On the generation of Arveson weakly continuous semigroups

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On the generation of Arveson weakly continuous semigroups

Jean Esterle

To cite this version:

Jean Esterle. On the generation of Arveson weakly continuous semigroups. 2017. �hal-01587949�

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On the generation of Arveson weakly continuous semigroups

Jean Esterle September 14, 2017

Abstract

We consider here one-parameter semigroupsT= (T(t))t>0of bounded operators on a Banach spaceXwhich are weakly continuous in the sense of Arveson. For such a semigroupTdenote byMω_Tthe convolution algebra consisting in those measuresµon(0,+∞)such thatR+∞

0 kT(t)kd|µ|(t)<

+∞.The Pettis integralR+∞

0 T(t)dµ(t)defines forµ∈ Mω_T a bounded operator φT(µ) onX.Identifying the spaceL¹_ω_T of (classes of) measurable functionsfsatisfyingR+∞

0 |f(t)kT(t)kdt <+∞to a closed subspace Mω_T in the usual way, we define the Arveson idealITof the semigroup to be the closure in B(X) of φT(L¹ω_T). Using a variant of a procedure introduced a long time ago by the author we introduce a dense idealUT

ofIT,which is a Banach algebra with respect to a suitable normk.kU_T, such thatlim sup_t→0+kT(t)kB(U_T)<+∞.The normalized Arveson ideal JT is the closure ofITinB(UT).The Banach algebraJThas a sequential approximate identity and is isometrically isomorphic to a closed ideal of its multiplier algebra M(JT). The Banach algebrasUT, IT and JT

are "similar", and the map Su/v → Sau/av defines when a generates a dense principal ideal ofUTa pseudobounded isomorphism from the alge- breQM(JT)of quasimultipliers onJTonto the quasimultipliers algebras QM(UT)andQM(IT).

We define the generator AT of the semigroup T to be a quasimultiplier onIT, or ,equivalently, onJT.Every character χon IT has an extensionχ˜ to QM(IT).Let Resar(AT) be the complement of the set {χ(A˜ T)}_χ∈d_I

T. The quasimultiplier A−µI has an inverse belonging to JT forµ∈Resar(AT),which allows to consider this inverse as a "regular" quasimultiplier on the Arveson idealIT.The usual resolvent formula holds in this context forRe(µ)>limt→+∞logkT(t)k

t .

Set Π⁺_α := {z ∈ C | Re(z) > α}. We revisit the functional calculus associated to the generator AT by defining F(−AT) ∈ JT by a Cauchy integral when F belongs to the Hardy space H¹(Π⁺α) for some α <−limt→+∞logkT(t)|

t .We then defineF(−AT)as a quasimultiplier on JT and IT when F belongs to the Smirnov class on Π⁺_α, and F(−AT) is a regular quasimultiplier on JT and IT if F is bounded on Π⁺α. If F(z) =e^−ztfor somet >0,thenF(−AT) =T(t),and ifF(z) =−z,we indeed haveF(−AT) =AT.

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keywords: semigroup of bounded operators, Arveson pairs, Arveson spectrum, Pettis integral, infinitesimal generator, resolvent, Laplace transform, holomorphic functional calculus

AMS classification: Primary 47A16; Secondary 47D03, 46J40, 46H20

1 Introduction

We study here the generator of a semigroupT= (T(t))t>0of bounded operators on a Banach space X which is weakly continuous with respect to a dual pair (X, X∗) satisfying the conditions introduced by Arveson in 1974 in his sem- inal paper [2] on group representations. Semigroups satisfying some weaker properties have been considered in 1953 by Feller, who studied semigroups T = (T(t))_t>0 of bounded operators on a Banach space X such there exists a nonzero continuous linear forml onX for which the mapt→< T(t)x, l >is measurable for everyx∈X,and his work involves the subspaceX_∗ of the dual spaceX⁰ generated by the family(l◦T(t))t>0.Feller’s Pettis integrals allow to construct operators taking values on some subspaceX˜ ofX_∗,and he shows that there exists a rather large closed subspaceX1ofX˜ such that the mapt→T(t)x is continuous on(0,+∞)forx∈X1.

Arveson’s conditions are more restrictive, but his approach allows to define a norm-decreasing homomorphismφTfrom the convolution Banach algebraMω_T

intoB(X),without enlarging the Banach spaceX.HereMω_Tdenotes the algebra of all Borel measuresµon(0,+∞)such thatR+∞

0 kT(t)kd|µ|(t)<+∞.

Denote byL¹_ω_T the convolution algebra of (classes) of measurable functions on the half-line satisfyingR+∞

0 |f(t)|kT(t)kdt <+∞,identified to a closed ideal of M_ω_T, and let A_T (resp. I_T) be the closure of φ_T(M_ω_T) (resp φ_T(L¹_ω

T)) in B(X) with respect to the operator norm. The idealIT of AT is called the Arveson ideal associated to the semigroupT,see sections 2 and 3.

In section 4 we introduce the infinitesimal generator AT,op as a weakly densely defined operator on the spaceX1:= [∪t>0T(t)(X)]⁻,following Feller’s approach in [?], and we also introduce the resolventRop(T, λ) = (AT,op−λI)⁻¹ as a weakly densely defined operator onX1.Iflim sup_t→0+kT(t)k<+∞,these partially defined operators are closed operators onX1.In the general case they can be interpreted as closed operators onX₀ :=X₁/[∩t>0Ker(T(t)],which of course equalsX₁ if∩t>0Ker(T(t) ={0}.

In section 5 we pave the way to a more intrinsic aopproach to the infinitesimal generator of a Arveson weakly continuous semigroup T. It follows from some results of [11] that L¹_ω

T contains a function g such that g∗ L¹_ω

T is dense in L¹_ω

T,and soI_T possesses dense principal ideals, and it follows from the weak^∗ continuity of the semigroup that uAT 6= {0} for every u ∈ AT\ {0}. This allows us to apply toIT the theory of quasimultipliers developped a long time ago by the author in [12]. Recall that if U is a Banach algebra such that uU 6={0} foru∈ U \ {0}and such that the setΩ(U) :={u∈ U |[uU]⁻ =U } is nonempty, a quasimultiplier onU is a closed densely defined operator onU of the form S = S_u/v, where u ∈ U, v ∈ Ω(U), and where the domain DS of

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S is the set of all x∈ U such that ux∈ vU. Ifx∈ D, then Sxis the unique y ∈ U such that ux = vy. A subset ∆ of the algebra QM(U) is said to be pseudobounded if ∩S∈∆Ds contains someu∈Ω(U) such thatsup_S∈∆kSuk <

+∞. A quasimultiplier S ∈ QM(U)is said to be regular if there existsλ > 0 such that the set{λⁿSⁿ}n≥1 is pseudobounded, and the familyQMr(U)of all regular quasimultipliers onU is a pseudo-Banach algebra in the sense of [1].

In section 6, we associate in a natural way to a quasimultiplier S on IT a weakly densely defined operatorS˜onX₁.If∩_t>0Ker(T(t)) ={0}thenS˜has an extensionS˜to a subspace ofX₁ which is a closed operator. In the general case the domain ofS˜can be considered as a subspace ofX₀:=X₁/∩_t>0Ker(T(t)) andS˜ has an extensionS˜to a subspace ofX₀which is a closed operator.

In section 7 we use a variant of a construction introduced in [?] to define the

"normalized Arveson ideal"JT. SetρT := lim_t→+∞kT(t)k^1/t.If λ > log(ρT), then the family{e^−λtT(t)}t>0}is pseudobounded. Setkukλ= sup_t>0ke^−λtT(t)uk ∈ [0,+∞] for u ∈ IT, set LT := {u ∈ IT | kukλ < +∞}, and denote by UT the closure of ∪t>0T(t)IT in (LT,k.kλ). Then (LT,k.k)λ is a Banach algebra, and the definition of LT and UT and the norm topology on LT and UT do not depend on the choice of λ > log(ρT). We can identify the usual multiplier algebraM(UT), which is a Banach algebra with respect to the operator norm kSkop,λ, to a subalgebra of QMr(IT). The closure JT of IT in (M(UT),kSkop,λ) is called the normalized Arveson ideal of the semigroup T.

The sequence(φ_T(f_n∗δ_n))_n≥1is a sequential bounded approximate identity for JTfor every Dirac sequence(f_n)_n≥1and every sequence(_n)_n≥1of positive real numbers which converges to0, and the mapS →S_|_U

T is an isometric isomorphism from(M(JT),k.k_M(J_T₎)onto(M(UT),k.kλ,op).The algebrasUT,ITand JTare similar in the sense of [12], and an explicit pseudobounded isomorphism between the quasimultiplier algebrasQM(JT)andQM(UT)orQM(IT)(resp.

between the algebrasQMr(JT)andQMr(UT)orQMr(IT)) is given by the mapSu/v →Sau/av wherea∈Ω(UT). We haveΩ(JT)∩ UT⊂Ω(UT),and the fact thatΩ(JT)∩ UT 6=∅ follows from a construction based on inverse Laplace transforms of outer functions on half-planes given by P. Koosis and the author in section 6 of [11].

In section 8 we consider the generator A_T of the semigroup as a quasimultiplier on the algebraI_T or J_T. To define the infinitesimal generator as a quasimultiplier onIT we use the formula

A_T,I_T=S_−φ_T_(f⁰

0)/φT(f0), wheref0∈ C¹([0,+∞))∩Ω L¹_ω_T

satisfiesf0= 0, f₀⁰ ∈L¹_ω_T.

This definition does not depend on the choice off0 and the quasimultiplier A_T,I_T is a closed operator on the Arveson idealIT.If u∈ IT, and if we have lim_t→0+k^T(t)u−u_t −vk= 0 for some v ∈ IT, then ubelongs to the domain of A_T,I_TandA_T,I_Tu=v.It is also possible to consider the infinitesimal generator of the semigroup as a quasimultiplier on the normalized Arveson idealJT.Set

˜

ωT(t) =kT(t)k_M(J_T),and defineφ˜T=Mω˜_T → M(JT)by using the formula

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φ˜_T(µ) :=

Z +∞

0

T(t)dµ(t),

where the Bochner integral is computed with respect to the strong operator topology onM(JT).To define the infinitesimal generator as a quasimultiplier onJT we use the formula

A_T,J_T=S_−φ_T_(f⁰

0)/φ_T(f₀), wheref0∈ C¹([0,+∞))∩Ω L¹_ω_˜_T

satisfiesf0= 0, f₀⁰ ∈L¹_ω_T. Notice that while the fact that C¹([0,+∞))∩Ω L¹_ω

T

6= ∅ follows from nontrivial results from section 6 of [11] it is very easy to find elements f₀ ∈ C¹([0,+∞))∩Ω L¹_ω_˜

T

satisfyingf₀= 0, f₀⁰ ∈L¹_ω

T : one can take for example f0 =vλ, for some λ > log(ρT),where vλ(t) =te^−λt for t ≥0. This definition agrees with the classical definition of the genarator of a strongly continous semigroup bounded in norm near0 : if u ∈ JT, then u belongs to the domain of A_T,J_Tif and only if there existsv∈ J_T such thatlim_t→0+k^T^(t)u−u_t −vk_J_T = 0, and in this situationA_T,J_Tu=v.

Identify the multiplier algebras QM(IT) and QM(JT), and set AT = A_T,I_T=A_T,J_T.Every characterχonIT extends in a unique way to a charac- terχ˜onQM(IT),and there exists a unique complex numberaχ= ˜χ(AT)such thatχ(T(t)) =e^ta^χ for t >0.If the Arveson ideal IT is not a radical algebra, then the family σar(S) := {χ(S)}˜ _χ∈

IcT wil be called the Arveson spectrum of a quasimultipliers∈ QM(IT), and we will use the conventionσar(AT) =∅ if the algebraIT is radical. If IT is not radical, the map χ → χ(A˜ T) defines a homeomorphism fromIcT ontoσar(AT),and we have, forf ∈ Mω_T,

˜

χ(φT(f)) =L(f)(−˜χ(AT)).

Similarly, we haveχ( ˜˜ φT(f)) =L(f)(−˜χ(AT))forf ∈ Mω˜_T,see section 9.

In section 10 we introduce the Arveson resolvent set Resar(AT) := {C\ σar(AT)},and observe that AT−µI has an inverse(AT−µI)⁻¹ in QM(IT) forµ∈Resar(AT).More precisely this quasimultiplier(AT−µI)⁻¹)belongs to JT⊂ QMr(IT),and the mapµ→(AT−µI)⁻¹)is an holomorphic map from Resar(AT) into JT. If Re(λ) > −log(ρT), then we have the usual resolvent formula

(A−λI)⁻¹=− Z +∞

0

e^−λtT(t)dt∈ JT,

where the Bochner integral is computed with respect to the strong operator topology onM(J_T).

We can also define in this case (A−λI)⁻¹ by using the formula (A−λI)⁻¹v=−

Z +∞

0

e^−λtT(t)vdt (v∈ UT),

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which defines a quasimultiplier onIT if we apply it withv∈ UT∩Ω(IT).

In section 11 we introduce a holomorphic functional calculus. Forα∈R,set Π⁺_α :={z∈C|Re(z)< α}.LetH¹(Π⁺_α)be the usual Hardy space consisting in those functionsF holomorphic onΠ⁺_α such thatkFk1:= sup_β>α _2π¹ R+∞

−∞ |F(β+ iy)|dy <+∞.Functions F ∈H¹(Π⁺_α)admit a.e. a nontangential limitF^∗(α+ iy)onα+iR,andkFk1:= _2π¹ R+∞

−∞ |F^∗(α+iy)|dy.The restriction toΠ⁺_β of a functionF ∈H¹(Π⁺_α)is bounded onΠ⁺_β forβ > α,and so∪α<−log(ρT)H¹(Π⁺_α) is an algebra.

Set, forF ∈H¹(Π⁺_α), α∈(−∞,−log(ρT)), F(−AT) =−_2π¹ R+∞

−∞ F^∗(α+iy)(AT+ (α+iy)I)⁻¹dy∈ JT⊂ QMr(IT).

ThenF G(−AT) =F(−AT)G(−AT), F_|

Π+ β

(−AT)forβ ∈(α,−log(ρT)),and F(−AT) =φT(L⁻¹(F))forF ∈ ∪_α<log(ρ_T₎H¹(Π⁺_α), G∈ ∪_α<log(ρ_T₎H¹(Π⁺_α).

After observing in section 11 that G(−AT) ∈ Ω(JT) if G ∈ H¹(Π⁺_α) is outer (this can be deduced from Nyman’s characterization of dense ideals of the convolution algebraL¹(R⁺),but we propose a direct proof), we defineF(−AT) forF∈ ∪α<log(ρ_T)H^∞(Π⁺_α)by using the formula

F(−A_T) =S_{F H(−A}_T_)/H(−A_T₎∈ QM_r(J_T) =QM_r(I_T),

where F ∈H^∞(Π⁺_α), F H ∈ H¹(Π⁺_α), and where H ∈H¹(Π⁺_α)is outer on Π⁺_α.

Again, this definition does not depend on the choice ofH.WhenF=L(f), withf ∈ ∪_λ>log(ρ_T₎L¹(R⁺, e^λt), thenF(−AT) = ˜φT(f)∈ JT. IfF(z) =e^−tz, witht >0,thenF(−AT) =T(t).A similar formula allows to defineF(−AT)∈ QM(IT) =QM(JT), ifF ∈ ∪_α<log(ρ_T₎N⁺(Π⁺_α), whereN⁺(Π⁺_α)denotes the Smirnov class onΠ⁺_α,which is the class of all holomorphic functions F onΠ⁺_α such thatF H∈H^∞(Π⁺_α)for some outer functionF ∈H^∞(Π⁺_α).We have again F G(−AT) =F(−AT)G(−AT),and, indeed,F(−AT) =AT ifF(z) =−z.

I. Chalendar, J. Partington and the author obtained in [7] lower estimates for small values ofof the norm of operators of the formF(−AT),whereF is the Laplace transform of a real-valued measure on[a, b],with0< a < b <+∞, and whereT= (T(t))_t>0is a norm-continuous semigroup, see also [15],[13], [3].

The motivation for this paper was to open the road to extensions of these lower estimates to much more general situations.

The idea of considering the infinitesimal generator of a semigroup as a quasi- multipier on a suitable Banach algebra associated to the semigroup is due to Galé and Miana, who used this approach in [19] for the infinitesimal generator of some one-parameter groups of regular quasimultipliers, see also [18]. Chal- endar, Partington and the author also used this approach for the infinitesimal generator on analytic semigroups. We deal here with a more general situation, and the author hopes that this approach centered on quasimultipliers and regular quasimultipliers on the Arveson idealIT will be helpful to mathematicians who use holomorphic functional calculus associated to semigroups.

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2 Arveson weakly continuous one-parameter semi- groups

LetX = (X,k.k)be a Banach space. We denote byB(X)the Banach algebra of bounded linear operatorsR:X →X with composition, we denote byGL(X) the group of invertible elements ofB(X)and we denote byI=I_X the identity map onX.We also denote byk.kthe operator norm onB(X)associated to the given norm onX, and we denote byρ(R) the spectral radius ofR ∈ B(X). If Y is a subspace of the dual spaceX⁰ ofX we will denote byσ(X, Y)the weak topology onX associated to Y.

We will use the following notion, introduced by Arveson in [2].

Definition 2.1 Let X be a Banach space, and let X_∗ be a subspace of the dual space X⁰. We will say that (X, X_∗) is a dual pairing if the two following conditions are satisfied:

1. kxk= sup{|hx, li| : l∈X∗,klk ≤1} for every x∈X.

2. The σ(X, X_∗)-closed convex hull of every σ(X, X_∗)-compact subset of X isσ(X, X_∗)-compact.

For example,(X, X⁰)is a dual pairing. Also ifX =Y⁰ for some Banach space Y,and if we identify Y to a subspace ofX⁰ = (Y⁰)⁰ in the obvious way, then (X, Y)is a dual pairing, see [2]. Notice that condition 1 means that if we set

˜

x(l) =hx, liforx∈X, l∈X∗ then the map x7→x˜is an isometry fromX into the dual space (X∗)⁰. This means that the spaces X and X∗ are reciprocal in the sense of definition 2 of [16]. An operator T ∈ B(X) is said to be weakly continuous with respect to a dual pairing(X, X_∗)ifl◦T ∈X_∗ for everyl∈X_∗. LetS be a locally compact space, and let(X, X_∗)be a dual pairing. A map u:S →X is said to be weakly continuous with respect to(X, X_∗)if the map s 7→ hu(s), li is continuous on S for everyl ∈ X_∗, and we will often just say thatuis weakly continuous when no confusion may occur. In this situation it follows from the Banach-Steinhaus theorem and from condition 1 of definition 2.1 that we have, for every compact subsetKofS,

sup

s∈K

ku(s)k= sup

s∈K

ku(s)kg = sup

s∈K

sup

l∈X∗,klk≤1

|hu(s), li|<+∞. (1) Sinceku(s)k= sup_l∈X_∗|hu(s), li|fors∈S, the functionωu:s7→ ku(s)k is lower semicontinuous onS, which allows to compute the upper integral

Z ∗

S

ku(s)kd|µ|(s) := sup

f∈C+ c(S) f≤ωu

Z

S

f(s)d|µ|(s)∈[0,+∞] (2)

for every regular measureµonS,whereC⁺_c(S)denotes the space of all nonneg- ative compactly supported continuous functions onS.The following proposition is an immediate generalization of proposition 1.2 of [2]. The details of the proof are given in [14].

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Proposition 2.2 LetSbe a locally compact space, let(X, X_∗)be a dual pairing, and let u: S → X be a weakly continuous map. Set ωu(s) = ku(s)k for s ∈ S, and denote by Mω_u(S) the set of all regular measures µ on S such that kµkωu := R∗

Sku(s)kd|µ|(s) < +∞. Then for every µ ∈ Mωu(S) there exists x∈X satisfying

hx, li= Z

S

hu(s), lidµ(s) (l∈X_∗). (3) Proof: SinceR

S|hu(s), li|d|µ|(s)≤ klkkµkω_u <+∞for everyl∈X_∗,the formula fµ(l) :=R

Shu(s), lidµ(s)forl∈X_∗ defines an elementfµ∈(X_∗)⁰,and we have to show thatfµ = ˜xfor some x∈ X.It follows from condition 1 of definition 2.1 that we have

kf_µk= sup

l∈X∗,klk≤1

Z

S

hu(s), lidµ(s)

≤ kµk_ω_u.

Denote byMc(S)the space of all regular measures onSsupported by some compact subset ofS. It follows from (2) thatMc(S)⊂ Mω_u(S),and the fact that property (2.2) holds for everyµ∈ Mc(S)follows directly from [2], proposition 2.1. SetX˜ :={˜x:x∈X}.It follows from condition 1 of definition 2.1 thatX˜ is closed in(X_∗)⁰.Letµ∈ Mωu(S).There exists a sequence (µ_n)_n≥1 of elements ofMc(S)such thatlim_n→+∞kµ−µ_nkωu = 0.Hencelim_n→+∞kfµ−f_µ_nk= 0,

andfµ∈X.˜

When the conditions of proposition 2.2 are satisfied, we will use the notation x=

Z

S

u(s)dµ(s), (4)

where the integral is a Pettis integral, that is a Bochner integral computed with respect to theσ(X, X_∗)topology, which defines an element ofX since(X, X_∗) is a dual pairing.

Notice that since ωu is lower semicontinuous, it follows from the theory of integration on locally compact spaces, see [4], chapter 4, that Mω_u(S) is the space of regular measuresµsuch thatωu is integrable with respect to the total variation|µ|of µ,and it follows from [4], proposition 1, that we have for µ∈ Mω_u(S)

kµkω_u = Z

S

ku(s)kd|µ|(s) = Z ∗

S

ku(s)kd|µ|(s) = sup

K⊂S,Kcompact

Z ∗

K

ku(s)kd|µ|(s).

Let X be a Banach space. A mapT : t → T(t) from (0,+∞)into B(X) is called a (one-parameter) semigroup ifT(s+t) =T(s)T(t) fors > 0, t > 0.

We will often write T= (T(t))t>0. A semigroup will be said to be normalized if∪t>0T(t)(X)is dense inX.

We now introduce the following definition, which is an obvious extension to one-parameter semigroups of the notion of weakly continuous group representations associated to a dual pairing, due to Arveson [2].

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Definition 2.3 Let (X, X_∗)be a dual pairing, and let Bw(X)the the space of weakly continuous elements of B(X). A semigroup T = (T(t))t>0 ⊂ Bw(X) is said to be weakly continuous with respect to the dual pairing (X, X∗) if the function t→< T(t)x, l > is continuous on(0,+∞) for every x∈X and every l∈X∗.

Let(X, X_∗)be a dual pairing, and letT= (T(t))_t>0⊂ B_w(X)be a weakly continuous semigroup. We have, fort >0,

kT(t)k= sup{|hT(t)x, li|:x∈X,kxk ≤1, l∈X_∗,klk ≤1}

and so the weight ω_T : t 7→ kT(t)k is lower semicontinuous on (0,+∞). Let K ⊂ (0,+∞) be compact. Since sup_t∈KkT(t)xk < +∞ for every x ∈ X, it follows again from the Banach-Steinhaus theorem thatsup_t∈KkT(t)k<+∞.A standard argument shows then thatlim_t→+∞kT(t)k¹^t =lim_n→+∞kT(sn)k^sn¹ = inf_n≥1kT(sn)k^sn¹ for everys >0.In this situation we can define the weighted space Mω_T =Mω_T[(0,+∞)] consisting of all regular measuresµ on (0,+∞) such that the upper integral R+∞

0 kT(t)kd|µ|(t) is finite. Since ωT(t1+t2) ≤ ωT(t1)ωT(t2) fort1>0, t2 >0, we see thatµ and ν are convolable forµ, ν ∈ Mω_T and that (Mω_T,k.kω_T) is a Banach algebra with respect to convolution which contains the convolution algebra Mc[(0,+∞)] of compactly supported regular measures on(0,+∞)as a dense subalgebra.

Denote byL¹_ω_T =L¹_ω_T(R⁺)the convolution algebra of all Lebesgue-measurable (classes of) functions f on [0,+∞) such that f ωT is integrable with respect to the Lebesgue measure on [0,+∞), identified to the space of all measures µ∈ MωT which are absolutely continuous with respect to Lebesgue measure.

Also denote byC_c^∞[(0,+∞)] the space of all infinitely differentiable functions f ∈ C_c[(0,+∞)]. We will say as usual that a sequence (f_n)_n≥1 of elements of L¹(R⁺)is a Dirac sequenceiffn(t)≥0 a.e. fort≥0, R+∞

0 fn(t)dt= 1and if there exists a sequence(an)n≥1 of positive real numbers which converges to 0 such thatfn(t) = 0 a.e. on[an,+∞)forn≥1.

The following result is an easy analog of [2], proposition 1.4.

Proposition 2.4 Let(X, X_∗)be a dual pairing, and letT= (T(t)_t>0⊂ B_w(X) be a weakly continuous semigroup. The Pettis integral

< φ_T(µ)x, l >=

Z +∞

0

< T(t)x, > dµ(t) (l∈X_∗) (5) defines for everyx∈X and every µ∈ MωT an element of X, φ_T(µ)∈ B(X) for every µ∈ MωT, and φ_T :µ 7→φ_T(µ) is a norm-decreasing unital algebra homomorphism from the convolution algebraM_ω_T intoB(X).

Moreover if(f_n)_n≥1 is a Dirac sequence, then we have, for t >0 (i)lim_n→+∞kφT(fn∗δt)φT(f)−T(t)φT(f)k= 0∀f ∈L¹_ω

T, (ii)lim_n→+∞hφT(fn∗δt)x−T(t)x, li= 0∀x∈X,∀l∈X_∗.

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Proof: SincekT(t)xk ≤ kT(t)kkxk,the fact that formula 5 defines an element ofX forx∈X andµ∈ Mω_T follows directly from proposition 2.2. We have

kφT(µ)xk = sup

l∈X∗,klk≤1

Z +∞

0

hT(t)x, lidµ(t)

≤ sup

l∈X∗,klk≤1

Z +∞

0

|hT(t)x, li|d|µ|(t)≤ kxkkµkωT,

and soφ_T(µ)∈ B(X)forµ∈ MωT and kφT(µ)k ≤ kµkωT.As observed in [2], a routine application of Fubini’s theorem shows thatφ_T(µ∗ν) = φ_T(µ)φ_T(ν) forµ, ν ∈ Mc((0,+∞)). SinceMc((0,+∞))is dense in MωT,this shows that φ_T is an algebra homomorphism.

The last assertions pertain to folklore, but we give the details for the sake of completeness. Let n ≥ 1, and let (f_n)_n≥1 be a Dirac sequence. We can assume without loss of generality thatfn(t) = 0a.e. on[1,+∞)forn≥1.The sequence(fn)_n≥1is a sequential bounded approximate identity forL¹(R⁺).So iff ∈L¹_ω_T,and ifsupp(f)⊂[a, b],with0< a < b <+∞,we have

limsup_n→+∞kfn∗δt∗f−δt∗fk_L1

ωT ≤limM sup_n→+∞kfn∗δt∗f−δt∗fk_L1(R⁺)= 0, whereM =supa+t≤s≤b+1+tkωT(s)k<+∞.

Since kfn∗δ_tkL¹_ω

T ≤sup_t≤s≤t+1kT(s)kR1

0 f_n(s)ds=sup_t≤s≤t+1kT(s)k for n≥1, a standard density argument shows thatlimsup_n→+∞kf_n∗δ_t∗f−δ_t∗ fk_L1

ωT

= 0,and (i) holds.

We have, forx∈X, l∈X_∗, t >0

|hφ_T(f_n∗δ_t)x−T(t)x, li| =

Z an

0

hT(s+t)x, lif_n(s)ds− hT(t)x, li Z an

0

f_n(s)ds

≤ sup

0≤t≤an

|hT(s+t)x−T(t)x, li|

Z a_n

0

fn(t)dt)

= sup

0≤t≤an

|hT(s+t)x−T(t)x, li|.

Hence lim_n→+∞|hφT(f_n∗δ_t)x−T(t)x, li| = 0, since the semigroup is weakly continuous with respect to the dual pairing(X, X_∗).

In the following (X, X_∗) will denote a dual pairing and T = (T(t))_t>0 ⊂ B_w(X) will denote a semigroup of bounded weakly continuous operators on X which is weakly continuous with respect to this dual pairing (X, X_∗). The weak topologyσ(X, X_∗)associated to the dual pairing will be called the weak topology onX.

We will use the following notations, forx∈X, µ∈ Mω_T, φT(µ)x=

Z +∞

0

T(t)xdµ(t), φT(µ) = Z +∞

0

T(t)dµ(t). (6)

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In particular we will write, forf ∈L¹_ω_T, Z +∞

0

f(t)T(t)dt=φ_T(f). (7)

3 The Arveson ideal

Denote by X1 = [∪t>0T(t)(X)]⁻ the closure of the space ∪t>0T(t)(X) in (X,k.k). Since Mc((0,+∞)) is dense in MT, we have φ_T(µ)x ∈ X₁ for µ ∈ MωT, x∈X.So we may considerφ_Tas an element ofB(X1)for everyµ∈ MωT, andkφ_T(µ)k_B(X) ≥ kφ_T(µ)k_B(X₁₎ for everyµ∈ M_T. The trivial example obtained by settingT(t) = P for t > 0, where P : X →Im(P) is a projection satisfyingkPk_B(X)>1,shows that equality does not hold in general.

Definition 3.1 We will denote byAT the closed subalgebra ofB(X1)generated byφT(Mω_T),and by IT the closed subalgebra of B(X1)generated by φT(L¹_ω_T), so thatIT is a closed ideal of AT,called the Arveson ideal of the semigroupT.

The fact thatIT is an ideal ofAT follows from the fact thatL¹_ω_T is an ideal of the convolution algebraMω_T,so thatφT(L¹_ω_T)is an ideal ofφT(Mω_T).

Proposition 3.2 (i) The map t →T(t)u is norm-continuous on(0,+∞) for everyu∈ IT.

(ii) The ideal IT equals the whole algebra AT if and only if the semigroup (T(t))t>0 is continuous with respect to the norm of B(X).

(iii) If theσ(X_∗, X)-closed convex hull of every σ(X_∗, X)-compact subset of X is σ(X_∗, X)-compact, then all elements of A_T are weakly continuous with respect toX_∗.

Proof: The map t → δt∗f is continuous on (0,+∞) for every f ∈ L¹_ω_T. Hence the map t → T(t)φT(f) is continuous on (0,+∞) for every f ∈ L¹_ω_T. Sincesup_a≤t≤bkT(t)k<+∞for0< a < b <+∞,this shows that (i) holds.

Now assume that the semigroup is continuous with respect to the norm on B(X). A well-known argument, see for example proposition 6.1 in [11], shows then that T(t)∈ IT for every t >0. Hence φT(µ)∈ IT for every µ∈ Mc((0,+∞)).SinceMc((0,+∞))is dense inMω_T,this shows thatAT=IT. Conversely it follows immediately from (i) that if IT =AT then the semigroup(T(t))t>0 is continuous with respect to the norm ofB( ˜X).

Now assume that theσ(X_∗, X)-closed convex hull of everyσ(X_∗, X)-compact subset ofX isσ(X_∗, X)-compact. Then(X_∗, X)is a dual pairing, and we see as in the proof of the second assertion of proposition 1.4 of [2] thatl◦φ_T(µ)∈X_∗ forµ∈ M_ω_T, l∈X_∗. Hencel◦u∈X_∗ foru∈ A_T, l∈X_∗.

Proposition 3.3 Let u∈ AT.If uv= 0 for everyv∈ IT,thenu= 0.

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Proof: Let(fn)_n≥1be a Dirac sequence, and letnbe a sequence of positive real numbers such thatlimn→+∞n= 0.We have, forx∈X, l∈X∗, t >0,

< uT(t)x, l >=< T(t)ux, l >=lim_n→+∞< φT(fn∗δ_n∗δt)ux, l >=

= lim

n→+∞< uφ_T(f_n∗δ_n)T(t)x, l >= 0.

Hence uT(t)x= 0for everyx∈X,and sou= 0since∪t>0T(t)X is dense inX1.

Recall that a sequence(en)_n≥1of elements of a commutative Banach algebra Ais said to be a sequential bounded approximate identity forAifsup_n≥1kenk<

+∞and iflim_n→+∞kaen−ak= 0 for everya∈ A.

Proposition 3.4 If limsup_t→0+kT(t)k < +∞, then the sequence φ_T(f_n)_n≥1 is a bounded sequential approximate identity for IT for every Dirac sequence (fn)_n≥1.

Proof: Assume thatlimsup_t→0+kT(t)k<+∞,and setM =sup_0<t≤1kT(t)k.

Thenkφ_T(f_n)k ≤M whennis sufficiently large. Since(f_n)_n≥1 is a sequential bounded approximate identity for L¹(R⁺), and since sup_a≤t≤bkT(t)k < +∞

for 0 < a < b < +∞, we see that lim_n→+∞kf_n ∗f −fk_L1 ωT

= 0, so that lim_n→+∞kφT(fn)∗φT(f)−φT(f)k = 0 for f ∈ Cc((0,+∞)). Since the sequence(φT(fn))_n≥1is bounded, this shows thatlim_n→+∞kφT(fn)∗u−uk= 0 foru∈ IT.

4 The generator as a weakly densely defined op- erator

We consider as above the operators(T(t))t>0as operators on the Banach space X₁:= [∪t>0T(t)X]⁻.We set

X˜ :={x∈X₁ | lim

t→0⁺< T(t)x−x, l >= 0∀l∈X_∗}.

We will use the conventionT(0) =I_X₁,the identity map onX₁.

Clearly,∪t>0T(t)(X)⊂X,˜ and[∩t>0Ker(T(t)]∩X˜ ={0}.Notice that ifx∈ X,˜ it follows from the Banach-Steinhaus theorem that lim sup_t→0+kT(t)xk=lim sup_t→0+kT(t)xk_(X_∗₎0 <+∞.Also if lim sup_t>0kT(t)k<+∞,thenX˜ is closed, so thatX˜ =X1.

The following definition is a variant of a definition introduced by Feller in [16] for semigroups satisfying some weak measurability conditions.

Definition 4.1 Denote byDT,op the space of allx∈X1 such that there exists y∈X˜ satisfying

lim

t→0⁺<T(t)x−x

t −y, l >= 0 ∀l∈X_∗. (8)

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In this situation we set AT,opx=y.

Notice that if x∈ DT,op, then AT,opx∈ X1. Also l◦T(t) ∈ X∗ for every t >0 and every l ∈X∗, and so when condition (8) is satisfied by xand y we havelim_t→0+ <^T(t+s)x−T(s)x

t −y, l >= 0for every l∈X_∗ and everys >0.

Ifx∈ DT,op,and ify=AT,opx,then we have, forl∈X_∗, t >0,

< T(t)x−x, l >=

Z t

0

< T(s)y, l > dt, and so, according to the notations of formula (4), we have

T(t)x−x= Z t

0

T(s)yds. (9)

Conversely, if y∈X,˜ and ifT(t)x−x=Rt

0T(s)ydsfort >0,then xandy obviuously satisfy (8) and sox∈ DT,op,andy=AT,opx.

Proposition 4.2 Set ρT= lim_t+∞kT(t)k¹^t, and for Re(λ)> log(ρT), y∈X,˜ defineRop(T, λ)y by the formula

< R_op(T, λ)y, l >:=

Z +∞

0

e^−λs< T(s)y, l > ds ∀l∈X_∗. (10) .

In other terms, according to the notations of formula (6), we have

R_op(T, λ)y= Z +∞

0

e^−λsT(s)yds.

ThenRop(T, λ)y∈ DT,op for y ∈X, R˜ op(T, λ) : ˜X → DT,op is one-to-one and onto, and we have

(λI_X₁−A_T,op)◦R_op(T, λ) =IX˜, Rop(T, λ)◦(λIX₁−AT,op) =ID_T,op.

Proof: The proof is standard, and we give the details for the sake of completeness. Let y ∈ X.˜ Since R+∞

0 e^−Re(λ)skT(s)ykds < +∞, Rop(T, λ)y is well-defined. Setx=R_op(T, λ)y.Sincel∗T(t)∈X_∗ for everyl∈X_∗and every t >0, we have

T(t)x= Z +∞

0

e^−λsT(t+s)yds=e^λt Z +∞

t

T(s)yds, T(t)x−x

t =e^λt−1 t x−e^λt

Rt

0T(s)yds

t ,

where the integral is computed with respect to the weak operator topology on X associated to the dual pairing(X, X_∗).

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We obtain, forl∈X_∗, lim

t→0⁺< T(t)x−x

t , l >=< λx−y, l >, and sox∈ DT,op,andAT,opx=λx−y, y=λx−AT,opx.

This shows thatR_op(T, λ)( ˜X)⊂ DT,op,and that(λI_X₁−A_T,op)◦Rop(T, λ) = IX˜,and soRT,op(λ) : ˜X → DT,opis one-to-one.

Conversely letx∈ DT,op,setu=AT,opx,and setS(t) =e^−λtT(t)fort >0.

Then x∈ DS,op, and if we sety =AS,opx we have y = u−λx. We have, for t >0,

S(t)x−x= Z t

0

S(s)yds= Z t

0

e^−λsT(s)yds.

Sincelim_t→+∞kS(t)k= 0,we obtain

x=− Z +∞

0

e^−λsT(s)yds=−RT,op(λ)y=λRT,op(λ)x−(RT,op(λ)◦AT,op)x.

HenceRT,op(λ) : ˜X→ DT,opis onto, and we haveRT,op(λ)◦(λIX₁−AT,op) = I_D_T,op.

Proposition 4.3 The domainDT,op ofAT,opis weakly dense in X1.

Proof: Letf ∈ C^∞_c ([0,+∞)),and denote byk.k1the usual norm onL¹(R⁺).

Then

lim

t→0⁺

δt∗f −f t −f⁰

_ω

T

= lim

t→0⁺

δt∗f−f t −f⁰

₁

= 0,

and so lim_t→0+

T(t)∗φT(f)−φT(f)

t −φT(f⁰)

= 0. Since φT(f⁰) ⊂ T(t)AT for some t > 0, we have φT(f)(X) ⊂ X,˜ and so φT(f)(X) ⊂ DT,op. Now let (fn)n≥1 ⊂ C_c^∞([0,+∞)) be a Dirac sequence. It follows from proposition 2.4 that we have forx∈X, l∈X∗, t >0,

n→+∞lim < φT(fn)T(t)x−T(t)x, l >= 0.

Hence DT,opis weakly dense in∪t>0T(t)(X),and soDT,op is weakly dense inX₁.

Recall that a partially defined operatorS:D →E on a Banach spaceE is said to be closed if the graph{x, Sx:x∈ D}ofS is closed. Notice that if lim sup_t→0+kT(t)k < +∞, then X˜ = X₁, and so R(T, λ) : ˜X → DT,op ⊂X˜ is a bounded operator. Hence its inverseλIX˜ −AT,op :DT,op →X˜ is closed, and AT,opis also closed. Now assume that(X_∗, X)is also a dual pair, which means that theσ(X_∗, X)-closed convex hull of everyσ(X_∗, X)-compact subset ofX_∗is σ(X_∗, X)-compact. Setuλ(t) =e^−λtforλ∈C, t≥0.ThenRT,op(λ) =φT(uλ) for Re(λ) > ρT, and so it follows from proposition 3.2(iii) that RT,op(λ) is

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weakly continuous with respect to X_∗, which shows as above that AT,op is weakly closed.

Check that the infinitesimal generator is not closed if lim sup_t→0+kT(t)k= +∞.

5 Quasimultipliers on the Arveson ideal

IfU be a commutative Banach algebra, setΩ(U) :={u∈ U | [uU]⁻ =U },and set U^⊥ ={u ∈ U |uU = {0}}. We recall the definition of quasimultipliers, a notion introduced by the author in [12].

Definition 5.1 [12] Assume that Ω(U)6=∅ and that U^⊥={0}.A quasimultiplier on U is a pair(S_u/v,DS_u/v), where u∈ U, v ∈Ω(U), where DS_u/v is the ideal ofU consisting of thosex∈ U such that ux∈vU,and whereS_u/vxis the uniquey ∈ U such thatux=vy forx∈ DS_u/v. The set of quasimultipliers on U will be denoted by QM(U), and a set B ⊂ QM(U) will be said to be pseudobounded if there existsx∈[∩_S∈BDS]∩Ω(U)such that sup_S∈BkSxk<+∞.

The quasimultipliers onU form an algebra, which is isomorphic to the algebra of fractionsU/Ω(U). Notice that since Ω(U) is stable under products, the product of two pseudobounded sets is pseudobounded, which givesQM(U) a structure of "algèbre à bornés".

The algebra QM(U) is in some sense too large, since u in invertible in QM(U)for every u∈Ω(U),so it is natural to consider the following class.

Definition 5.2 [12] A quasimultiplierR∈ QM(U)is said to be regular if there existsλ >0such that the family(λⁿRⁿ)_n≥1is pseudobounded. The set of regular quasimultipliers onU will be denoted byQM_r(U), and a setB⊂ QM_r(U)will be said to be multiplicatively pseudobounded if it is contained in some set of the form λV, where λ > 0 and where V is a pseudobounded subset of QMr(U) which is stable under products.

Recall that a multiplier on U is a bounded linear map S : U → U such that(Su)v=S(uv)for(u, v)∈ U. The setM(U)of multipliers on M(U)is a closed subalgebra of B(U), and U can be identified to an ideal ofM(U)in an obvious way. We also have the obvious identification M(U) :={S = Su/v ∈ QM(U)|DS_u/v) =U } ⊂ QMr(U).

Equipped with the family of all its multiplicatively pseudobounded subsets, the setQMr(U)forms an algebra which is a pseudo-Banach algebra in the sense of Allan, Dales and McClure [1]. In particular every maximal ideal ofQMr(U) is the kernel of a character of QM_r(U), andQM_r(U)is an inductive limit of commutative Banach algebras. In fact regular quasimultipliers on U can be turned into multipliers in the usual sense by modifying the algebraU,as will be seen later.

Precise estimates on inverse Laplace transforms, obtained by the author in section 6 of [11] in collaboration with Paul Koosis, combined with Nyman’s

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characterization of dense ideals ofL¹(R⁺),show that if ω is a positive, continuous, nonincreasing submultiplicative weight on(0,+∞) the convolution alge- braL¹_ω := L¹_ω(R⁺) contains an infinitely differentiable convolution semigroup (a(t))t>0 such thata(t)∗L¹_ω is dense in L¹_ω for everyt >0, [11], theorem 6.8.

This result uses the fact that iff is a continuous function on(0,+∞)such that lim sup_t→∞f(t)<+∞,then there exists a continuously differentiable function h on (0,+∞) such that h(t) ≥ f(t) for t > 0, and the rather sophisticated contruction, applied to the functionf :t →e^−λtω(t)for some suitable λ∈R, consists in finding a convolution semigroup (a(t))_t>0 of functions in L¹(R⁺) such that lim sup_t→0+h(t)ka(t)k < +∞ and such that the convolution ideal generated by a(t) is dense in L¹(R⁺) for every t > 0 (which implies that the Laplace tranform ofa(t)is an outer function of the right-hand half-plane). The fact thata(t)∗L¹_ωis dense inL¹_ωfollows from Cohen’s factorization theorem for modules over commutative Banach algebras with bounded approximate identi- ties, which implies that everyf ∈L¹_ωcan we written under the formf =g∗h, whereg∈L¹(R⁺)andh∈L¹_ω.

Now assume thatf is positive and lower semicontinuous on(0,+∞),so that f is bounded on [a, b] for 0 < a < b < +∞, and that lim sup_t→+∞f(t) <

+∞. Set f1(t) =sup_s≥tf(s), f2(t) = ²_tRt

t/2f1(s)ds. Then f1 is nonincreasing, f₁(s/2)≥f₂(s)≥f₁(s)≥f(s),and applying the previous result to f₂ we see that there exists a continuously differentiable functionhon(0,+∞)such that h(t) ≥ f(t) for t > 0. This shows that the construction used in the proof of theorem 6.8 of [11] extends to lower-semicontinuous weights on (0,+∞). Set u_λ(t) = e^λt for t > 0. Since the map f → f u_−λ is a norm-preserving homomorphism from the convolution algebra L¹_ω(R⁺) onto the convolution algebra L¹_u_λ_ω(R⁺)forλ∈R,we obtain the following result.

Theorem 5.3 Letωbe a positive, lowersemicontinuous, submultiplicative weight on(0,+∞), and assume that ω˜ :t→e^λtω(t)is nonincreasing for some λ∈R. Then the convolution algebraL¹_ω:=L¹_ω(R⁺)contains an infinitely differentiable semigroup(a(t))t>0 such thata(t)∗L¹_ω is dense inL¹_ω for every t >0.

We obtain the following result, which shows in particular that the ideal IT possesses dense principal ideals, and that the family (e^−λtT(t))t>0 is pseu-

dobounded inQM(I_T)forλ >lim_t→+∞kT(t)k¹^t.We set againρ_T:= lim_t→+∞kT(t)k¹^t. Proposition 5.4 Let λ > log(ρ_T), and set ω_λ(t) = e^λtsup_s≥te^−λskT(s)k for

λ > 0. Let g ∈ L¹_ω_λ. Then kφT(g)T(t)k ≤ e^λtkgk_L1

ωλ for every g ∈ L¹_ω_λ and everyt >0,andφ_T(g)⊂Ω(IT)for everyg∈Ω L¹_ω

λ

. Proof: Notice that kT(t)k ≤ ω_λ(t) for t > 0, so that L¹_ω

λ ⊂ L¹_ω

T. Also ω_λ(s)≥ eλssup_t≥0e^−λ(s+t)kT(s+t)k, and so kT(s+t)k ≤ e^λtω_λ(s) for s ≥ 0, t >0.We have, fort >0, g∈L¹_ω

λ, kφT(g)T(t)k=kφT(g∗δt)k ≤

Z +∞

t

g(s−t)T(s)ds

≤ Z +∞

t

|g(s−t)|kT(s)kds

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≤ Z +∞

0

|g(s)|kT(s+t)kds≤e^λtkgk_L1 ωλ. Since L¹_ω

T contains C_c^∞[(0,+∞)], L¹_ω

λ is dense in L¹_ω

T, and so Ω(L¹_ω

λ) ⊂ Ω(L¹_ω_T),which shows thatφT Ω(L¹_ω_λ)

⊂Ω(IT).

6 Quasimultipliers on the Arveson ideal as weakly densely defined operators

We now associate to every quasimultiplier onIT a weakly densely defined operator onX1.Set againX˜ :={x∈X1| lim_t→0+< T(t)x−x, l >= 0∀l∈X∗}.

Lemma 6.1 Let v ∈Ω(IT). ThenKer(v)⊂ ∩t>0Ker(T(t), andv : ˜X →X1

is one-to-one.

Proof: Letx∈Ker(v), and let (f_n)_n≥1 ⊂ C_c((0,∞))be a Dirac sequence.

Since vI_T is dense in I_T, there exists a sequence (w_n)_n≥1 of elements of I_T such that lim_n→+∞kθ_T(f_n)−vw_nk = 0. It follows then from proposition 2.4 (ii) that we have, forx∈X, l∈X_∗, t >0,

lim_n→+∞< vT(t)wnx−T(t)x, l >= 0.

SincevT(t)wnx=T(t)wnvx= 0,we havex∈ ∩t>0Ker(T(t)).Since(∩t>0Ker(T(t))∩

X˜ ={0},this shows thatv is one-to-one onX.˜ .

Proposition 6.2 Let S =S_u/v ∈ QM(IT). Set DS˜ :={x∈X | ux∈v( ˜X)}, and denote by S(x)˜ the unique y ∈ X˜ satisfying ux =vy. ThenDS˜ is weakly dense inX1. If lim sup_t→0+kT(t)k<+∞, then S˜ is a closed partially defined operator on X1, andS˜ is weakly closed if, further, the σ(X_∗, X)-closed convex hull of everyσ(X_∗, X)-compact subset ofX_∗ isσ(X_∗, X)-compact.

Proof: It follows from the lemma thatS˜is well-defined onDS˜.SincevIT is dense inIT,we see as above there exists a sequence(wn)n≥1of elements ofIT

such that we have, forx∈X, l∈X∗, t >0,

limn→+∞< vT(t)wnx−T(t)x, l >= 0.

SinceuvT(t)w_n =vuT(t)w_n ∈v( ˜X), we haveT(t)w_nx∈ DS˜ fort >0, n≥ 1, x ∈ X, and we see that the closure of DS˜ in X₁ with respect to the weak topology σ(X, X_∗) containsT(t)(X) for every t > 0, which shows that DS˜ is weakly dense inX₁.

If lim sup_t→0+kT(t)k < +∞, then X˜ = X₁, and DS˜ = {x ∈ X₁ | ux ∈ v(X1)}.Let(x, y)∈X1×X1,and assume that there exists a sequence(xn)_n≥1 of elements ofDS˜such that limn→+∞kx−xnk=limn→+∞ky−Sx˜ nk= 0.Then ux =lim_n→+∞ux_n =lim_n→+∞vy_n = vy. Hence x ∈ DS˜ and y = ˜Sx, which shows that S˜ is a closed operator. Now if the σ(X_∗, X)-closed convex hull of