On the generation of Arveson weakly continuous semigroups · if kxk 1=2;then kxk>1=4:Concerning (nonzero) strongly continuous semigroups T = (T(t)) t>0 of bounded operators

On the generation of Arveson weakly continuous semigroups

Jean Esterle

University of Oulu - July 9, 2017

Introduction

The author observed in [11] that if a Banach algebra A does not possess anynonzero idempotent then inf x∈A

‖x‖≥1/2‖x2 − x‖ ≥ 1/4. If x is quasinilpotent, and

if ‖x‖ ≥ 1/2, then ‖x‖ > 1/4. Concerning (nonzero) strongly continuoussemigroups T = (T (t))t>0 of bounded operators on a Banach space X , theseelementary considerations lead to the following results, obtained in 1987 byMokhtari

1 If lim supt→0+ ‖T (t)− T (2t)‖ < 1/4, then the generator of the semigroup isbounded, and so lim sup ‖T (t)− T (2t)‖ = 0.

2 If the semigroup is quasinilpotent, then ‖T (t)− T (2t)‖ > 1/4 when t issufficiently small.

If the semigroup is norm continuous, and if there exists a sequence (tn)n≥1 ofpositive real numbers such that limn→+∞ tn = 0 and ‖T (tn)− T (2tn)‖ < 1/4,then the closed subalgebra AT of B(X ) generated by the semigrouppossesses an exhaustive sequence of idempotents, i.e. there exists asequence (Pn)n≥1 of idempotents of AT such that for every compact setK ⊂ AT there exists nK > 0 satisfying χ(Pn) = 1 for χ ∈ K , n ≥ nK .

Jean Esterle On the generation of Arveson weakly continuous semigroups

Introduction, continued

More sophisticated arguments allowed A. Mokhtari and the author to obtainin 2002 the following more general results [15], valid for every integer p ≥ 1,

1 If lim supt→0+ ‖T (t)− T ((p + 1)t)‖ < p(p+1)1+1/p , then the generator of the

semigroup is bounded, and so lim sup ‖T (t)− T ((p + 1)t)‖ = 0.2 If the semigroup is quasinilpotent, then ‖T (t)− T ((p + 1)t)‖ > p

(p+1)1+1/p

when t is sufficiently small.3 If the semigroup is norm continuous, and if there exists a sequence (tn)n≥1

of positive real numbers such that limn→+∞ tn = 0 and‖T (tn)− T ((p + 1)tn)‖ < p

(p+1)1+1/p for n ≥ 1, then the closed subalgebra AT

of B(X ) generated by the semigroup possesses an exhaustive sequenceof idempotents.



These results led the author to consider in [12] the behavior of the distance‖T (s)− T (t)‖ for s > t near 0. The following results were obtained in [12]

1 If there exist for some δ > 0 two continuous functions r → t(r ) andr → s(r ) > t(r ) on [0, δ] such that

‖T (t(r ))− T (s(r ))‖ < (s(r )− t(r ))s(r)s(r)

s(r)−t(r)

t(r)t(r)

s(r)−t(r), then the generator of the

semigroup is bounded, and so ‖T (t)− T (s)‖ → 0 as 0 < t < s, s → 0+.2 If the semigroup is quasinilpotent, there exists δ > 0 such that

‖T (t)− T (s)‖ > (s − t)ss

s−t

tt

s−tfor 0 < t < s ≤ δ.

3 If the semigroup is norm continuous, and if there exists two sequences ofpositive real numbers such that 0 < tn < sn, limn→+∞ sn = 0, such that

‖T (tn)− T (sn)‖ < (sn − tn)ssn

sn−tnn

ttn

sn−tnn

, then the closed subalgebra AT of B(X )

generated by the semigroup possesses an exhaustive sequence ofidempotents.



The quantities appearing in these statements are not mysterious: considerthe Hilbert space L2([0,1], and for t > 0 define T0(t) : L2([0,1]→ L2([0,1] bythe formula T0(t)(f )(x) = x t f (x) (0 < x ≤ 1). Then

‖T0(t)− T0(s)‖ = (s − t)ss

s−t

tt

s−t. This remark also shows that the assertions (1)

and (3) in these statements are sharp, and examples show that theassertions (2) are also sharp.One can consider T (t) as defined by the formula

∫ +∞0 T (x)dδt(x), where δt

denotes the Dirac measure at t . Heuristically, T (t) = etA, where A denotesthe generator of the semigroup, and since the Laplace transform of δt isdefined by the formula L(δt)(z) =

∫ +∞0 e−zxdδt(x) = e−zt , it is natural to write

L(δt)(−A) = T (t). More generally, if an entire function F has the formF = L(µ), where µ is a measure supported by [a,b], with 0 < a < b < +∞,we can set

F (−A) =

∫ +∞

0T (x)dµ(x),

and consider the behavior of the semigroup near 0 in this context.



I.Chalendar, J.R. Partington and the author used this point of view in [6].Denote byMc(0,+∞) the set of all measures µ supported by some interval[a,b], where 0 < a < b < +∞. For the sake of simplicity we restrict attentionto statements analogous to assertion 2. We have the following resultQuasinilpotent semigroupsTheorem: Let µ ∈Mc(0,+∞) be a nontrivial real measure such that∫ +∞

0 dµ(t) = 0 and let T = (T (t))t>0 be a quasinilpotent semigroup ofbounded operators.Then there exists δ > 0 such that‖F (−sA)‖ > maxx≥0 |F (x)| for 0 < s ≤ δ.

When µ = δ1 − δ2 this gives assertion 3 of Mokhtari’s result, and whenµ = δ1 − δp+1 this gives assertion 3 of the Esterle-Mokhtari result (but toobtain extensions of the results of [12] one would need several variablesextensions of this functional calculus).This theory applies, for example, to quantities of the form‖T (t)− 2T (2t) + T (3t)‖, or Bochner integrals ‖

∫ 21 T (tx)dx −

∫ 32 T (tx)dx‖,

which are not accessible by the methods of [21] or [15]. This was themotivation to consider the following question



question 1Find a good approach to the definition of F (−A), when F belongs to asuitable class of holomorphic functions on a suitable class of open subsetsof the complex plane and where A is the generator of a semigroup (T (t))t>0of bounded operators satisfying some suitable continuity properties.

question 0Find a good general notion of continuity for semigroups (T (t))t>0 of boundedoperators leading to a notion of generator suitable for functional calculus.

The situation with respect to this program that there is a draft giving areasonable answer to question 1, and that an answer to the preliminaryquestion 0 is written in detail. The remainder of the talk will be devoted tothis answer to question 0. The details will be available in [13].


Semigroups and weak continuity properties

W. Feller considered a long time ago in [16] semigroups satisfying some veryweak continuity properties: he only assumes to develop his construction thatthere exists a continuous linear form l 6= 0 on X such that the functiont →< T (t)x , l > is measurable for every x ∈ X . Since he has to enlarge thedomain of the semigroup and/or go to quotient space we will not use thisextremely general approach but use the notion of weak continuity withrespect to a dual pair introduced by Arveson in [2] to study grouprepresentations. We will also use the notions of quasimultipliers introducedby the author in the second of the Long Beach Conference papers [11]: LetA be a commutative Banach algebra such that the set Ω(A) of elementsa ∈ A such that aA is dense in A is nonempty and such thatA⊥ := a ∈ A | ax = 0∀x ∈ A reduces to 0. A quasimultiplier on A is a pair(Su/v ,Du/v), where u ∈ A, v ∈ Ω(A) and where Du/v is the space of all x ∈ Asuch that ux ∈ vA, and the closed operator Su/v : Du/v → A is defined by theformula vSu/vx = ux . The algebra of quasimultipiers on A is denoted QM(A),and a family U ⊂ QM(A) is said to be pseudobounded if there existsa ∈ Ω(A) such that supS∈U ‖Sa‖ < +∞.



A quasimultiplier S on A is said to be regular if the sequence (λnSn)n≥1 ispseudobounded for some λ > 0, and the family of regular quasimultipliers onA, denoted QMr(A), is a pseudo-Banach algebra in the sense of Allan, Dalesand Mc Clure [1]. We will interpret the generator of a semigroup weaklycontinuous in the sense of Arveson as a quasimutiplier on the "Arvesonideal" IT associated to the semigroup. The interpretation of the generator asa quasimutiplier was already proposed by Gale and Miana [19] for groups ofregular quasimultipliers and by Chalendar, Partington and the author foranalytic semigroups [5].A nontrivial result obtained by the author in collaboration with Koosis [10], th.6.8 shows that if ω is a continuous nonincreasing submultiplicative weight on(0,+∞) and if we denote by L1

ω the convolution algebra of all measurablefunctions f on (0,+∞) such that ‖f‖1

ω :=∫ +∞

0 |f (x)|ω(x)dx < +∞, then(L1

ω, ‖.‖1ω) has dense principal ideals.

The "Arveson ideal" IT is defined as follows: set ωT := ‖T (t)‖, so that ωT islower semicontinuous, and define L1

ωTas above. Let T = (T (t))t>0 be a

semigroup of weakly continuous bounded operators on X , an assume thatthe semigroup is "weakly continuous" with respect to some "dual pair"(X ,X∗).



Then a homomorphism ΦT : L1ωT→ B(X ) is defined by the formula

< ΦT(f )x , l >=

∫ +∞

0< f (x)T (x), l > dx (f ∈ L1

ωT, x ∈ X , l ∈ X∗).

Denote byMωT the convolution algebra of all measures µ on (0,+∞) suchthat

∫ +∞0 ‖T (t)‖d |µ|(t) < +∞, and extend ΦT toMωT by using a similar

formula. The "Arveson ideal" IT is the closure in B(X ) of ΦT(IT). It is a closedideal of the "Arveson algebra" AT defined to be the closure of ΦT(MωT) inB(X ).The result about existence of dense ideals in the convolution algebras L1

ω

extends to the case where ω is lower semicontinuous, and it follows from thisfact that the Arveson ideal IT possesses dense principal ideals, which allowsto consider the algebra QM(IT) of quasimultipiers on IT and the algebraQMr(IT) of regular quasimultipiers on IT.


The generator as a partially defined operator

We now consider the operators (T (t))t>0 as operators on the Banach spaceX1 := [∪t>0T (t)X ]− . We will use the convention T (0) = IX1, the identity mapon X1. Set

X := x ∈ X1 | limt→0+

< T (t)x − x , l >= 0 ∀l ∈ X∗.

Clearly, ∪t>0T (t)(X ) ⊂ X , and [∩t>0Ker (T (t)] ∩ X = 0. Notice that if x ∈ X ,it follows from the Banach-Steinhaus theorem that lim supt→0+‖T (t)x‖ =limsupt→0+‖T (t)x‖(X∗)′ < +∞. Also if lim supt>0‖T (t)‖ < +∞, then X is closed,so that X = X1.The following definition is a variant of a definition introduced by Feller in [16]for semigroups satisfying some weak measurability conditions.An operator theoretical approach to the generatorDefinition: Denote by DT,op the space of all x ∈ X1 for which there existsy ∈ X satisfying

limt→0+ < T (t)x−xt − y , l >= 0 ∀l ∈ X∗. (*)

In this situation we set AT,opx = y . The partially defined operator AT,op will becalled the infinitesimal generator of the semigroup T = (T (t))t>0.


The generator as a partially defined operator, continued

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

t − y , l >= 0 for every l ∈ X∗ and every s > 0.If x ∈ DT,op, and if y = AT,opx , then we have, for l ∈ X∗, t > 0,

< T (t)x − x , l >=

∫ t

0< T (s)y , l > dt ,

and so, according to the usual notations for Pettis integrals, we have

T (t)x − x =∫ t

0 T (s)yds.

Conversely, if y ∈ X , and if T (t)x − x =∫ t

0 T (s)yds for t > 0, then x and yobviuously satisfy (*) and so x ∈ DT,op, and y = AT,opx .Since sup‖T (t)‖a≤t≤b < +∞ for 0 < a < b < +∞, we havelimt→+∞ ‖T (t)‖1

t = limn→+∞‖T (n)‖1n .

An operator theoretical approach to the resolvent

Definition Set ρT = limt+∞ ‖T (t)‖1t , and for Re(λ) > log(ρT), y ∈ X , define

Rop(AT,op, λ)y by the formula

< Rop(AT,op, λ)y , l >:= −∫ +∞

0e−λs < T (s)y , l > ds ∀l ∈ X∗. (1)

.In other terms, according to the previous notations, we have

Rop(AT,op, λ)y = −∫ +∞

0e−λsT (s)yds.


The generator as a partially defined operator, end

According to previous notations, we have, for y ∈ X , λ > ρT,

Rop(AT,op, λ)y = −∫ +∞

0e−λsT (s)yds.

Properties of the resolventProposition: Rop(AT,op, λ)y ∈ DT,op for y ∈ X , Rop(AT,op, λ) : X → DT,op isone-to-one and onto, and we have(

AT,op − λIX1

) Rop(AT,op, λ) = IX ,

Rop(AT,op, λ) (AT,op − λIX1

)= IDT,op.

We defined above Rop(AT,op, λ)y for y ∈ X . Of course, the same formulaallows to define Rop(AT,op, λ)y for all y ∈ X1 such that∫ +∞

0 | < T (t)y , l > |dt < +∞ for every l ∈ X∗.


Quasimultipliers on IT as weakly densely defined operators

We saw before that the set Ω(IT) := u ∈ IT | [uIT]− = IT is nonempty. Let(Su/v ,Du/v) ∈ QM(IT), where u ∈ IT, v ∈ Ω(IT) and where Du/v is the spaceof all x ∈ IT such that ux ∈ vIT. The closed operator Su/v : Du/v → IT isdefined by the formula vSu/vx = ux .

A weakly densely defined operator on X1 associated to a quasimultiplier onIT.

Lemma: Let v ∈ Ω(IT). Then Ker (v) ⊂ ∩t>0Ker (T (t), and v : X → X1 isone-to-one.Proposition: Let S = Su/v ∈ QM(IT). Set DS := x ∈ X | ux ∈ v(X ), anddenote by S(x) the unique y ∈ X satisfying ux = vy . Then DS is weaklydense in X1. If lim supt→0+‖T (t)‖ < +∞, then S is a closed partially definedoperator on X1, and S is weakly closed if, further, the σ(X∗,X )-closed convexhull of every σ(X∗,X )-compact subset of X∗ is σ(X∗,X )-compact.

It would be better to find a closed extension of S to some subspace of X1.This is easy if ∩t>0Ker (T (t) = 0, but in the general case we have toconsider partially defined operators on X1/[∩t>0Ker (T (t)].


A closed operator associated to a quasimultiplier on QM(IT)

For u ∈ AT, x ∈ X1, set

u (x + ∩t>0Ker (T (t))) = ux + ∩t>0Ker (T (t)),

so that u ∈ B (X1/[∩t>0Ker (T (t))]) . An elementary well-known argumentshows that ∩t>0Ker (T (t)) = 0, which implies that v is one-to-one onX1/ ∩t>0 Ker (T (t) if v ∈ Ω(IT).This suggests the following definition.A closed operator on X1/[∩t>0Ker (T (t)] associated to a quasimultiplier onQM(IT)

Definition: Let S = Su/v ∈ QM(IT). SetD

S:= z ∈ X1/ ∩t>0 T (t)/ | uz ∈ v (X1/ ∩t>0 T (t))). The operator

S : DS→ X1/ ∩t>0 Ker (T (t)). is defined by the formula

v(

Sz)

= uz, (z ∈ DS

).

We see that S is a closed operator on X0 := X1/[∩t>0Ker (T (t)], which maybe considered as an extension of S since X ∩ [∩t>0Ker (T (t))] = 0.


The normalized Arveson ideal

We now apply to the Arveson ideal IT and to the family (T (t))t>0 a slightmodification of construction performed by the author in [11].

Theorem: Set LT := u ∈ IT | lim supt→0+ ‖T (t)u‖ < +∞ ⊃ ∪t>0T (t)IT,choose λ > log(ρT), set ‖u‖λ := sups≥0e−λs‖T (s)u‖ for u ∈ LT, denote by DTthe closure of ∪t>0T (t)IT in (LT, ‖.‖λ), and set‖R‖λ,op := sup‖Ru‖λ | u ∈ DT, ‖u‖λ ≤ 1 = ‖R‖M(DT) for R ∈M(DT).Then (LT, ‖.‖λ) is a Banach algebra, the norm topology on LT does notdepend on the choice of λ, DT = u ∈ LT | limt→0+ ‖T (t)u − u‖λ = 0, DT is adense ideal of IT, and the following properties hold(i) ‖T (t)u‖λ ≤

(supt≥0 e−λs‖T (t + s)‖

)‖u‖ for u ∈ IT, t > 0.

(ii) ‖T (t)‖λ,op ≤ eλt for every t > 0.(iii) R(LT) ⊂ LT and R(DT) ⊂ DT for every R ∈M(IT), and‖R‖λ,op ≤ ‖R‖M(IT). In particular ‖u‖λ,op ≤ ‖u‖ for every u ∈ AT.(iv) ab ∈ DT for a ∈ LT, b ∈ LT.(v) Ω(IT) ∩ LT 6= ∅, aLT is dense in DT for every a ∈ Ω(IT) ∩ LT, andab ∈ Ω(DT) for a ∈ Ω(IT) ∩ LT,b ∈ Ω(IT) ∩ LT.


The normalized Arveson ideal, continued

Theorem, continuedTheorem (continued): Now denote by JT the closure of IT, identified to asubset ofM(DT), in (M(DT), ‖.‖λ,op). Then DT is a dense ideal of (JT, ‖.‖λ,op)and the following properties hold(vi) (φT(fn ∗ δεn))n≥1 is a bounded sequential approximate identity for JT forevery Dirac sequence (fn)n≥1 and every sequence (εn)n≥1 such thatlimn→+∞ εn = 0, and lim supn→+∞ ‖φT(fn ∗ δεn)‖λ,op = 1.(vii) Ru ∈ DT for every R ∈M(JT) and every u ∈ DT, and the map R → R|DTis a norm-preserving isomorphism fromM(JT) ontoM(DT).(viii) If a ∈ Ω(DT) then the map Tu/v → Tau/av defines a pseudboundedisomorphism from QM(JT) onto QM(IT) and from QM(JT) onto QM(DT).

The Banach algebra JT, which is a closed ideal of its multiplier agebraM(JT), will be called the normalized Arveson ideal of the semigroup T.

Notice that if a ∈ Ω(DT) then the map Tu/v → Tau/av defines alsopseudbounded isomorphism from QM r(JT) onto QM r(IT) and from QM r(JT)onto QM r(DT), and that bc ∈ Ω(DT) if b ∈ Ω(IT) ∩ LT, c ∈ Ω(IT) ∩ LT. Forexample set ωλ(t) = sups≥te−λs‖T (s)‖, where λ > log(ρT), and setωλ(t) = eλtωλ(t). It follows from theorem 6.8 of [10] that Ω(L1

ωλ) 6= ∅.


The generator as a quasimultiplier on the Arveson ideal

Strictly speaking, theorem 6.8 of [10] is proved in the case where the weightω under consideration is continuous and nondecreasing on (0,+∞), but theproof extends easily to the case where this weight is lower semicontinuousand nondecreasing, which is the case for the weight ωλ. Set uλ(t) = e−λt fort > 0. Since the map f → uλf is an isomorphism from (L1

ωλ, ∗) onto (L1

ωλ, ∗), it

follows from theorem 6.8 of [10] that the convolution algebra L1ωλ

contains an

indefinitely differentiable semigroup (bt)t>0 such that bt ∈ Ω(

L1ωλ

)for t > 0,

and so φT(bt) ∈ Ω(DT) for t > 0.The generator as a quasimultiplier on the Arveson idealDefinition: Let λ > log(ρT), let f0 ∈ C1([0,+∞)) ∩ Ω(L1(R+,eλt) be such thatf (0) = f ′(0) = 0 and such that f ′0 ∈ L1(R+,eλt), and let b ∈ Ω(DT). Theinfinitesimal generator AT,IT of a Arveson weakly continuous semigroup T isthe quasimultiplier on IT defined by the formula

AT,IT = TbφT(f ′0)/bφT(f0).

This definition does not depend on the choice of λ,b or f0.


The domain of the generator

For example we can take f0(t) = t2e−λt for t ≥ 0, where λ > log(ρT), so thatf ′0(t) = −2λte−λt . More generally it follows from Nyman’s theorem [22] that iff0 ∈ L1(R+,e−λt), then f0 ∗ L1(R+,e−λt) is dense in L1(R+,e−λt) if and only ifthe two following conditions are satisfied

1 L(f0)(z) 6= 0 for Re(z) ≥ λ,2 f0 does not vanish a.e. on any interval of the form [0, δ] with δ > 0.

The domain of the generator as a quasimultiplier on IT

Proposition: Let u ∈ IT. If limh→0+

∥∥∥T (h)u−uh − v

∥∥∥ = 0 for some v ∈ IT, thenu ∈ DT,IT, and AT,ITu = v .

In order to consider the generator as a partially defined operator on X1 or onX1/ ∩t>0 Ker (T (t)), is seems more convenient to define the generator as aquasimultiplier on IT and then to apply the procedure used to associate toany element of QM(IT) a partially defined operator on X1 or on X0.We can also consider the generator of the semigroup as a quasimultiplier onthe normalized Arveson ideal JT by setting AT,JT = SφT(f ′0)/φT(f0), and in this

case u ∈ DT,JT and v = AT,JTu if and only if limh→0+

∥∥∥T (h)u−uh − v

∥∥∥JT

= 0.


The Arveson spectrum

Denote by IT the character space of the Arveson ideal IT, equipped with theusual Gelfand topology induced bu the weak-∗ topology on the unit ball of thedual space of IT. If χ ∈ IT, then there exists a unique character χ on QM(IT)

such that χ|IT = χ, defined by the formula χ(Su/v

)= χ(u)

χ(v) foru ∈ IT, v ∈ Ω(IT).

Arveson spectrum of elements of QM(IT).

Definition: The Arveson spectrum of S = Su/v ∈ QM(IT) is the set

σar(S) := χ(S) : χ ∈ IT = χ(u)/χ(v) : χ ∈ IT.

Proposition: The map χ→ χ(AT,IT) is a homeomorphism from IT ontoσar(AT,IT) = σar(AT,JT).

Notice that the set z ∈ σar(AT,IT) | Re(z) ≥ λ is compact for every λ ∈ R.A character η on AT satisfies η(T (t)) = χ(T (t)) for some χ ∈ IT if and only ifthe map t → η(T (t)) is continuous on (0,+∞), and in this situation(χ φT)(f ) =

∫ +∞0 η(T (t))f (t)dt for f ∈ L1

ωT. Also the maps χ→ χ|DT

and

χ→ χ|JTare homeomorphisms from IT onto DT and JT.


The resolvent

We now set AT = AT,IT.

The resolvent takes values in the normalized Arveson idealProposition: AT − λI is invertible in QM(IT), (AT − λI)−1 ∈ JT ⊂ QM r(IT) forλ ∈ C \ σar(AT), the map λ→ (AT − λI)−1 is an holomorphic map fromC \ σar(AT) into JT, and we have, for Re(λ) > log(ρT), u ∈ DT,

(AT − λI)−1u = −∫ +∞

0e−λtT (t)udt .

In other terms we have (AT − λI)−1 = −∫ +∞

0 e−λtT (t)dt for Re(λ) > log(ρT)where the Bochner integral is computed with respect to the strong operatortopology onM(DT). Alternatively we can identify isometrically JT to an idealofM(JT) and use the same formula to compute the Bochner integral withrespect to the strong operator topology onM(JT).This approach is certainly well-known for strongly continuous semigroups ofoperators such that lim supt→0+ ‖T (t)‖ < +∞, but might be helpful forsemigroups which are not bounded in norm near 0. It is easy to check thatthe partially defined operator on X1 associated to the quasimultiplier(AT − λI)−1 ∈ QM r(IT) equals Rop(T, λ).


Holomorphic functional calculus

H∞− functional calculusProposition: Assume that F is holomorphic on some open half-planeΠα := z ∈ C | Re(z) > α, where α < −log(ρT).

1 If supRe(z)>α(1 + |z|2)|F (z)| < +∞, set

F (−AT) =1

2π

∫ +∞

−∞F (β + iy)(AT + (β + iy)I)−1dy ,

where α < β < −log(ρT), and where the Bochner integral is computed inJT ⊂ QM r(IT).

2 If F is bounded for Re(z) > α, set

F (−AT) = SFG(−AT)/G(−AT) ∈ QM r(JT) ≈ QM r(IT),

and where G is an outer function on Πα such thatsupRe(z)>α(1 + |z|2)|F (z)| < +∞, so that G(−AT) ∈ Ω(JT).

Then these definitions do not depend on the choice of β and G, and the mapF → F (−AT) is a one-to-one algebra homomorphism from ∪α<−log(ρT)H∞(Πα)into QM r(JT) ≈ QM r(IT) such that exp(tAT) = T (t) for t > 0.


A functional calculus for the Smirnov class

Recall that a function F ∈ Πα is said to be outer if there exists a real-valuedfunction φ ∈ L1(R) such that F (z) = exp(1

π

∫ +∞−∞

1+it(z−α)it+z−α

11+t2φ(t)dt) for z ∈ Πα.

The Smirnov class N+(Πα) on Πα is the space of holomorphic functions F onΠα which can be written in the form F = G/H where G ∈ H∞(Πα) andH ∈ H∞(Πα) is outer. It is possible to deduce from Nyman’s characterizationof dense principal ideals of L1(R+) that F (−AT) is invertible in QM(JT) ifF ∈ H∞(Πα) is outer, where α < −log(ρT). We obtain the following result.Functional calculus for the Smirnov classProposition: For F = G/H ∈ N+(Πα), where α < −log(ρT),G ∈ H∞(Πα), andwhere H ∈ H∞(Πα) is outer, set

F (−AT) = G(−AT)H(−AT)−1 ∈ QM(JT) ≈ QM(IT).

Then this definition does not depend on the choice of G,H and α, and themap F → F (−AT) is a one-to-one algebra homomorphism from∪α<−log(ρT)N+(Πα) into QM(JT) ≈ QM(IT) such that F (−AT) = AT ifF (z) = −z for z ∈ C.


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