Algebras of Operators on Holomorphic Functions and Applications

Mathematical Physics, Analysis and Geometry 5: 65–76, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

65

Algebras of Operators on HolomorphicFunctions and Applications

M. BEN CHROUDA and H. OUERDIANEDepartment of Mathematics, Faculty of Sciences of Tunis, Université de Tunis El Manar,1060 Tunis, Tunisia. e-mail: [email protected]

(Received: 6 March 2001; in final form: 17 August 2001)

Abstract. We develop the theory of operators defined on a space of holomorphic functions. First, wecharacterize such operators by a suitable space of holomorphic functions. Next, we show that everyoperator is a limit of a sequence of convolution and multiplication operators. Finally, we define theexponential of an operator which permits us to solve some quantum stochastic differential equations.

Mathematics Subject Classifications (2000): primary 60H40; secondary 46A32, 46F25, 46G20.

Key words: symbols of operators, infinite dimensional holomorphy, convolution product of opera-tors, quantum stochastic differential equations.

1. Introduction

Let N be a complex nuclear Fréchet space. Assume that its topology is defined byan increasing family of Hilbertian norms {|.|p, p ∈ N}. Then N is represented asN =⋂

p∈NNp , where for p ∈ N the space Np is the completion of N with respect

to the norm |.|p. For simplicity, we denote by H the complex Hilbert space N0

and by N−p the dual space of Np, then the dual space N ′ of N is represented asN ′ = ⋃

p∈NN−p , and it is equipped with the inductive limit topology. We denote

by 〈., .〉 the C-bilinear form on N ′ × N connected to the inner product 〈.|.〉 of H ,i.e.

〈z, ξ 〉 = 〈z̄|ξ 〉, z ∈ H, ξ ∈ N.

For any n ∈ N we denote by SnN the nth symmetric tensor product of N equippedwith the π -topology and by SnNp the nth symmetric Hilbertian tensor product ofNp. We will preserve the notation |.|p and |.|−p for the norms on SnNp and SnN−p,respectively.

Let n,m ∈ N and 0 � k � m ∧ n. We denote by 〈., .〉k the bilinear map fromSmN ′ × SnN into Sm−kN ′⊗̂Sn−kN defined by⟨

x⊗m, y⊗n⟩k:= 〈x, y〉kx⊗(m−k) ⊗ y⊗(n−k), x ∈ N ′, y ∈ N.

The bilinear map 〈., .〉k is continuous, then using the density of the vector spacegenerated by {x⊗m, x ∈ N ′} in SmN ′ and the density of the vector space generated

66 M. BEN CHROUDA AND H. OUERDIANE

by {y⊗n, x ∈ N} in SnN , we can extend 〈., .〉k to SmN ′ ×SnN . Let φm ∈ SmN ′ andϕn ∈ SnN ; then 〈φm, ϕn〉k is called the right contraction of φm and ϕn of degree k.

Let θ be a Young function on R+, i.e. θ is continuous, convex, increasing func-tion and satisfies lim+∞ θ(x)/x = +∞. We define the conjugate function θ∗ of θ

by

∀x � 0, θ∗(x) := supt�0

(tx − θ(t)). (1)

For a such Young function θ , we denote by Gθ (N) the space of holomorphic func-tions on N with exponential growth of order θ and of arbitrary type, and by Fθ (N

′)the space of holomorphic functions on N ′ with exponential growth of order θ andof minimal type. For every p ∈ Z and m > 0, we denote by exp(Np, θ,m) thespace of entire functions f on the complex Hilbert space Np such that

nθ,p,m(f ) := supz∈Np

|f (z)|e−θ(m|z|p) < +∞.

Then the spaces Fθ (N′) and Gθ (N) are represented as

Fθ (N′) =

⋂p∈N

m>0

exp(N−p, θ,m),

Gθ (N) =⋃p∈N

m>0

exp(Np, θ,m),

and equipped with the projective limit topology and the inductive limit topology,respectively. Let p ∈ N and m > 0, we define the Hilbert spaces

Fθ,m(Np)

={f = (fn)

∞n=0, fn ∈ SnNp; ‖f ‖θ,p,m :=

∑n�0

θ−2n m−n|fn|2p < +∞

},

Gθ,m(N−p)

={φ = (φn)

∞n=0, φn ∈ SnN−p; ‖φ‖θ,−p,m :=

∑n�0

(n!θn)2mn|φn|2−p < +∞},

where θn = infr>0 eθ(r)/rn, n ∈ N. The sequences θn and θ∗n are connected by thefollowing relation

LEMMA 1. For every n ∈ N\{0} we have en � nnθnθ∗n � e2n.

Proof. We can assume that θ(x) = ∫ x

0 µ(t) dt where µ is a continuous, in-creasing function which satisfies lim+∞ µ(x) = +∞ (see [4]). Then θ∗(x) =∫ x

0 ω(t) dt , where ω is the inverse function of µ, i.e. µ ◦ ω = ω ◦ µ = id. A directcalculation shows that

θn = eθ(tn)

tnnand θ∗n =

eθ∗(xn)

xnn

,

ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS 67

where tn and xn are the solutions of tµ(t) = n and tω(t) = n, respectively, andsatisfy tnxn = n. Hence,

nnθnθ∗n =

(n

tnxn

)n

eθ(tn)eθ∗(xn)

� e2n.

On the other hand, for every t, x > 0 we have

etx

(tx)n� eθ(t)

tn

eθ∗(x)

xn, ∀n � 1.

Then, using the fact that inft>0etx

(tx)n= en

nn , we obtain en/nn � θnθ∗n . ✷

Put

Fθ(N) =⋂p∈N

m>0

Fθ,m(Np),

Gθ (N′) =

⋃p∈N

m>0

Gθ,m(N−p).

Then the space Fθ(N) equipped with the projective limit topology is a nuclearFréchet space [4], and Gθ(N

′) carries the dual topology of Fθ(N) with respect tothe C-bilinear form (., .):

(φ,f ) =∑n�0

n!〈φn, fn〉, φ = (φn) ∈ Gθ(N′), f = (fn) ∈ Fθ(N).

For simplicity, we put

Fθ (N′) = Fθ , Gθ∗(N) = Gθ∗ , Fθ(N) = Fθ , Gθ(N

′) = Gθ

and we denote by F ′θ the strong dual of the space Fθ . It was proved in [4] that

the Taylor series map S.T yields a topological isomorphism between Fθ (respec-tively Gθ∗) and Fθ (respectively Gθ ). The nuclear Fréchet space Fθ and its dualF ′

θ are called the test function space and the distribution space, respectively. TheC-bilinear form on F ′

θ×Fθ is denoted by 〈〈., .〉〉. We denote by L(Fθ ,Fθ ) the spaceof continuous linear operators from Fθ into itself, equipped with the topology ofbounded convergence.

In this paper, we do not restrict ourselves to the theory of Gaussian (whitenoise) and non-Gaussian analysis studied, for example, in [1, 6, 8, 9] and [10]but we develop a general infinite-dimensional analysis. First, we give a decom-position of convolution operators from Fθ into itself into a sum of holomorphicderivation operators. Second, we establish a topological isomorphism between thespace L(Fθ ,Fθ ) of operators and the space Fθ ⊗̂Gθ∗ of holomorphic functions.


Next, we develop a new convolution calculus over L(Fθ ,Fθ ) and we give senseto the expression eT := ∑

n�0 Tn/n! for some class of operators T . Finally, as

an application of this operator theory we solve some linear quantum stochasticdifferential equations.

2. Some Properties on the Distribution Space

Let θ be a Young function. For every ξ ∈ N , the exponential function eξ : z �→e〈z,ξ 〉, z ∈ N ′ belongs to Fθ . Then we define the Laplace transform of a distributionφ ∈ F ′

θ by

φ̂(ξ ) := 〈〈φ, eξ 〉〉, ξ ∈ N.

PROPOSITION 1 ([4]). The Laplace transform realizes a topological isomorphismbetween F ′

θ and Gθ∗ .

By composition of the Taylor series map with the Laplace transform, we deducethat φ ∈ F ′

θ if and only if there exists a unique formal series φ = (φn)n�0 ∈ Gθ

such that

φ̂(ξ ) =∑n�0

〈ξ⊗n, φn〉.

Then, the action of the distribution φ on a test function ϕ(z) = ∑n�0〈z⊗n, ϕn〉 is

given by

〈〈φ, ϕ〉〉 =∑n�0

n!〈φn, ϕn〉.

In particular, for every z ∈ N ′, the Dirac mass δz defined by

〈〈δz, ϕ〉〉 = ϕ(z), (2)

belongs to F ′θ . Moreover, δz coincides with the distribution associated to the formal

series

δz :=(z⊗n

n!)

n�0

.

Now, we recall some properties of translation operators and convolution productof distributions studied in [2]. Let z ∈ N ′, the translation operator τ−z is defined by

τ−zϕ(λ) = ϕ(z+ λ), λ ∈ N ′.

For every z ∈ N ′, the linear operator τ−z is continuous from Fθ into itself. Wedefine the convolution product of a distribution φ ∈ F ′

θ with a test function ϕ ∈ Fθ

as follows

φ ∗ ϕ(z) := 〈〈φ, τ−zϕ〉〉, z ∈ N ′.


A direct calculation shows that φ ∗ ϕ ∈ Fθ . Let φ1, φ2 ∈ F ′θ , we define the

convolution product of φ1 and φ2, denoted by φ1 ∗ φ2, by

〈〈φ1 ∗ φ2, ϕ〉〉 := [φ1 ∗ (φ2 ∗ ϕ)](0), ϕ ∈ Fθ .

Moreover, ∀φ1, φ2 ∈ F ′θ we have φ̂1 ∗ φ2 = φ̂1φ̂2.

3. Convolution Operators

In infinite-dimensional complex analysis, a convolution operator on the test spaceFθ is a continuous linear operator from Fθ into itself which commutes with trans-lation operators. It was proved in [2, 5] that T is a convolution operator on Fθ ifand only if there exists φT ∈ F ′

θ such that

T ϕ = φT ∗ ϕ, ∀ϕ ∈ Fθ . (3)

Moreover, if the distribution φT is given by

φT = (φm)m�0 ∈ Gθ and ϕ(z) =∑n�0

〈z⊗n, ϕn〉 ∈ Fθ ,

then

φT ∗ ϕ(z) =∑m�0

∑n�0

(n+m)!n! 〈z⊗n, 〈φm, ϕm+n〉m〉. (4)

In particular, we have

T (eξ )(z) = φT ∗ eξ (z) = φ̂(ξ )eξ (z).

Let θ be a Young function, y ∈ N ′ and ϕ(z) = ∑n�0〈z⊗n, ϕn〉 ∈ Fθ . We define

the holomorphic derivative of ϕ at a point z ∈ N ′ in a direction y by

Dyϕ(z) :=∑n�0

(n+ 1)〈z⊗n, 〈y, ϕn+1〉1〉.

LEMMA 2. The operator Dy is continuous from Fθ into itself. Moreover, for everyϕ ∈ Fθ , p ∈ N and m > 0, we have

‖Dyϕ‖θ,p,m �√mθ1|y|−py

‖ϕ‖θ,py∨p, m16,

where py = min{p ∈ N; y ∈ N−p} and py ∨ p = max(py, p).Proof. By definition of the norm ‖.‖θ,p,m defined on the space Fθ of formal

series, we have

‖Dyϕ‖θ,p,m =( ∑

n�0

(n+ 1)2θ−2n m−n|〈y, ϕn+1〉1|2p

)1/2


� |y|−py

( ∑n�0

(n+ 1)2θ−2n m−n|ϕn+1|2p∨py

)1/2

�√m|y|−py

( ∑n�0

θ−2n+1

(m

16

)−n−1

|ϕn+1|2p∨py

[(n+ 1)θn+1

22n+2θn

]2)1/2

�√m|y|−py

supn�1

[θn+1

2n+1θn

]‖ϕ‖θ,p∨py,

m16.

Finally, the desired inequality follows immediately using the fact that 2−l−kθlθk �θl+k � 2l+kθlθk, ∀l, k ∈ N\{0}. ✷

In view of Lemma 2, for each m ∈ N the m-linear operator D defined by

D: N ′ × · · · × N ′ → L(Fθ ,Fθ )

(y1, . . . , ym) �→ Dy1 . . . Dym

is symmetric and continuous, hence, it can be continuously extended to SmN ′, i.e.D: φm ∈ SmN ′ �→ Dφm

∈ L(Fθ ,Fθ ). The action of the operator Dφmon a test

function ϕ(z) =∑n�0〈z⊗n, ϕn〉 given by

Dφm(ϕ)(z) =

∑n�0

(n+m)!n! 〈z⊗n, 〈φm, ϕn+m〉m〉. (5)

Then, in view of (3), (4) and (5), we give an expansion of convolution operatorsin terms of holomorphic derivation operators.

PROPOSITION 2. Let T ∈ L(Fθ ,Fθ ), then T is a convolution operator if andonly if there exists φ = (φm)m�0 ∈ Gθ such that T =∑

m�0 Dφm.

Remark. Let Tφ = ∑m�0 Dφm

be a convolution operator and n ∈ N. Thenequality (3) shows that

T nφ := Tφ ◦ · · · ◦ Tφ︸︷︷︸

n

= Tφ∗n . (6)

In particular,

T nφ (eξ )(z) = Tφ∗n(eξ )(z) = (φ̂(ξ))neξ (z), z ∈ N ′, ξ ∈ N.

4. Symbols of Operators

In this section we define the symbol map on the space L(Fθ ,Fθ ). Then we give anexpansion of such operators in terms of multiplication and derivation operators.


DEFINITION 1. Let T ∈ L(Fθ ,Fθ ), the symbol σ (T ) of the operator T is aC-valued function defined by

σ (T )(z, ξ) := e−〈z,ξ 〉T (eξ )(z), z ∈ N ′, ξ ∈ N.

Similar definitions of symbols have been introduced in various contexts ([7, 10–12]).

In the general theory ([13]), if we take two nuclear Fréchet spaces X and D ,then the canonical correspondence T ↔ KT given by

〈T u, v〉 = 〈KT , u⊗ v〉, u ∈ X, v ∈ D ′,

yields a topological isomorphism between the spaces L(X,D) and X′ ⊗̂D . Inparticular, if we take X = D = Fθ which is a nuclear Fréchet space, then we get

L(Fθ ,Fθ ) ∼= F ′θ ⊗ Fθ . (7)

So, the symbol σ (T ) of an operator T can be regarded as the Laplace transform ofthe kernel KT

σ(T )(z, ξ) = KT (eξ ⊗ δz), z ∈ N ′, ξ ∈ N. (8)

Moreover, with the help of equalities (2), (7), (8) and Proposition 1 we obtain thefollowing theorem

THEOREM 1. The symbol map yields a topological isomorphism betweenL(Fθ ,Fθ ) and Fθ⊗̂Gθ∗ . More precisely, we have the following isomorphisms:

L(Fθ ,Fθ )σ→ Fθ⊗̂Gθ∗

S.T→ Fθ⊗̂Gθ,

T �→ σ (T )(z, ξ) =∑l,m

〈Kl,m, z⊗l ⊗ ξ⊗m〉 �→ K = (Kl,m)l,m�0.

EXAMPLES. (1) Let φm ∈ SmN ′. Then

σ (Dφm)(z, ξ) = e−〈z,ξ 〉Dφm

(eξ )(z)

= e−〈z,ξ 〉〈φm, ξ⊗m〉e〈z,ξ 〉

= 〈φm, ξ⊗m〉.

In particular, the symbol of a convolution operator Tφ =∑m�0 Dφm

is given by

σ (Tφ)(z, ξ) = e−〈z,ξ 〉∑m�0

Dφm(eξ )(z) =

∑m�0

〈φm, ξ⊗m〉 = φ̂(ξ ).

Hence, the operator Tφ can be expressed in an obvious way by

Tφ =∑m�0

Dφm:=

∑m�0

〈φm,D⊗m〉 = σ (Tφ)(z,D), z ∈ N ′.


(2) Let f ∈ Fθ . We denote by Mf the multiplication operator by the testfunction f . Its symbol is given by

σ (Mf )(z, ξ) = e−〈z,ξ 〉(f eξ )(z)

= e−〈z,ξ 〉f (z)eξ (z)

= f (z).

By the same argument, the multiplication operator is also expressed by Mf =σ (Mf )(z,D). We note that the symbol of a convolution (respectively, multiplica-tion) operator σ (T )(z, ξ) depends only on ξ (respectively, z).

Let K ∈ Fθ⊗̂Gθ and assume that K = f ⊗ φ = (fl ⊗ φm)l,m�0. Then theoperator T associated to K (see Theorem 1) satisfies

T = MfTφ, (9)

where f (z) = ∑l�0〈z⊗l , fl〉 and Tφ is the convolution operator associated to the

distribution φ given by φ. Moreover, we have

T = MfTφ = σ (Mf )(z,D)σ (Tφ)(z,D) = σ (T )(z,D).

Thus, using the density of Fθ ⊗Gθ in Fθ ⊗̂Gθ , we obtain the following result:

PROPOSITION 3. The vector space generated by operators of type (9) is densein L(Fθ ,Fθ ).

5. Convolution Product of Operators

Let T1, T2 be two operators in L(Fθ ,Fθ ); the convolution product of T1 and T2,denoted by T1 ∗ T2, is uniquely determined by

σ (T1 ∗ T2) = σ (T1)σ (T2).

If the operators T1 and T2 are of type (9), i.e. T1 = Mf1Tφ1 and T2 = Mf2Tφ2 , then

T1 ∗ T2 = Mf1f2Tφ1∗φ2 .

In particular, if T = MfTφ, then for every n ∈ N we have

T ∗n = MfnTφ∗n . (10)

Remark. Let Tφ (resp. Mf ) be a convolution (resp. multiplication) operator.Then for every n ∈ N

T ∗nφ = Tφ∗n = T nφ and M∗n

f = Mfn = Mnf .


LEMMA 3. Let γ1, γ2 two Young functions and (Fn) a sequence belonging toFγ1⊗̂Gγ2 . Then (Fn) converges in Fγ1⊗̂Gγ2 if and only if

(1) (Fn) is bounded in Fγ1⊗̂Gγ2 .(2) (Fn) converges simply.

Proof. The proof is based on the nuclearity of the spaces Fγ1 and Gγ2 . A similarproof is established with more details in [3], Theorem 2. ✷PROPOSITION 4. Let T ∈ L(Fθ ,Fθ ); then the operator

e∗T :=∑n�0

T ∗n

n!belongs to L(F(eθ∗)∗,Feθ ).

Proof. Let T ∈ L(Fθ ,Fθ ) and put

Sn =n∑

k=0

T ∗k

k! .

Then, using Lemma 3, we show that σ (Sn) converges in Feθ ⊗̂Geθ to eσ(T ), fromwhich the assertion follows. ✷COROLLARY 1. Let T ∈ L(Fθ ,Fθ ), and assume that σ (T )(z, ξ) is a polyno-mial in z and ξ of degree k and k/(k− 1), respectively, k � 2. Then e∗T belongs toL(Fk,Fk), where Fk is the test space associated to the Young function θ(x) = xk .

Let T ∈ L(Fθ ,Fθ ) and consider the linear differential equation

dE

dt= TE, E(0) = I.

Then the solution is given informally by

E(t) = etT , t ∈ R.

In the particular case, where T is a convolution or a multiplication operator; thesolution E(t) = etT is well defined since eT = e∗T . If T is not a convolution or amultiplication operator then the following theorem gives a sufficient condition onT to insure the existence of its exponential eT .

THEOREM 2. Let K = (Kl,m) ∈ Fθ⊗̂Gθ satisfying 〈Kl,m,Kl′,m′ 〉k = 0 for everym, l′ � 1,m′, l � 0 and 1 � k � m ∧ l′ and denote by T the operator associatedto K (see Theorem 1). Then,

T n = T ∗n, ∀n ∈ N.

Moreover, eT = e∗T ∈ L(F(eθ∗)∗,Feθ ).


Proof. Using Proposition 3, it will be sufficient to assume that Kl,m = (fl⊗φm),i.e.

T = MfTφ =∑l,m�0

MflDφm

,

where fl(z) = 〈z⊗l , fl〉. Assume that

fl = η⊗l , η ∈ N and φm = y⊗m, y ∈ N ′.

Then it is easy to see that

DφmMfl

= MflDφm

+m∧l∑k=0

k!Ckl C

km〈y, η〉kMfl−k

Dφm−k,

an equality on Fθ . The assumption 〈Kl,m,Kl′,m′ 〉k = 0 implies that 〈y, η〉 = 0.Then

DφmMfl

= MflDφm

. (11)

Thus, using the density of the vector space generated by {η⊗l , η ∈ N} in the spaceSlN and the density of the vector space generated by {y⊗m, y ∈ N ′} in SmN ′, wecan extend equality (11) to every fl ∈ SlN and φm ∈ SmN ′ such that 〈φm, fl〉k =0,∀1 � k � l ∧m. Hence, we obtain

MfTφ =∑l,m�0

MflDφm

=∑l,m�0

DφmMfl

= TφMf .

Using equalities (6) and (10), for every n ∈ N we have

T n = (Mf Tφ)n = (Mf )

n(Tφ)n = MfnTφ∗n = T ∗n.

This completes the proof. ✷Remark. The condition of Theorem 2 is not satisfied by convolution or multi-

plication operators. In fact, let K = (Kl,m) ∈ Fθ⊗̂Gθ and let T be the operatorassociated to K .

If T is a convolution operator then K = (Kl,m)l,m�0 = (K0,m)m�0 ∈ Gθ , seeProposition 2. Hence, the right contraction 〈Kl,m,Kl′,m′ 〉k = 0 with 1 � k � m∧ l′can not be defined since l′ = 0.

If T is a multiplication operator then K = (Kl,m)l,m�0 = (Kl,0)l�0 ∈ Fθ . Thus〈Kl,m,Kl′,m′ 〉k = 0 with 1 � k � m ∧ l′ can not be defined since m = 0.

Now we give an example of family of kernels K ∈ Fθ⊗̂Gθ which satisfies thecondition of Theorem 2.

EXAMPLE. Let N = S(R) ↪→ H = L2(R, dt) ↪→ N ′ = S ′(R) and K =(Kl,m)l,m�0 ∈ Fθ ⊗̂Gθ , i.e. Kl,m ∈ Sl(S(R))⊗̂Sm(S ′(R)). Assume that there exists


t ∈ R such that for every l, m ∈ N the support of Kl,m is included in ]−∞, t]l×]t ,+∞[m. Then K satisfies the condition of Theorem 2.

Remark. In Theorem 2 we assume that N is a C-vector space of dimensionn � 2. However, if N = C then for every m, l � 0; SlN⊗̂SmN ′ = C. Thus theassumption 〈Kl,m,Kl′,m′ 〉k = 0 for every m, l′ � 1, m′, l � 0 and 1 � k � m∧l′ isequivalent to Kl,m = 0, ∀l, m ∈ N and the set of operators satisfying the conditionof Theorem 2 is reduced to the null operator.

6. Applications to Quantum Stochastic Differential Equations

A one-parameter quantum stochastic process with values in L(Fθ ,Fθ ) is a familyof operators {Et, t ∈ [0, T ]} ⊂ L(Fθ ,Fθ ) such that the map t �→ Et is continuous.For a such quantum process Et we set

En = t

n

n−1∑k=0

Etkn, n ∈ N\{0}, t ∈ [0, T ].

Then we prove using Lemma 3 that the sequence (En) converge in L(Fθ ,Fθ ). Wedenote its limit by∫ t

0Es ds := lim

n→+∞En in L(Fθ ,Fθ ).

Moreover, we have

σ

(∫ t

0Es ds

)=

∫ t

0σ (Es) ds, ∀t ∈ [0, T ].

THEOREM 3. Let t ∈ [0, T ] �→ f (t) ∈ Fθ and t ∈ [0, T ] �→ φ(t) ∈ F ′θ be

two continuous processes and put Lt = Mf(t)Tφ(t). Then the linear differentialequation

dEt

dt= Mf(t)EtTφ(t), E0 = I (12)

has a unique solution Et ∈ L(F(eθ∗)∗,Feθ ) given by

Et = e∗(∫ t

0 Ls ds).

Proof. Applying the symbol map to Equation (12) we get

dσ (Et )

dt= σ (Lt)σ (Et), σ (I ) = 1.

Then σ (Et) = e∫ t

0 σ(Ls) ds which is equivalent to Et = e∗(∫ t

0 Ls ds). Finally, weconclude by Proposition 4 that Et ∈ L(F(eθ∗)∗,Feθ ). ✷


THEOREM 4. Let Lt be a quantum stochastic process with values in L(Fθ ,Fθ )

such that

σ

(∫ t

0Ls ds

)(z, ξ) =

∑l,m�0

〈Kl,m(t), z⊗l ⊗ η⊗m〉,

and assume that for every t ∈ [0, T ],m′, l � 0 and m, l′ � 1 we have

〈Kl,m(t),Kl′,m′(t)〉k = 0, ∀1 � k � m ∧ l′.

Then the following differential equation

dE

dt= LtE, E(0) = I, (13)

has a unique solution in L(F(eθ∗)∗,Feθ ) given by E(t) = e∫ t

0 Ls ds .

Acknowledgement

We are grateful to the Professor Luis Boutet de Monvel for many stimulatingremarks and useful suggestions.

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Algebras of Operators on Holomorphic Functions and Applications