Multipliers and Wiener-Hopf operators on weighted L^p spaces

0.19 MB·English

by  Violeta Petkova

MULTIPLIERS AND WIENER-HOPF OPERATORS ON WEIGHTED
Lp SPACES
VIOLETA PETKOVA
2
1
0
2
Abstract. WestudythemultipliersM (boundedoperatorscommutingwiththetrans-

n lations) on weighted spaces Lp(R). We establish the existence of a symbol µ for M
a ω M
and some spectralresults for the translations S and the multipliers. We also study the
J t
operatorsT ontheweightedspaceLp(R+)commutingeitherwiththerighttranslations
9 ω
1 St, t∈R+, orleft translationsP+S−t, t∈R+, andwe establishthe existence ofa sym-
bol µ of T. We characterize completely the spectrum σ(St) of the operator St proving
] that
A
σ(S )={z ∈C:|z|≤etα0},
t
F
. where α0 is the growth bound of (St)t≥0. We obtain a similar result for the spectrum
h of (P+S ), t≥0. Moreover,for an operatorT commuting with S , t≥0, we establish
−t t
at the inclusion µ(O)⊂σ(T), where O={z ∈C:Imz <α0}.
m
[
1. Introduction

2
v Let E be a Banach space of functions on R. For t ∈ R, define the translation by t on
5 E by
8
9 Stf(x) = f(x−t),a.e., ∀f ∈ E.
4
We call a multiplier on E, every bounded operator on E commuting with S for every
. t
2 t ∈ R. For the multipliers on a Hilbert space we have the existence of a symbol and some
1
1 spectral results concerning the translations and the multipliers are obtained by using this
1 property of the multipliers (see [7], [8]). In the arguments exploited in [7], [8] the spectral
:
v mapping theorem of Gearhart [3] for semigroups in Hilbert spaces plays an essential role.
i
X The first purpose of this paper is to extend the main results in [8], [7] concerning the
r existence of the symbol of a multiplier as well as the spectral results in the case where
a E is a weighted Lp(R) space. For general Banach spaces the characterization of the
ω
spectrum of the semigroup V(t) = etG by the resolvent of its generator G is much more
complicated than for semigroups in Hilbert spaces (see for instance [4]). In particular,
the statements of Lemma 1, 2 and 3 (see Section 2) are rather difficult to prove and for
general Banach spaces this problem remains open. In this paper we restrict our attention
to Lp(R), 1 ≤ p < ∞, weighted spaces. The advantage that we take account is that the
ω
semigroupofthetranslations(S )preserves thepositive functions. Forsemigroups having
t
this special property in the spaces Lp(R) we have a spectral mapping theorem (see [1],
ω
[12], [13]). We obtain Theorems 1-4 for multipliers on Lp(R) and in this work we explain
ω
only these parts of the proofs which are based on spectral mapping techniques and which
are different from the arguments used to establish Theorems 1-4 in the particular case
1
2 VIOLETA PETKOVA
p = 2 (see for more details [8], [7]).
′ ′
For a Banach space E denote by E the dual space of E. For f ∈ E, g ∈ E , denote
by < f,g > the duality. Let p ≥ 1, and let ω be a weight on R. More precisely, ω is a
positive, continuous function such that
ω(x+t)
sup < +∞,∀t ∈ R.
x∈R ω(x)
Let Lp(R) be the set of measurable functions on R such that
ω
1/p
kfk = |f(x)|pω(x)pdx < +∞, 1 ≤ p < +∞.
p,ω (cid:16)ZR (cid:17)
Let C (R) (resp. C (R+) ) be the space of continuous functions on R (resp. R+) with
c c
compact support in R (resp. R+). Notice that C (R) is dense in Lp(R).
c ω
In the following we set E = Lp(R) and we consider only Banach spaces having this form
ω
for 1 ≤ p < +∞. In this case
hf,gi = f(x)g¯(x)ω2(x)dx
Z
R
and
|hf,gi| ≤ kfk kgk , for1 < p < +∞,
p,ω q,ω
where 1 + 1 = 1. For p = 1, we have
p q
E′ = L∞(R) = {f ismeasurable : |f(x)|ω(x) < ∞,a.e.}
ω
and
kgk = esssup{|f(x)|ω(x), x ∈ R}.
∞,ω
If M is a multiplier on E then, there exists a distribution µ such that
Mf = µ∗f, ∀f ∈ C (R+).
c
For φ ∈ C (R+), the operator
c
M : Lp(R) ∋ f −→ φ∗f
φ ω
is a multiplier on E. Introduce
1 1
α0 = lim lnkStkt, α1 = lim lnkS−tkt.
t→+∞ t→+∞
It is easy to see that α +α ≥ 0. Consider
1 0
U = {z ∈ C, Imz ∈ [−α , α ]}.
1 0
For an operator T denote by ρ(T) the spectral radius of T and by σ(T) the spectrum of
T. It is well known that ρ(S ) = eα0t, for t ≥ 0.
t
Given a function f and a ∈ C, denote by (f) the function
a
R ∋ x −→ f(x)eax
WIENER-HOPF OPERATORS 3
anddenote by M the algebra of the multipliers onE. Wenote by gˆthe Fourier transform
of a function g ∈ L2(R). Our first result is a theorem saying that every multiplier on E
has a representation by a symbol.
Theorem 1. Let M be a multiplier on E. Then
1)For a ∈ [−α , α ], we have (Mf) ∈ L2(R), for every f ∈ E such that (f) ∈ L2(R).
1 0 a a
2) For a ∈ [−α , α ], there exists a function ν ∈ L∞(R) such that
1 0 a
\
(Mf) (x) = ν (x)(f) (x), ∀f ∈ E, with(f) ∈ L2(R), a.e.
a a a a
Moreover, we have kν k ≤ CkMdk, ∀a ∈ [−α , α ].
a ∞ 1 0
◦ ◦
3) If U 6= ∅, there exists a function ν ∈ H∞(U) such that
◦
Mf(z) = ν(z)fˆ(z), z ∈ U, ∀f ∈ C∞(R),
c
where Mf(ia+x) = (\Mdf) (x), for a ∈]−α ,α [, f ∈ C∞(R).
a 1 0 c
Thedfunction ν is called the symbol of M. The above result is similar to that
established in [8], [7] and the novelty is that we treat Banach spaces Lp(R) and not only
ω
Hilbert spaces. Define A as the closed Banach algebra generated by the operators M ,
φ
for φ ∈ C (R). Notice that A is a commutative algebra. Our second result concerns the
c
spectra of S and M ∈ M.
t
Theorem 2. We have
i) σ(S ) = {z ∈ C, e−α1t ≤ |z| ≤ eα0t}, ∀t ∈ R. (1.1)
t
Let M ∈ M and let µ be the symbol of M.
M
ii) We have
µ (U) ⊂ σ(M). (1.2)
M
iii) If M ∈ A, then we have
µ (U) = σ(M). (1.3)
M
The equality (1.3) may be considered as a weak spectral mapping property (see [2])
for operators in the Banach algebra A. On the other hand, it is important to note that if
M ∈ M, but M ∈/ A, ingeneral wehave µ (U) 6= σ(M). ForthespaceE = L1(R), there
M
existsacounter-example(seesection2and[2]). Thustheinclusionin(1.2)couldbestrict.
In section 3, we obtain similar results for Wiener-Hopf operators on weighted Lp(R+)
ω
spaces. In the analysis of Wiener-Hopf operators some new difficulties appear in com-
parison with the case of multipliers.
Let E be a Banach space of functions on R+. Let p ≥ 1 and let ω be a weight on R+.
It means that ω is a positive, continuous function such that
ω(x+t) ω(x+t)
0 < inf ≤ sup < +∞,∀t ∈ R+.
x≥0 ω(x) x≥0 ω(x)
4 VIOLETA PETKOVA
Let Lp(R+) be the set of measurable functions on R+ such that
ω
∞
|f(x)|pω(x)pdx < +∞.
Z
0
Notice that C (R+) is dense in Lp(R+).
c ω
Let P+ be the projection from L2(R−) ⊕ Lp(R+) into Lp(R+). From now we will
ω ω
denote by S the restriction of S on Lp(R+) for a ≥ 0 and, for simplicity, S will be
a a ω 1
denoted by S. Let I be the identity operator on Lp(R+).
ω
Definition 1. A bounded operator T on Lp(R+) is called a Wiener-Hopf operator if
ω
P+S TS f = Tf, ∀a ∈ R+, f ∈ Lp(R+).
−a a ω
As in [5] we can show that every Wiener-Hopf operator T has a representation by a
convolution. More precisely, there exists a distribution µ such that
Tf = P+(µ∗f), ∀f ∈ C∞(R+).
c
If φ ∈ C (R), then the operator
c
Lp(R+) ∋ f −→ P+(φ∗f)
ω
is a Wiener-Hopf operator and we will denote it by T . Moreover, we have
φ
(P+S S )f = f, ∀f ∈ Lp(R+),
−a a ω
but it is obvious that
(S P+S )f 6= f,
a −a
for all f ∈ Lp(R+) with a support not included in ]a,+∞[. The fact that S is not
ω a
invertible leads to many difficulties in contrast to the case when we deal with the space
Lp(R).
ω
Let E be the space Lp(R+). As above define
ω
a0 = lim lnkStk1t, a1 = lim lnkS−tk1t
t→+∞ t→+∞
and set J = [−a ,a ]. The next theorem is similar to Theorem 1.
1 0
Theorem 3. Let a ∈ J and let T be a Wiener-Hopf operator. Then for every f ∈ Lp(R+)
ω
such that (f) ∈ L2(R+), we have
a
(Tf) = P+F−1(h (f) ) (1.4)
a a a
with ha ∈ L∞(R) and d
kh k ≤ CkTk,
a ∞
where C is a constant independent of a. Moreover, if a +a > 0, the function h defined
1 0
◦
on U = {z ∈ C : Imz ∈ J} by h(z) = h (Rez) is holomorphic on U.
Imz
Definition 2. The function h defined in Theorem 3 is called the symbol of T.
WIENER-HOPF OPERATORS 5
We are able to examine the spectrum of the operators in the space W of bounded
operators on E commuting with (S ) or (P+S ) .
t t≥0 −t t≥0
Let O = {z ∈ C, Imz < a } and V = {z ∈ C, Imz < a }.
0 1
Theorem 4. We have
i) σ(S ) = {z ∈ C, |z| ≤ ea0t}, ∀t > 0. (1.5)
t
ii) σ(P+S ) = {z ∈ C, |z| ≤ ea1t}, ∀t > 0. (1.6)
−t
Let T ∈ W and let µ be the symbol of T.
T
iii) If T commutes with S , ∀t ≥ 0, then we have
t
µ (O) ⊂ σ(T). (1.7)
T
iv) If T commutes with P+S , ∀t ≥ 0, then we have
−t
µ (V) ⊂ σ(T). (1.8)
T
The equalities (1.5),(1.6) generalize the well known results for the spectra of the right
and left shifts in the space of sequences l2 (see for instance, [10]). However, our proofs
are based heavily on the existence of symbols for Wiener-Hopf operators and having in
mind Theorem 3, we follow the arguments in [9].
In section 4, we obtain a sharp spectral result for Wiener-Hopf operators having the
form T with φ ∈ C (R). This result is established here for operators in spaces Lp(R+).
φ c ω
It is important to note that even for p = 2 and for the Hilbert space L2(R+) our result
ω
below is new.
Theorem 5. Let φ ∈ C (R). Then
c
i) if supp(φ) ⊂ R+, we have
ˆ
φ(O) = σ(T ).
φ
ii) if supp(φ) ⊂ R−, we have
ˆ
φ(V) = σ(T ).
φ
The above result yields a weak spectral mapping property and can be compared with
the equality (1.3) in Theorem 2, however the proof is more complicated.
2. Multipliers on Lp(R)
ω
Recall that we use the notation E = Lp(R). We start with the following
ω
Lemma 1. Let λ ∈ C be such that eλ ∈ σ(S) and let Reλ = α . Then there exists a
0
sequence (fn)n∈N of functions of E and an integer k ∈ Z so that
lim etA −e(λ+2πki)t f = 0, ∀t ∈ R, kf k = 1, ∀n ∈ N. (2.1)
n n
n→∞ (cid:16) (cid:17)
(cid:13) (cid:13)
(cid:13) (cid:13)
6 VIOLETA PETKOVA
Proof. Let A be the generator of the group (St)t∈R. It is clear that the group (St)t∈R
preserves positivefunctions. SinceE = Lp(R)theresultsof[12], [13]saythatthespectral
ω
mapping theorem holds and
σ(etA)\{0} = etσ(A) = {etλ : λ ∈ σ(A)}.
In particular, for the spectral bound s(A) of A we get
s(A) := sup{Rez : z ∈ σ(A)} = α .
0
Thus eλ ∈ σ(S)\{0} = eσ(A) yields λ+2πki = λ ∈ σ(A) for some k ∈ Z. On the other
0
hand, Reλ = α , and we deduce that λ is on the boundary of the spectrum of A. By
0 0 0
a well known result, this implies that λ is in the approximative point spectrum of A.
0
Let µ be a sequence such that µ → λ , Reµ > λ , ∀n ∈ N. Then
n n n→∞ 0 n 0
(µ I −A)−1 ≥ (dist(µ ,σ(A)))−1,
n n
(cid:13) (cid:13)
hence k(µ I −A)−1k → ∞(cid:13). Applying th(cid:13)e uniform boundedness principle and passing to
n
a subsequence of µ (for simplicity also denoted by µ ), we may find f ∈ E such that
n n
lim (µ I −A)−1f → ∞.
n
n→∞
(cid:13) (cid:13)
(cid:13) (cid:13)
Introduce f ∈ D(A) defined by
n
(µ I −A)−1f
n
f = .
n k(µ I −A)−1fk
n
The identity
(λ+2πki−A)f = (λ −µ )f +(µ −A)f
n 0 n n n n
implies that (λ+2πki−A)f → 0 as n → ∞. Then the equality
n
t
(etA −et(λ+2πki))f = e(λ+2πki)(t−s)eAsds (A−λ−2πki)f
n (cid:16)Z (cid:17) n
0
(cid:3)
yields (2.1).
Now we prove the following important lemma.
Lemma 2. For all φ ∈ C∞(R) and λ such that eλ ∈ σ(S) with Reλ = α we have
c 0
|φˆ(iλ+a)| ≤ kM k, ∀a ∈ R. (2.2)
φ
Proof. Let λ ∈ C be such that eλ ∈ σ(S) and Reλ = α0 and let (fn)n∈N be the
sequence constructed in Lemma 1. We have
1 = kf k = sup | < f ,g > |.
n n
′
g∈E ,kgk ′≤1
E
′
Then, there exists g ∈ E such that
n
1
| < f ,g > −1| ≤
n n
n
WIENER-HOPF OPERATORS 7
and kgnkE′ ≤ 1. Fix φ ∈ Cc∞(R) and consider
1
ˆ ˆ ˆ
|φ(iλ+a)| ≤ |φ(iλ+a)hf ,g i|+ |φ(iλ+a)|
n n
n
1
≤ φ(t) e(λ+2πki)t −S e−i(a+2πk)tf ,g dt + |φˆ(iλa)|
(cid:12)ZR(cid:10) (cid:16) t(cid:17) n n(cid:11) (cid:12) n
(cid:12) (cid:12)
(cid:12) (cid:12)
+ hφ(t)S e−i(a+2πk)tf ,g idt .
(cid:12)ZR t n n (cid:12)
(cid:12) (cid:12)
The first two terms on the rig(cid:12)ht side of the last inequality(cid:12) go to 0 as n → ∞ since by
Lemma 1 we have
e−i(a+2πk)t e(λ+2πki)t −S f −→ 0.
t n n→∞
(cid:13) (cid:16) (cid:17) (cid:13)
On the other hand,(cid:13) (cid:13)
I = < φ(t)S e−i(a+2πk)tf ,g > dt
n (cid:12)ZR t n n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= h φ(t)e−i(a+2πk)tf (.−t)dt ,g i
(cid:12) hZR n i n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= h (φ(.−y)ei(a+2πk)yf (y)dy,ei(a+2πk).g i
(cid:12) ZR n n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= h M (ei(a+2πk).f ) ,ei(a+2πk).g i
φ n n
(cid:12) (cid:16) (cid:17) (cid:12)
(cid:12) (cid:12)
and In ≤ kMφkkfnkkgnkE′ ≤(cid:12)kMφk. Consequently, we deduce(cid:12)that
ˆ
|φ(iλ+a)| ≤ kM k.
φ
(cid:3)
Notice that the property (2.2) implies that
|φˆ(λ)| ≤ kM k, ∀λ ∈ C, provided Imλ = α .
φ 0
Lemma 3. Let φ ∈ C∞(R) and let λ be such that e−λ¯ ∈ σ((S )∗) with Reλ = −α .
c −1 1
Then we have
|φˆ(iλ+a)| ≤ k(M )k, ∀a ∈ R. (2.3)
φ
Proof. Consider the group (S )∗ acting on E′. Let λ ∈ C be such that e−λ¯ ∈
−t t∈R
σ((S )∗) and
−1
|e−λ¯| = ρ(S ) = ρ((S )∗) = eα1.
−1 −1
The group (S )∗ preserves positive functions. To prove this, assume that g(x) ≥ 0, a.e.
−t
is a positive function and let h ∈ E be such that h(x) ≥ 0, a.e. Then
hh,(S )∗gi = hS h,gi ≥ 0.
−t −t
If F(x) = ((S )∗g)(x) < 0 for x ∈ Λ ⊂ R and Λ has a positive measure, we choose
−t
h(x) = 1 (x). Then h1 ,Fi ≤ 0 and we conclude that F(x) = 0 a.e. in Λ which is
Λ Λ
a contradiction. For the group (S )∗ the spectral mapping theorem holds and, by the
−t
8 VIOLETA PETKOVA
same argument as in Lemma 1, we prove that there exists a sequence (gk)k∈N of functions
of E′ and an integer m so that for all t ∈ R,
lim k(etB −e(−λ¯+2πmi)t))gkkE′ = 0
k→∞
and kgkkE′ = 1.
Since S S = I, we have (S )∗(S )∗ = I. This implies that
−t t t −t
(S )∗g −e(λ¯−2πmi)tg = (S )∗ −e(λ¯−2πmi)t(S )∗(S )∗ g
t k k E′ (cid:16) t t −t (cid:17) k E′
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
≤ k(St)∗kE′→E′ (cid:16)e(−λ¯+2πmi)t −(S−t)∗(cid:17)gk E′
(cid:13) (cid:13)
and we deduce that for every t ∈ R w(cid:13)e have (cid:13)
lim (S )∗ −e(λ¯−2πmi)t g = 0.
k→∞(cid:13)(cid:16) t (cid:17) k(cid:13)E′
For 1 < p < +∞ the space E =(cid:13) Lp(R) is reflexive an(cid:13)d the dual to E′ can be identified
ω
with E. Consequently, since kgkkE′ = 1, there exists fk ∈ E such that
1
| < f ,g > −1| ≤ , kf k ≤ 1. (2.4)
k k k E
k
For p = 1 the space L1(R) is not reflexive and to arrange (2.4), we use another argument.
ω
In this case the dual to L1ω(R) is L∞ω (R). Let kgkL∞ω(R) = 1. Fix 0 < ǫ < 1 and consider
the set
M = {x ∈ R : |g(x)|ω(x) ≥ 1−ǫ, m ≤ x < m+1}, m ∈ Z.
ǫ,m
If µ(M ) (the Lebesgue measure of M ) is zero for all m ∈ Z, we obtain a contra-
ǫ,m ǫ,m
diction with kgkL∞(R) = 1. Thus there exists r ∈ Z such that µ(Mǫ,r) > 0. Now we
ω
take
1 (x)eiarg(g(x))
f(x) = Mǫ,r .
µ(M )ω2(x)
ǫ,r
Then
r+1
1 ≥ hf,gi = f(x)g¯(x)ω2(x)dx ≥ 1−ǫ
Z
r
and we can obtain (2.4) choosing ǫ = 1/k. Passing to the proof of (2.3), we get
1
ˆ ˆ ˆ
|φ(iλ+a)| ≤ |φ(iλ+a) < f ,g > |+ |φ(iλ+a)|
k k
k
≤ < φ(t)e−i(a+2πm)tf , e(¯λ−2πmi)t −(S )∗ g > dt
(cid:12)ZR k (cid:16) t (cid:17) k (cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12)1
+ < φ(t)S e−i(a+2πm)tf ,g > dt + |φˆ(iλ+a)| = J′ +I′ + |φˆ(iλ+a)|.
(cid:12)ZR t(cid:16) k(cid:17) k (cid:12) k k k k
(cid:12) (cid:12)
Fro(cid:12)m the argument above we deduce t(cid:12)hat J′ → 0 as k → ∞. For I′ we apply the
k k
same argument as in the proof of Lemma 2 and we deduce
|φˆ(iλ+a)| ≤ kM k. (cid:3)
φ
WIENER-HOPF OPERATORS 9
For the proof of Theorem 1 we apply the argument in [7] and Lemmas 2-3. There
exists eλ0 ∈ σ(S) such that Reλ = α . Then for every z ∈ C with Imz = α we have
0 0 0
|ϕˆ(z)| ≤ kM k.
ϕ
Alsothereexistse−λ1 ∈ σ((S )∗)withReλ = −α andforeveryz ∈ CwithImz = −α
−1 1 1 1
we have
|ϕˆ(z)| ≤ kM k.
ϕ
Applying Phragmen-Lindel¨off theorem for the Fourier transform of ϕ ∈ C∞(R) in the
c
domain {z ∈ C : −α ≤ Imz ≤ α }, we deduce
1 0
|ϕˆ(z)| ≤ kM k
ϕ
for z ∈ U. Next we exploit the fact that M can be approximated by M with respect
ϕ
to the strong operator topology (see [6] for a very general setup covering our case). We
complete the proof repeating the arguments from [6], [7] and since this leads to minor
modifications, we omit the details. To obtain Theorem 2 we follow the same argument
as in [8] and the proof is omitted.
To see that in (1.2) the inclusion may be strict, consider a measure η on R such that
the operator
M : f −→ S (f)dη(x)
η Z x
R
is bounded on L1(R). For this it is enough to have d|η|(x) < ∞. Then M is a
R η
multiplier on L1(R) with symbol R
ηˆ(t) = e−ixtdη(x).
Z
R
On the other hand, there exists a bounded measure η on R such that
ηˆ(R) 6= σ(M )
η
(see for details [2]). In L1(R) we have α = α = 0 and U = R. So we have not the
0 1
property (1.2) in Theorem 2 for every multiplier even in the case L1(R).
3. Wiener-Hopf operators
We need the following lemmas.
Lemma 4. Let φ ∈ C (R+). The operator T commutes with S , ∀t > 0, if and only if
c φ t
the support of φ is in R+.
Proof. Consider φ ∈ C (R+) and suppose that T commutes with S , t ≥ 0. We write
c φ t
φ = φχR− +φχR+.
10 VIOLETA PETKOVA
If Tφ commutes with St, t ≥ 0, then the operator TφχR− commutes too. Let ψ = φχR−
and fix a > 0 such that ψ has a support in [−a,0]. Setting f = χ , we get S f = χ .
[0,a] a [a,2a]
For x ≥ 0 we have
0 min(x−a,0)
P+(ψ ∗S f)(x) = ψ(t)χ dt = ψ(t)dt.
a Z {a≤x−t≤2a} Z
−a max(−a,−2a+x)
Since P+(ψ ∗S f) = S P+(ψ ∗f), for x ∈ [0,a], we deduce P+(ψ ∗S f)(x) = 0 and
a a a
x−a
ψ(t)dt = 0, ∀x ∈ [0,a].
Z
−a
This implies that ψ(t) = 0, for t ∈ [−a,0] hence supp(φ) ⊂ R+.
(cid:3)
Next we establish the following
Lemma 5. Let T , φ ∈ C (R). Then T commutes with P+(S ), ∀t > 0 if and only if
φ c φ −t
supp(φ) ⊂ R−.
Proof. For φ ∈ C (R), suppose that T commutes with P+(S ), ∀t > 0. Set ψ =
c φ −t
φχ . There exists a > 0 such that supp(ψ) ⊂ [0,a]. We have P+(ψ ∗P+S χ ) = 0
R+ −a [0,a]
and then P+S (P+ψ ∗χ ) = 0. This implies that
−a [0,a]
(ψ ∗χ )(x) = 0, ∀x > a.
[0,a]
On the other hand, for x > a we have
min(a,x) a
(ψ ∗χ )(x) = ψ(t)χ (x−t)dt = ψ(t)dt = ψ(t)dt.
[0,a] Z [0,a] Z Z
R max(0,x−a) x−a
Hence aψ(t)dt = 0,∀a > ǫ > 0andwegetψ = 0. Thus weconcludethatsupp(φ) ⊂ R−.
ǫ
(cid:3)
R
It is clear that (S ) and (P+(S )) form continuous semigroups and these semi-
t t≥0 −t t≥0
groups preserve positive functions. Moreover, by using the equality
h(P+S )h,gi = hh,(P+S )∗gi,
t −t
weconcludethatthesemigroup(P+S )∗ preservepositivefunctions. Theissueisthatfor
−t
S and (P+S )∗ the spectral mapping theorem holds and we may repeat the arguments
t −t
used in section 2. Thus we obtain the following
Lemma 6. 1) For all φ ∈ C∞(R) such that supp(φ) ⊂ R+, for λ such that eλ ∈ σ(S)
c
and Reλ = a , we have
0
|φˆ(iλ+a)| ≤ kT k, ∀a ∈ R.
φ
2) For all φ ∈ C∞(R) such that supp(φ) ⊂ R− and for λ such that e−λ¯ ∈ σ((S )∗) and
c −1
Reλ = −a , we have
1
|φˆ(iλ+a)| ≤ kT k, ∀a ∈ R.
φ