spaces are
weakly sequentially complete, T(t) applies bounded
sets onto weakly compact sets, due to the
Eberlein-Smulian theorem. Therefore T(t) is weakly
compact. Since AL-spaces have the Dunford-Pettis property, the square of a
weakly compact operator is compact; hence for t>0 we have (3.4)T(t)=T(t2)2; with T(t/2) being weakly  compact, T(t) is
compact.Corollary 3.6.
Every
infinite-dimensionalAL-space contains a copy of   l1.Proof.
If not, we
consider the semigroup defined, for eacht by T(t):=I, the identity on X, which is strongly continuous and hence compact, and,
by the Riesz lemma, X is
finite-dimensional.We finish this
section with two results related to the aim of finding a characterization of
the spaceX⊙⊙ of strong
continuity of {T∗∗(t)}t≥0.Theorem 3.7.
IfX   is   ⊙-reflexive
under   {T(t)}t≥0   and   X⊙
is isomorphic
to a dual Banach space, then   {T∗(t)}t≥0   is  C>0.Proof.
X is ⊙-reflexive if
and only if X⊙ is ⊙-reflexive (see
[2]). If X is ⊙-reflexive,
then the resolvent R(λ,A) is weakly
compact. Since (3.5)X=D(A)¯=linspanR(λ,A)BX¯, we can conclude that each ⊙-reflexive
Banach space is necessarily weakly compactly generated.
ThenX⊙ is weakly
compactly generated and therefore X⊙ has the
Radon-Nikodym property since weakly compactly generated dual Banach spaces have
that property (see [13]). Now  [2, Corollary 6.2.4] is applied.Theorem 3.8.
IfX
is an AM-space
or an AL-space, and  {T(t)}t≥0   is a   C0-semigroup on  X, then   {T∗∗(t)}t≥0   is strongly
continuous if and only if   {T(t)}t≥0   has bounded
generator.Proof.
IfX is an AM-space,
then X∗ is an AL-space
and therefore there exists (Ω,Σ,μ) such that X is
isometrically isomorphic to L1(μ) (see [14, page 18] for details). So X∗∗ is
isometrically isomorphic to L∞(μ), which is a Grothendieck space with the
Dunford-Pettis property. Since {T∗∗(t)}t≥0 is a C0-semigroup, it
has bounded generator, by the Lotz theorem. So {T(t)}t≥0 has bounded
generator.
Now we suppose thatX is an AL-space
and {T∗∗(t)}t≥0 is strongly
continuous. By  [12, Lemma II.3.2], {T∗(t)}t≥0 is strongly
continuous. Since X∗ is a
Grothendieck space with the Dunford-Pettis property, it has bounded generator.
Therefore {T(t)}t≥0 has bounded
generator.


## 3.1. The Weak Measurability of the Adjoint Semigroup

Definition 3.9.
LetK
be a compact
Hausdorff space. A function   ϕ:K→ℂ    is said to be
universally
measurable  if it is   μ-measurable for
all finite positive regular Borel measures   μ   on   K.   A function   ψ:K→X, with   X
a Banach space,
is universally weakly measurable  if   〈x∗,ψ(⋅)〉
is universally
measurable for all   x∗∈X∗.ψ   is called
universally
Pettis integrable
if it is Pettis
integrable with respect to every  μ.The following
theorem of Riddle-Saab and Uhl (see [15]) is useful in providing examples of C0-semigroups for
which X⊙⊙=kX⊙⊙.Theorem 3.10.
LetX
be a separable
Banach space and suppose that   ψ:K→X∗ is a bounded,
universally weakly measurable function. Then   ψ
is universally
Pettis integrable.Remark 3.11.
If{T∗(t)}t≥0   is  weakly
Borel measurable,
that is, for
all   x∗∈X∗   and   τ>0   the map   t→T∗(t)x∗
is weakly Borel
measurable on   [0,τ],
it  follows from
Theorem 3.10 that   {T∗(t)}t≥0   is Pettis
integrable. Combining this with Corollary 2.5 and the notes immediately following it, we get.Corollary 3.12.
Suppose thatX
is separable.
If   {T∗(t)}t≥0   is weakly Borel
measurable, then   X⊙⊙=kX⊙⊙.Remark 3.13.
It should be pointed
out that weak measurability does not imply strong measurability. For example,
let   JF
be theJames
function space, defined as the completion of the linear span of the
characteristic functions of subintervals of   [0,1]
with respect to
the norm  (3.6)∥f∥=sup(∑i=0k−1|∫titi+1f(t)dt|2)1/2, where the supremum runs over all partitions 0=t0<t1<⋯<tk=1 of [0,1]. Define T(t) on JF by (3.7)T(t)f=f(x+tmod1). It is shown [2, pages 159-160] that the adjoint semigroup {T∗(t)}t≥0 is weakly Borel
measurable (and hence Pettis integrable) but not C>0. In particular, it cannot be strongly
measurable.As it has been
noted by van Neerven, as a consequence of the theorem of Odell-Rosenthal (see [16]), if X is separable
and does not contain a closed subspace isomorphic to l1, then each x∗∗∈X∗∗ is the weak∗-limit of some sequence in X.If{T(t)}t≥0 is a C0-semigroup on
such a space, then 〈x∗∗,T∗(t)x∗〉 is the
pointwise limit of the continuous functions 〈T∗(t)x∗,xn〉, which implies
that it is Borel measurable. From this, we can deduce that X⊙⊙=kX⊙⊙.Now, using the WRNP of the dual
space of any space with no copy ofl1, we will prove
that if T(t) factors through
a separable space without copy of l1, then {T∗(t)}t≥0 is weakly
measurable, thus giving some generality to the former results.We will begin with a weak∗ version of the theorem of
Riddle-Saab and Uhl.Theorem 3.14.
LetX
be a Banach
space with no copy of   l1   and suppose
that   ψ:K→X∗   is a bounded,
universally weakly measurable function. Then   ψ   is weak*-equivalent to a Pettis
integrable function.Proof.
Letγψ:Σ→X∗ be the weak∗-integral of ψ, that is, (3.8)〈x,γψ(E)〉=∫E〈x,ψ〉dμ  for each x∈X and E∈Σ.  γψ is weakly∗-countably additive. X does not
contain a subspace isomorphic to l1. Then, by a
classic result of Bessaga and Pelczynski (see [17, pages 48-49]), X∗ does not
contain a copy of l∞. By [5, Theorem 4.2, pages 197-198] (see also [3, I.4.7]), γψ is a measure in
the  strong topology of X∗.
Now we use the WRNP ofX∗: there exists an μ-Pettis
integrable function g such that (3.9)〈x,γψ(E)〉=∫E〈x,g〉dμ.  By (3.8) and (3.9),
we conclude that 〈x,ψ〉=〈x,g〉  μ-almost
everywhere, for all x∈X.Theorem 3.15.
LetX
be a Banach
space and let   {T(t)}t≥0   be a   C0-semigroup
defined on  X. If for every   t>0,T(t)
factors through
a separable Banach space with no copy of   l1,   then   {T∗(t)}t≥0   is weakly Borel
measurable.Proof.
Let[a,b] be any closed and
bounded interval contained in (0,∞). There exists a
Banach space Ya with no copy of l1, and bounded
linear operators u:X→Ya and v:Ya→X  such that the
following diagram is commutative: (3.10) For every t∈[a,b], the following
diagram also commutes, due to semigroup property: (3.11) Then v∗T∗(t−a) is weak∗-continuous and hence Gelfand
integrable with respect to the Lebesgue measure on [a,b].
For eachE⊆[a,b], we define (3.12)γ(E)=∫Ev∗T∗(t−a)x∗dt,  where γ(E) is a countably
additive vector measure which is also a vector measure in the strong topology,
by the same argument used in the proof of Theorem 3.14. On the other hand, we
have that γ is absolutely
continuous with respect to the Lebesgue measure on [a,b].
By the WRNP ofYa∗, there exists a
Pettis-integrable function f such that (3.13)γ(E)=∫Ef(t)dt.  Let {yn}n∈ℕ be a dense
subset in Ya. From (3.12) and
(3.13), we have (3.14)ynγ(E)=∫Eynf(t)dt=∫Eynv∗T∗(t−a)x∗dt  for each n∈ℕ.
Letμ denote the
Lebesgue measure on [a,b]. According to
(3.14), there exists Nn,  μ(Nn)=0 such that (3.15)ynf(t)=ynv∗T∗(t−a)x∗  for every t∈[a,b]∖Nn.
Now, takeN=⋃n=1∞Nn; then μ(N)=0 and (3.16)ynf(t)=ynv∗T∗(t−a)x∗  for every t∈[a,b]∖Nn and n∈ℕ.
Since{yn}n∈ℕ is dense in Ya, we have that (3.17)f(t)=v∗T∗(t−a)x∗ for every t∈[a,b]∖N, and hence v∗Tt−a∗ is weakly Borel
measurable. Finally (3.18)u∗(T(t−a)v)∗=(T(t−a)vu)∗=(T(t−a)T(a))∗=T∗(t) is also weakly Borel measurable.Theorem 3.16.
LetX   and   {T(t)}t≥0   be as in
Theorem 3.15. Then X⊙⊙=kX⊙⊙.Proof.
We are going
to prove that{T∗(t)}t≥0 is Pettis
integrable.
As in Theorem3.15, Let [a,b]⊆(0,∞) by any closed
bounded interval. We have the following factorization scheme: (3.19)we also have (3.20)Then, using the same argument, we conclude that there
exists a Pettis-integrable function f:[a,b]→X∗ such that (3.21)v∗T∗(t−a)x∗=f(t)almost everywhere but then (3.22)T∗(t)x∗=u∗(f(t)) and it is easy to see that u∗(f(t)) is Pettis integrable.Remark 3.17.
A Banach spaceX is said to have
the Lotz property if every strongly continuous semigroup on X has bounded
generator. It is noticeable that Leung [18]  has found a Lotz space without the
Dunford-Pettis property, and it is also noticeable that to be a Grothendieck
space is not enough to ensure the Lotz property, since in lp, 1<p<∞, we can easily
define a C0-semigroup
which is not uniformly continuous by means of (3.23)T(t):lp→lp,x:=∑n=1∞αnxn↦T(t)x:=∑n=1∞αne−ntxn, where {xn} is the standard
unit basis.
Reworking on van Neerven results, we can show the
relationship that exists among some of them (note that they were stated
separately), thus giving necessary and sufficient conditions to ensure the
uniform continuity of the adjoint of a strongly continuous semigroup defined on
a Banach space whose dual has the Lotz property. This can be done as follows.
LetX   be a complex
Banach space such that   X∗   has the Lotz
property, and let   {T(t)}t≥0   be a   C0-semigroup on   X
with
infinitesimal generator  A.
The following statements are equivalent.(1)
{T(t)}t≥0   is uniformly
continuous.(2)
{T∗(t)}t≥0
is strongly
continuous.(3)
limλ→∞∥λR(λ,A∗)x∗−x∗∥=0   for all  x∗∈X∗.(4)
limt↓0∥(1/t)(weak∗∫otT∗(σ)x∗dσ)−x∗∥=0   for all   x∗∈X∗,   where    weak∗∫otT∗(σ)x∗dσ
is the Gelfand
integral of   T∗(σ)   on   [0,t]
with respect to
the Lebesgue measure.(5)
The quotient semigroup onX∗/X⊙
is strongly
continuous.



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*Source: 651294-2008-03-02.xml*