TAUBERIAN AND COTAUBERIAN MULTIPLIERS
OF THE GROUP ALGEBRAS
L_{1}(G)
MANUELGONZÁLEZ
ABSTRACT. We describe somerecentresults on themultipliersof the groupalgebras
L_{1}(G)
whicharetauberianorcotauberian,where G isalocallycompactabeliangroup. We show theconnectionsbetween thoseresults, andwe statesomeopenquestions onthetopic.
1. INTRODUCTION
Tauberian
operators
wereintroducedby
Kalton andWilansky
in[13]
inorder tostudy
a
problem
ofsummability
fromanabstractpoint
ofview,
and cotauberianoperators
wereintroducedin
[18]
as thoseoperators
whoseconjugate
is tauberian. Thesetwo classes ofoperators
have foundmanyapplications
inBanach spacetheory
(see [10,
Chapter
5 The tauberianoperators
fromL_{1}( $\mu$)
intoa Banachspacewere studied in[9],
where thequestion
whether all tauberianoperators
T :L_{1}( $\mu$)
\rightarrowL_{1}( $\mu$)
are upper semi‐Fredholm(have
closed range and finite dimensionalkernel)
was raised. Anegative
answer tothisquestion
wasgiven
in[12]:
there exists a tauberianoperator
T :L_{1}(0,1)
\rightarrowL_{1}(0,1)
with non‐closed range. The
corresponding
question
for themultipliers
of the Banachalgebra
L_{1}(G)
which aretauberianorcotauberian,
where Gis alocally
compact
abehangroup, was studied in
[4]
and[5].
Observe that themultipliers
ofL_{1}(G)
coincide withthe convolution
operators
T_{ $\mu$}
associated to the Borel measures $\mu$ on G[
14,Chapter
0].
It was
proved
in[4]
that the tauberianoperators
T_{ $\mu$}
are invertible when the group G isnon‐compact,
and thatthey
areFredholmwhenGiscompact
andthey
haveclosed rangeorthe
singular
continuouspart
(with
respect
tothe Haarmeasure onG)
ofthe associatedmeasure $\mu$ is zero.
Moreover,
it wasproved
in[5]
that the cotauberianoperators
T_{ $\mu$}
areinvertible whenGis
non‐compact,
and thatthey
areFredholm whenGiscompact.
Theseresults
provide
new characterizations ofthe Fredholmmultipliers
ofthe groupalgebras
L_{1}(G)
described in[1,
Theorem5.97].
Herewepresent
the results of[4, 5],
describe therelations between
them,
andpoint
out someproblems
that remain open.Throughout
the paper X and Yare(complex)
Banachspaces,weconsider(continuous
linear)
operators
T:X\rightarrow Y, andwedenoteby
R(T)
andN(T)
the range and the kernel of T. Anoperator
T:X\rightarrow Y is called taubentan ifitssecondconjugate
T^{**}:X^{**}\rightarrow Y^{**}satisfies
T^{**-1}(Y)
= X, and the
operator
T is called cotauberian when itsconjugate
$\tau$* : \mathrm{Y}^{*}\rightarrow X^{*} is tauberian. Moreoveranoperator
K : X\rightarrow \mathrm{Y} is calledweakly
compact
if
K^{**}(X^{**})
\subset \mathrm{Y}. Note thatgiven
T : X\rightarrow Y tauberian(cotauberian)
and K : X\rightarrow Yweakly
compact,
the sumT+K istauberian(cotauberian).
SupportedinpartbyMINECO
(Spain),
Grant MTM2016‐76958.2010 Mathematics SubjectClassification. Primary: 47\mathrm{A}53,43\mathrm{A}22.
MANUELGONZÁLEZ
An
operator
T : X \rightarrow Y is upper semi‐Fredholm if it has closed range and finitedimensional
kernel,
and T is Fredholm if it has closed finite codimensional range and finite dimensional kernel.Upper
semi‐Fredholmoperators
are tauberian[10,
Theorem2.1.5],
andfrom ourpoint
ofview canbe considered as $\zeta$trivial tauberianoperators.
Forbasicresults onFredholm
theory,
tauberianoperators,
multipliers
of Banachalge‐
bras and Fourier
analysis
wereferto[1], [10] [14]
and[17].
2. PRELIMINARY RESULTSWe denote
by
G alocally
compact
abelian group(a
LCA group, forshort),
m is theHaarmeasure on
G,
L_{1}(G)
isthe space ofm‐integrable complex
functionsonG endowedwith the
L_{1}
‐norm\Vert. \Vert_{1}
, andM(G)
denotes thespace ofcomplex
Borelmeasures on G endowed with the variation norm. The spaceL_{1}(G)
can beidentifiedwith thesubspace
of those $\mu$ \in
M(G)
that areabsolutely
continuous withrespect
to mby associating
tof\in L_{1}(G)
themeasurem_{f}\in M(G)
definedby
m_{f}(A)=\displaystyle \int_{A}f(x)dm(x)
. The spaceL_{1}(G)
with the convolution
(f\displaystyle \star g)(x)=\int_{G}f(x-y)g(y)dm(y)
isacommutativeBanachalgebra.
Given$\mu$\in M(G)
andf\in L_{1}(G)
theexpression
( $\mu$\displaystyle \star f)(x)=\int_{G}f(x-y)d $\mu$(y)
defines$\mu$\star f
\inL_{1}(G)
satisfying
\Vert $\mu$\star f\Vert_{1}
\leq\Vert $\mu$\Vert\cdot\Vert f\Vert_{1}
. Thus forevery $\mu$\inM(G)
we obtainaconvolution
operator
T_{ $\mu$}
onL_{1}(G)
definedby
T_{ $\mu$}f
= $\mu$\star f
, andsatisfying
\Vert T_{ $\mu$}\Vert
=\Vert $\mu$\Vert.
Moreover, given
$\mu$,\mathrm{v}\in M(G)
, theconvolution ofmeasures$\mu$\star \mathrm{v}\in M(G)
is commutative[17].
ThereforeT_{ $\mu$\star $\nu$}=T_{ $\mu$}T_{ $\nu$}=T_{ $\nu$}T_{ $\mu$}
. For eachr\in G the translationoperator T_{r}
onL_{1}(G)
isdefined
by
(T_{r}f)(x)=f(x-r)
. Note thatT_{r}
isthe convolutionoperator
associatedtothe unitmeasure
$\delta$_{r}
concentrated at\{r\}.
The convolution
operators
acting
onL_{1}(G)
can be characterized as thoseoperators
T :
L_{1}(G)
\rightarrowL_{1}(G)
that commute with translations(
T_{r}T=TT_{r}
for each r \in G),
and coincide with the convolutionoperators
T_{ $\mu$},
$\mu$\in M(G) [
14,Chapter
0].
Let $\Gamma$ denote the dual group of G. Given
f
\inL_{1}(G)
and $\mu$ \inM(G)
, the Fourier
transform
\hat{f}: $\Gamma$\rightarrow \mathbb{C}
off
and theFourier‐Sieltjes
transform\hat{ $\mu$}
: $\Gamma$\rightarrow \mathbb{C} of $\mu$ aredefinedby
\displaystyle \hat{f}( $\gamma$)=\int_{G}f(x) $\gamma$(-x)dm(x)
and\displaystyle \hat{ $\mu$}( $\gamma$)=\int_{G} $\gamma$(-x)d $\mu$(x)
.Let A be a Banach
algebra
and let B be a subset ofa A. Theleft
annihilator of Bisthe set
l(B)
:=\{x\in A : xB=\{0\}\}
, and theright
annihilatorof B is the setr(B)
:=\{x\in A: Bx=\{0\}\}
.Following
[14],
we say thata Banachalgebra
A is without order ifl(A)=\{0\}
orr(A)=\{0\}
,andamapping
T:A\rightarrow Aisamultiplier
of
A ifx(Ty)=(Tx)y
for allx,
y\in A
. IfG is anLCAgroup, thenL_{1}(G)
iswithout order.The second dual space A^{**} of a Banach
algebra
A is also a Banachalgebra
endowed with the(first)
Arensproduct
[3].
Specifically, given M,
N\in A^{**}, f\in A^{*}
and a,b\in A,
wedefine the
product
M\cdot N in threesteps
as follows:f\cdot a\in A^{*}: \langle f\cdot a, b\rangle :=\langle f, ab\rangle
N\cdot f\in A^{*}: \langle N\cdot f, a\rangle :=\langle N, f\cdot a\rangle
M\cdot N\in A^{**} :
\langle M\cdot N, f\rangle :=\langle M, N\cdot f\rangle.
Thus,
for GanLCAgroup,L_{1}(G)^{**}
is aBanachalgebra.
Given
f\in L_{\infty}(G)\equiv L_{1}(G)^{*}
and$\phi$\in L_{1}(G)
, anddenoting
\overline{ $\phi$}(x)= $\phi$(-x)
, wehaveL_{1}(G)
Since Gis
commutative,
the centerofL_{1}(G)^{**}
isL_{1}(G) [
15,Corollary
3]
i.e.,
L_{1}(G)= {m\in L_{1}(G)^{**}
: m\cdot n=n\cdot mforeachn\in L_{1}(G)^{**}}.
When G is
compact,
L_{1}(G)
is \mathrm{a}(closed)
ideal ofL_{1}(G)^{**} [
19,Proposition
4.2].
ThusL_{1}(G)^{**}/L_{1}(G)
is aBanachalgebra.
For
f\in L_{1}( $\mu$)
, wedenote\mathrm{D}(f)=\{t:f(t)\neq 0\}
. Wesay that asequence(f_{n})
inL_{1}( $\mu$)
is
disjoint
if$\mu$(\mathrm{D}(f_{k})\cap \mathrm{D}(f_{l}))
=0fork\neq l.
The
following
resultwasproved
in[9] (see
also[10,
Chapter
4])
when $\mu$ isanon‐atomicfinite measure, but the
arguments
given
there arevalid when $\mu$ is $\sigma$‐finite.Theorem 2.1.
[9,
Theorems 2 and6]
Let $\mu$ be a $\sigma$‐finite
measure. For anoperator
T:L_{1}( $\mu$)\rightarrow Y
thefollowing
assertions areequivalent:
(1)
T istauberian;
(2) \displaystyle \lim\inf_{n\rightarrow\infty}\Vert Tf_{n}\Vert
>0for
everydisjoint
normalized sequence(f_{n})
inL_{1}( $\mu$)
;(3)
there exists a numberr > 0 such that\displaystyle \lim\inf_{n\rightarrow\infty}\Vert Tf_{n}\Vert
> rfor
everydisjoint
normalizedsequence
(f_{n})
inL_{1}( $\mu$)
;(4) \displaystyle \lim\inf_{n\rightarrow\infty}\Vert Tf_{n}\Vert
> 0for
every normalized sequence(f_{n})
inL_{1}( $\mu$)
satisfying
\displaystyle \lim_{n\rightarrow\infty} $\mu$(D(f_{n}))=0.
The convolution
operators
with closed rangeweredescribedby
Host and Parreau.Theorem 2.2.
[11,
Théorème1]
Let G be a LCA group and let $\mu$ \inM(G)
. Then theconvolution
operator
T_{ $\mu$}
has closedrangeif
andonly if $\mu$= $\nu$\star $\xi$
, where v,$\xi$\in M(G)
, $\nu$ is invertible and$\xi$
isidempotent.
Corollary
2.3. Let G be a LCA group and let$\mu$\in M(G)
.Suppose
that the convolutionoperator
T_{ $\mu$}
has closed range. ThenL_{1}(G)=R(T_{ $\mu$})\oplus N(T_{ $\mu$})
.Proof.
Thefactorization$\mu$= $\nu$\star $\xi$
inTheorem 2.2 and thecommutativity
of the convolu‐tion
product
ofmeasuresgive
T_{ $\mu$}=T_{ $\nu$}T_{ $\xi$}=T_{ $\xi$}T_{ $\nu$}
. So the resultfollows from thefact thatT_{ $\xi$}
isaprojection,
sinceN(T_{ $\xi$})=N(T_{ $\mu$})
andR(T_{ $\xi$})=R(T_{ $\mu$})
. \squareThe
point spectrum
$\sigma$_{p}(T)
ofanoperator
T : X\rightarrow X is the set ofthose $\lambda$ \in \mathbb{C} suchthat T- $\lambda$ I is not
injective.
When G iscompact,
thepoint spectrum
ofa convolutionoperator
admits thefollowing
description.
Proposition
2.4.[16,
Example
4.6.2]
Let G be acompact
group with dual group $\Gamma$, andlet $\mu$\in M(G)
. Then$\sigma$_{p}(T_{ $\mu$})=\hat{ $\mu$}( $\Gamma$)
.In
particular,
sinceT_{r}=T_{$\delta$_{r}}
and(2)
\displaystyle \hat{$\delta$_{r}}( $\gamma$)=\int_{G} $\gamma$(-x)d$\delta$_{r}(x)= $\gamma$(-r)
foreach
$\gamma$\in $\Gamma$
, we obtainthat$\sigma$_{p}(T_{r})=\{ $\gamma$(-r) : $\gamma$\in $\Gamma$\}.
Recall that r\in Ghas
finite
order ifthere existsm\in \mathrm{N} such that mr=0(where
mrdenotes the sumofm
copies
ofr in G).
Otherwise we say that r hasinfinite
order. It followsfromProposition
2.4andformula(2)
that$\sigma$_{\mathrm{p}}(T_{r})
isafinite subset ofthe unitcircle $\Gamma$when r has finite order.MANUELGONZÁLEZ
Ameasure
$\mu$\in M(G)
is said to be discrete if it is concentrated inacountable subset of G;i.e.,
if there exist sequences(x_{i})
in G and($\beta$_{i})
in \mathbb{C} so that\displaystyle \sum_{i=1}^{\infty}|$\beta$_{i}|
< \infty and$\mu$=\displaystyle \sum_{i=1}^{\infty}$\beta$_{i}$\delta$_{x_{i}}.
Proposition
2.5.[16,
Theorem4.11.1]
LetG be a LCA group and let $\mu$ \inM(G)
be adiscrete measure. Then the
spectrum
$\sigma$(T_{ $\mu$})
of
the convolutionoperator
T_{ $\mu$}
coincides withthe closure
of
theset\hat{ $\mu$}( $\Gamma$)
.3. TAUBERIAN OPERATORS
Here we show
that,
under certainconditions,
tauberian convolutionoperators
acting
on
L_{1}(G)
are Fredholm.Theorem 3.1. LetG be a
non‐compact
LCA group. Then every tauberian convolutionoperator
T_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
is invertible.In the
proof
ofTheorem3.1weshow first thatT_{ $\mu$}
isboundedbelow,
and thenwederivefrom
Corollary
2.3thatT_{ $\mu$}
is invertible.The
prof
of the result for thecaseGcompact
(Theorem 3.5)
isdonein inseveralsteps.
Recall that
given
r\in G, the translationoperator T_{r}
isaninvertibleoperator
onL_{1}(G)
that satisfies
T_{r}T_{s}=T_{r+s}
for everys\in G. Intheproof
ofthe nextresult wedistinguish
thecasesinwhich r has finiteorinfiniteorder.
Proposition
3.2. Let G be acompact
group, andlet$\lambda$\in$\sigma$_{p}(T_{r})
for
some r\in G. ThenT_{r}- $\lambda$ I
is not tauberian.Next we consider the case of a discrete measure concentrated in a finite number of
points
of G;namely
$\mu$=\displaystyle \sum_{l=1}^{k}$\alpha$_{l}$\delta$_{r_{l}}
. In this caseT_{ $\mu$}=\displaystyle \sum_{l=1}^{k}$\alpha$_{l}T_{r_{l}}.
Theorem 3.3. LetG bea
compact
group with dual group $\Gamma$, and let$\mu$=\displaystyle \sum_{l=1}^{k}$\alpha$_{l}$\delta$_{r_{l}}
wherer_{1},...
,r_{k} are distinct
points
in G. ThenT_{ $\mu$}- $\lambda$ I
is not tauberian when$\lambda$\in $\sigma$(T_{ $\mu$})
.To prove this
result,
we consider first the case $\lambda$\in$\sigma$_{p}(T_{ $\mu$})
, which coincides with\hat{ $\mu$}( $\Gamma$)
(Proposition 2.4),
and then thegeneral
case.Nowweconsiderthecaseofan
arbitrary
discretemeasure onG;namely
$\mu$=\displaystyle \sum_{i=1}^{\infty}$\beta$_{i}$\delta$_{x}i
where(x_{i})
isasequenceofpoints
ofGand($\beta$_{i})
isasequence in\mathbb{C}satisfying
\displaystyle \sum_{i=1}^{\infty}|$\beta$_{i}|<\infty.
In this case
T_{ $\mu$}=\displaystyle \sum_{i=1}^{\infty}$\beta$_{i}T_{x $\iota$}.
Proposition
3.4. LetG be acompact
group with dual group$\Gamma$_{\mathrm{Z}}
and let$\mu$=\displaystyle \sum_{i=1}^{\infty}$\beta$_{i}$\delta$_{x_{i}},
where(x_{i})
is a sequenceof
distinctpoints
inG and($\beta$_{i})
\subset \mathbb{C}satisfying
\displaystyle \sum_{i=1}^{\infty}|$\beta$_{i}|
< \infty.Then
T_{ $\mu$}- $\lambda$ I
is not tauberian when$\lambda$\in $\sigma$(T_{ $\mu$})
.Nowwe canstateourresultforthecase G
compact.
Theorem 3.5. LetG be a
compact
group G, let $\mu$, $\mu$_{0}\in M(G)
with $\mu$_{0}discrete,
and letf\in L_{1}(G)
. Then(1)
If
T_{ $\mu$}
is tauberian with closed range, then it isFredholm.(2) T_{$\mu$_{0}}
\dot{u} tauberianif
andonly if
it isinvertible.4. COTAUBERIAN OPERATORS
Inthis sectionweshow that thecotauberian convolution
operators
T_{ $\mu$}
acting
onL_{1}(G)
are
always
Fredholm,
and thatT_{ $\mu$}
istauberian ifandonly
ifits natural extensiontothealgebra
of measuresM(G)
is tauberian. We derive some consequences for convolutionoperators
acting
onC_{0}(G)
andL_{\infty}(G)
, andwe answer aquestion
raised in[8]
about themeasures
$\mu$\in M(G)
suchthat$\nu$\in M(G)
and$\mu$\star $\nu$\in L_{1}(G)
imply
$\nu$\in L_{1}(G)
.Firstweshowthat the Banach
algebras
involved inourarguments
arewithout order.Proposition
4.1. Let G be a LCA group. Then thealgebra
(L_{1}(G)^{**}, \cdot)
admits a norm‐one
right identity;
hence it \dot{u} a Banachalgebra
without order.Moreover,
when the groupG is
compact,
thequotient
algebra
L_{1}(G)^{**}/L_{1}(G)
also admits a norm‐oneright identity
and it is a Banach
algebra
without order.The
multipliers
ofalgebras
without order have agood
behavior underduality:
Proposition
4.2. Let A a Banachalgebra
without order andlet T be amultiplier of
A.Then thesecond
conjugate
T^{**}:A^{**}\rightarrow A^{**} is amultiplier of
A^{**}.Given a Banach space X, we denote
by
X^{co} thequotient
spaceX^{**}/X
. The secondconjugate
T^{**}ofanoperator
T:X\rightarrow Y induces anotheroperator
T^{co}:X^{co}\rightarrow \mathrm{Y}^{\mathrm{c}o} whichis defined
by
T^{\mathrm{c}o}(m+X) :=T^{**}m+Y(m\in X^{**})
, and it is called the residuumoperator
ofT. Notethat T is tauberianif and
only
ifT^{co} isinjective,
and T is cotauberianif andonly
if T^{co} has dense range[10,
Proposition
3.1.8 andCorollary
3.1.12].
Corollary
4.3. LetG be acompact
LCA group and letT_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
be a convo‐lution
operator.
Thenthe residuumoperator
T_{ $\mu$}^{\mathrm{c}o}
is amultiplier of
thealgebra
L_{1}(G)^{co}.
Next we show that cotauberian convolution
operators
onL_{1}(G)
are tauberian. Thisresult contrasts with the fact that it is easy to find non‐trivial cotauberian
operators
on
L_{1}(G)
,just
take asurjective
operator
with non‐reflexivekernel,
but it is muchmoredifficult toobtainanon‐trivial tauberian
operator
(see
[12]).
Proposition
4.4. LetG be a LCA group. Then every cotauberian convolutionoperator
T_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
is tauberiian.Corollary
4.5. LetG be anon‐compact
LCA group. A convolutionoperator
onL_{1}(G)
is cotauberian
if
andonly
if
it is invertible.‐Let Ebe a
right identity
inL_{1}(G)^{**}
provided by Proposition
4.1. We consider themap$\Gamma$_{E}
:M(G)\rightarrow L_{1}(G)^{**}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{J}}\mathrm{e}\mathrm{d}
by
$\Gamma$_{E}( $\mu$):=T_{ $\mu$}^{**}(E) , $\mu$\in M(G)
.Themap
$\Gamma$_{E}
isanisometricalgebra homomorphism
ofM(G)
intoL_{1}(G)^{**}
which extends thenaturalembedding
ofL_{1}(G)
intoL_{1}(G)^{**}
[
7,Proposition
2.3].
Since
T_{ $\mu$}^{**}
is amultiplier
ofL_{1}(G)^{**}
, foreachm\in L_{1}(G)^{**}
wehave(3)
T_{ $\mu$}^{**}m=(T_{ $\mu$}^{**}m)\cdot E=m\cdot T_{ $\mu$}^{**}E=m\cdot$\Gamma$_{E}( $\mu$)
.Thus
T_{ $\mu$}^{**}
is aright multiplication
operator
(by $\Gamma$_{E}( $\mu$)).
Moreover(4)
E\cdot$\Gamma$_{E}( $\mu$)=T_{ $\mu$}^{**}(E)=$\Gamma$_{E}( $\mu$)
.MANUELGONZÁLEZ
Theorem 4.6. LetG be a LCA group. Then
T_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
is cotauberianif
andonly if
it isFredholmof
index zero.To prove Theorem
4.6,
wenote thatT_{ $\mu$}
cotauberianimplies
T_{ $\mu$}
tauberian(Proposition
4.4).
Then,
in thecase Gnon‐compact,
Theorem 3.1implies
thatT_{ $\mu$}
isinvertible.In thecaseG
compact,
L_{1}(G)^{\mathrm{c}o}
is aBanachalgebra,
andweprove thatT_{ $\mu$}
cotauberianimplies
that the residuumoperator
T_{ $\mu$}^{co}
acting
onL_{1}(G)^{co}
isbijective,
andfrom the inverseof
T_{ $\mu$}^{\mathrm{c}o}
weget
an inverse ofT_{ $\mu$}
modulo thecompact operators,
henceT_{ $\mu$}
isFredholm.Next we
study
the relation between aconvolutionoperator
T_{ $\mu$}
:L_{1}(G) \rightarrow L_{1}(G)
anditsextension
M_{ $\mu$}
:M(G)\rightarrow M(G)
definedby
M_{ $\mu$}( $\nu$)= $\mu$\star $\nu$.
Theorem4.7. LetG beaLCAgroup.
ThenT_{ $\mu$}
istauberianif
andonly
ifM_{ $\mu$}
istauberian.Proof. Suppose
thatT_{ $\mu$}
istauberian,
and let Ebearight identity
inL_{1}(G)^{**}
. Then thefollowing diagram
is commutative:Now
T_{ $\mu$}
tauberianimplies
T_{ $\mu$}^{**}
tauberian[10,
Theorem4.4.2].
ThereforeT_{ $\mu$}^{**}$\Gamma$_{E}=$\Gamma$_{E}M_{ $\mu$}
is
tauberian,
and henceM_{ $\mu$}
istauberian,
inboth casesby
[10,
Proposition
2.1.3].
Similarly,
denoting by
J:L_{1}(G)\rightarrow M(G)
the naturalisomorphic embedding,
wehaveJT_{ $\mu$}=M_{ $\mu$}J
.Hence,
by
[10,
Proposition
2.1.3],
ifM_{ $\mu$}
istauberian,
soisT_{ $\mu$}.
\squareRecall that an
operator
T :L_{1}(G)
\rightarrowL_{1}(G)
is tauberian if andonly
ifm \inL_{1}(G)^{**}
and T^{**}m\inL_{1}(G)
imply
m\inL_{1}(G)
. Inparticular,
ifT_{ $\mu$}
istauberian,
then $\nu$ \inM(G)
and
$\mu$\star $\nu$\in L_{1}(G)
imply
\mathrm{v}\in L_{1}(G)
.Observation4.8. Itwas askedin
[8]
whetheraconvolutionoperator
T :L_{1}(G)\rightarrow L_{1}(G)
is tauberian when themeasure $\mu$
satisfies
thefollowing
condition:(5)
$\nu$\in M(G) , $\mu$\star $\nu$\in L_{1}(G)\Rightarrow $\nu$\in L_{1}(G)
.Next we will show that the answerto this
question
isnegative.
Indeed,
it wasproved
in[4]
that there exists an atomic measure$\mu$_{0}\in M( $\Gamma$)
such thatT_{$\mu$_{0}}
is aninjective
non‐tauberianoperator,
where $\Gamma$ denotes theunit circle. It isenough
tochoose $\mu$_{0} such that its
Fourier‐Stieltjes
transform\hat{ $\mu$}_{0}
satisfies0\in\overline{\hat{ $\mu$}_{0}(\mathbb{Z})}\backslash \hat{ $\mu$}_{0}(\mathbb{Z})
. Thefollowing
argument,
duetoDoss[6],
shows thatT_{ $\mu$ 0}
satisfiesformula(5):
Every
$\nu$\in M( $\Gamma$)
can be written as $\nu$=$\nu$_{1}+$\nu$_{2} with $\nu$_{1}\ll mand$\nu$_{2}\perp m
, where mis the Haar measure on $\Gamma$. Since $\mu$_{0}\star$\nu$_{1}\in L_{1}( $\Gamma$)
and $\mu$_{0}\star \mathrm{v}_{2} issupported
ina m‐nullset,
T_{ $\mu$ 0} $\nu$\in L_{1}(G)
if andonly
if$\nu$_{2}=0.
\square Note that$\mu$\star\tilde{f}=\overline{ $\mu$}\star f
for$\mu$\in M(G)
andf\in L_{1}(G)
.Also,
ifasequence(f_{n}) \subset L_{1}(G)
is normalized and
disjoint,
thensois(fn).
Therefore,
itfollows from[10,
Theorem4.1.3]
that
T_{ $\mu$}
istauberianif andonly
ifsoisT_{\overline{ $\mu$}}
.Hence, by
Theorem4.7,
thesamehappens
forM_{ $\mu$}
andM_{\overline{ $\mu$}}
,andweget
thefollowing result,
whereS_{ $\mu$}
:C_{0}(G)\rightarrow C_{0}(G)
and its extensionProposition
4.9. LetG be anon‐compact
LCA group. Then(i)
L_{ $\mu$}
:L_{\infty}(G)
\rightarrowL_{\infty}(G)
is tauberianif
andonly if
it \dot{u}cotauberian,
and this \dot{u}equivalent
toL_{ $\mu$}
invertiblef
(ii) M_{ $\mu$}
:M(G)\rightarrow M(G)
is tauberianif
andonly if
it \dot{u}invertible;
(iii) S_{ $\mu$}
:C_{0}(G)\rightarrow C_{0}(G)
is cotauberianif
andonly if
it is invertible.5. SOME OPEN QUESTIONS
The main
question
that remains open isthefollowing
one.Question
1. LetGbeacompact
LCAgroup and letT_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
beatauberianoperator.
IsT_{m}u
Fredholm?This
question
admitsequivalent
formulations:Question
2. Let Gbeacompact
LCAgroup and letT_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
beatauberianoperator.
IsT_{ $\mu$}
cotauberian?Observation 4.8
gives
anegative
answer to aproblem
raised in[8],
but we can refor‐mulate it asfollows.
Question
3. Find a condition additional to $\nu$ \inM(G)
, $\mu$\star $\nu$ \inL_{1}(G)
\Rightarrow \mathrm{v} \inL_{1}(G)
implying
T_{ $\mu$}
tauberian.We have seeninTheorem 4.7 that
T_{ $\mu$}
:L_{1}(G)\rightarrow L_{1}(G)
is tauberianif andonly
ifsoisM_{ $\mu$}
:M(G)\rightarrow M(G)
. The second condition is muchstronger.
Question
4. Find characterizations ofT_{ $\mu$}
tauberian interms ofthe restrictionsM_{ $\mu$}|_{L_{1}(|\mathrm{v}|)}:L_{1}(|\mathrm{v}|)\rightarrow M(G)
for
special
measures$\nu$\in M(G) (different
from the Haarmeasure m).
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609‐626.DEPARTAMENTODE MATEMÁTICAS, UNIVERSIDAD DE CANTABRIA,E‐39071 SANTANDER, SPAIN