Weighted composition operators on the Dirichlet space: boundedness and spectral properties

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Chalendar, I, Gallardo-Gutierrez, EA and Partington, JR (2015) Weighted composition

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WEIGHTED COMPOSITION OPERATORS ON THE DIRICHLET SPACE: BOUNDEDNESS AND SPECTRAL PROPERTIES

I. CHALENDAR, E. A. GALLARDO-GUTI´ERREZ, AND J. R. PARTINGTON

Abstract. Boundedness of weighted composition operatorsWu,ϕacting on the

clas-sical Dirichlet spaceDasWh,ϕf=h(f◦ϕ) is studied in terms of themultiplier space

associated to the symbolϕ, i.e.,M(ϕ) ={u∈ D:Wu,ϕis bounded onD}. A

promi-nent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum ofWu,ϕ inDwheneverϕis an automorphism of the unit disc is studied,

extending a recent work of Hyv¨arinen, Lindstr¨om, Nieminen and Saukko [13] to the context of the Dirichlet space.

1. Introduction and Preliminaries

LetD_{denote the open unit disc in the complex plane}C_{. The Dirichlet space}_D_consists

of analytic functionsf onD_{such that the norm}

kfk2D=|f(0)|2+

Z

D

|f′(z)|2dA(z)

is finite. Here A stands for the normalized Lebesgue area measure of the unit disc. Observe that for a univalent functionf, the integral above is just the area of f(D_).

It is well known thatD ⊂ H2_{⊂ A}2_{, where}_H2_and_A2 _{denote respectively the Hardy}

and Bergman spaces onD_{, and that}_f _{∈ D}_{if and only if}_f′ _{∈ A}2_{. The recent monograph}

[8] is an excellent source to learn about the Dirichlet space and its particular issues. Ifϕis an analytic function onD_with_ϕ₍D₎_⊂D_{, then the equation}

Cϕf =f◦ϕ

defines a composition operatorCϕ on the space of all holomorphic functions on the unit disc H(D_{). On the Dirichlet space} _D_{, a necessary condition for} _C_ϕ _{to be bounded is}

that ϕ ∈ D. Nevertheless, not all the Dirichlet functions induce bounded composition operators onD. Such functions were characterized in 1980 by C. Voas [21] in his Ph.D. thesis.

In this work, we shall be concerned with weighted composition operators on D: for

u∈ D andϕa holomorphic self-map of D_{we define the weighted composition operator}

Date: September 2014, Revised: February 2015.

1991Mathematics Subject Classification. Primary 47B38.

Key words and phrases. Dirichlet space, weighted composition operator.

The authors are partially supported by Plan Nacional I+D grant no. MTM2013-42105-P.

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Wu,ϕ onDby

(Wu,ϕf)(z) =u(z)f(ϕ(z)),

noting that Wu,ϕ is not, in principle, a bounded operator on D. It is clear that if Cϕ is a bounded operator on D and u is a multiplier of D, that is, the Toeplitz operator

Tu:f 7→uf is defined everywhere on Dand hence bounded, the weighted composition operatorWu,ϕ onDis obviously bounded.

A well known fact about the Dirichlet space is that the algebraM(D) consisting of the multipliers ofD is not that easy to describe. Indeed, their elements were characterized by Stegenga [20] in a remarkable paper in terms of a condition involving the logarithmic capacity of their boundary values. In particular, the strict inclusion M(D)⊂ D ∩ H∞

holds. HereH∞_{denotes the space of bounded analytic functions in}_D_{endowed with the}

sup-norm. A straightforward reformulation in terms of Carleson measures forD(that is, there is a continuous injection fromDintoL2₍_D_{, µ}_{)), yields the fact that}_u_{∈ M}₍_D_{) if and}

only ifuis bounded and the measure µdefined bydµ(z) =|u′₍_z₎_|2_dA₍_z_{) is a Carleson}

measure forD. We refer to [22] for multipliers and Carleson measures in Dirichlet spaces (and to [1, 8] for more on the subject of multipliers).

Concerning boundedness of weighted composition operators onD, let us remark that one may construct self-maps of the unit disc ϕ such that ϕ 6∈ D and a multiplier u∈ M(D) such thatWu,ϕ is bounded in the Dirichlet space. For instance, letu(z) = (1−z)2 and letϕbe the infinite Blaschke product with zeroes (1−1/n2₎

n≥1. Nowϕ6∈ D, soCϕ is clearly unbounded. However, forf ∈ Dwe have

((1−z)2(f◦ϕ))′ ₌₋₂₍₁₋_z₎₍_f_◦_ϕ_{) + (1}₋_z₎2₍_f′_◦_ϕ₎_ϕ′_;

now the first term is clearly in the Bergman space A2_{, while for the second term we}

observe that (1−z)2_ϕ′ _{is a bounded analytic function in}_D_{and (}_f′_◦_ϕ₎_{∈ A}2_{, so that it}

also lies inA2 _{(with control of norms), showing that}_W

u,ϕ is bounded onD.

Therefore, facing the problem of describing the weighted composition operators taking

Dboundedly into itself deals not only with the multipliers ofDbut also with those self-maps of the unit disc that may induce unbounded composition operators inD.

At this regards, for a self-map ϕ of the unit disc D_{, we define the} _{multiplier space}

M(ϕ) associated toϕby

M(ϕ) ={u∈ D:Wu,ϕ is bounded onD}.

It is clear that ifCϕis bounded onD, thenM(D)⊆ M(ϕ)⊆ D. Moreover, ifϕinduces an unboundedCϕinD, thenM(D) is no longer contained inM(ϕ) since, in such a case, this latter space does not contain the constant functions.

The aim of this work is twofold. On one hand, we are interested in identifying the multiplier space M(ϕ) for self-maps ϕ of D_{. Indeed, we will be able to characterize}

the extreme cases whenever ϕ is a self-map of D _{belonging to} _D_{. Let us remark here}

that in the case of the Hardy space H2_{, where} _C

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that MH2(ϕ) =H2 if and only ifkϕk_∞<1 (see [9]) andM_H2(ϕ) =H∞ if and only if

ϕis a finite Blaschke product; this latter statement was showed previously in a different way in [5] and [17].

Let us also point out that in the course of our findings, we will prove aDecomposition Theorem for the Dirichlet space (cf. Theorem 2.1), which is interesting in its own and whose proof is based on the theory of model spaces for the shift operator in the Hardy space (see [18] for more information about model spaces). As far as we know, this is the first time model spaces come into play with the Dirichlet space.

On the other hand, we are interested in the spectral properties of weighted composition operators in D. In [12], Higdon computed the spectrum of composition operators inD

induced by linear fractional self-maps ofD_{. The techniques developed there were quite}

different from those carried over by Cowen in [6] in the corresponding case of the Hardy spaceH2_{(see also [7, Chapter 7]), due to the particular nature of}_D_.

In a very recent work, Hyv¨arinen, Lindstr¨om, Nieminen and Saukko [13] have described the spectra of invertible weighted composition operatorsWu,ϕ acting on a large class of analytic function spaces including the weighted Bergman and the weighted Hardy spaces; generalizing previous results obtained in [11]. Nevertheless, as they also remark, their results do not apply directly to the Dirichlet space since they rely on the fact that the algebra of the multipliers of the spaces considered is H∞_{. Our aim is to extend}

Hyv¨arinen, Lindstr¨om, Nieminen and Saukko’s results to the context of the Dirichlet spaceD, pointing out that their techniques are no longer working inD.

The last section of the paper gives a description of the spectra of invertible weighted composition operators. We first note (see Proposition 3.1) that a bounded weighted composition operator Wu,ϕ in the Dirichlet space D is invertible if and only if u is a multiplier bounded away from zero in D _and _ϕ _{is an automorphism of the unit disc.}

Consequently, three separate cases are considered, depending on the nature of the disc automorphismϕ: elliptic, parabolic or hyperbolic. Whenϕis parabolic, causal operators will play a prominent role in order to determine explicitly the spectrum ofWu,ϕ.

2. Boundedness of weighted composition operators

In this section, we study boundedness of weighted composition operators in the Dirich-let space. In order to show the results at this respect, we prove aDecomposition Theorem

forDbased on model spaces.

LetBbe a finite Blaschke product and writeKBfor the model spaceKB =H2⊖BH2, which is finite-dimensional; indeed dimKB = degB. Observe that ifgk ∈KB, then it does not matter which norm we use, since KB is finite-dimensional and all norms are equivalent. We proceed to state theDecomposition Theorem in its full generality, since the main arguments of the proof also work for the Bergman space.

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(1) f ∈ H2 _{if and only if} _f ₌P∞

k=0gkBk (convergence inH2 norm) with gk∈KB andP∞_k₌₀kgkk2<∞.

(2) f ∈ D if and only if f =P∞_k₌₀gkBk (convergence in D norm) with gk ∈ KB

andP∞_k₌₀(k+ 1)kgkk2<∞.

(3) f ∈ A2 _{if and only if}_f ₌P∞

k=0gkBk (convergence in A2 norm) with gk ∈KB andP∞_k₌₀kgkk2/(k+ 1)<∞.

Statement (1) is included for the sake of completeness since it is a standard fact that

H2₌_K

B⊕BKB⊕B2KB⊕...as an orthonormal direct sum.

A word about notation. Throughout this work, a . b will denote that there exists an independent constant C such that a ≤ C b; this constant may be different in each instance.

Proof. We proceed to prove (2) and (3). We claim that for finite sums

k

N

X

k=0

gkBkk2D .

N

X

k=0

(k+ 1)kgkk2 and k N

X

k=0

gkBkk2A2 .

N

X

k=0

kgkk2/(k+ 1)

with the implied constants independent ofN.

Let e1, . . . , en be a basis of the space KB. Then, writing gk =Pn_ℓ₌₁akℓeℓ we have, sinceCB is a bounded composition operator inD, that

N X k=0 n X ℓ=1

akℓeℓBk

D = n X ℓ=1 eℓ N X k=0 akℓBk

D ≤ n X ℓ=1

keℓkM(D) N X k=0 akℓBk

D . n X ℓ=1 N X k=0

(k+ 1)|akℓ|2

!1/2

. n X ℓ=1 N X k=0

(k+ 1)|akℓ|2

!1/2

, by equivalence of norms on Cn_,

. N

X

k=0

(k+ 1)kgkk2

!1/2 ,

where the notationkeℓkM(D) represents sup{keℓfkD :kfkD ≤1}. A similar calculation

can be made inA2_.

To obtain the converse inequality, we use the dual pairing betweenD, equipped with the equivalent norm kP∞_k₌₀akzkk2X =

P∞

k=0(k+ 1)|ak|2 and A2, equipped with the equivalent normkP∞k=0akzkk2Y =

P∞

k=0|ak|2/(k+ 1), namely

*_∞ X

k=0 akzk,

∞

X

k=0 bkzk

+

=

∞

X

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For a finite sumPN_k₌₀gkBk we takehk ∈KBwithhk = (k+ 1)gk for eachk. Now we use theH2 _{orthogonality of}_Br_K

B and BsKB for positive integersr and swithr6=s, to deduce that _*

N

X

k=0 gkBk,

N

X

k=0 hkBk

+

= N

X

k=0

(k+ 1)kgkk2H2.

Since N X k=0 hkBk

2 Y . N X k=0

khkk2/(k+ 1) = N

X

k=0

(k+ 1)kgkk2,

it follows that

N X k=0 gkBk

2 X & N X k=0

(k+ 1)kgkk2,

and so we have a uniform equivalence of the Dirichlet norm and the quantity

N

X

k=0

(k+ 1)2kgkk2

!1/2 ,

at least for finite sums. Since the Dirichlet space is contained in the Hardy space we may make the obvious extension to the whole ofDusing infinite sums.

The argument for the Bergman space is analogous: once more we have an equivalence of norms, and since the Hardy space is dense in the Bergman space we obtain the required

result.

Recall that ifCϕis bounded thenM(D)⊆ M(ϕ)⊆ D. Forϕa finite Blaschke product the space of weighted composition operators is as small as possible, as the following result shows.

Theorem 2.2. Let ϕbe an inner function. Then M(ϕ) =M(D) if and only if ϕis a finite Blaschke product.

Proof. If ϕ is inner but not a finite Blaschke product, then ϕ 6∈ D, and soCϕ is un-bounded. Thus, takingu(z)≡1, we have thatu∈ M(D) butu6∈ M(ϕ).

Now suppose thatϕ=B, a finite Blaschke product. LetTudenote the mapf ∈ D 7→

uf. We must show that TuCB is bounded if and only if u∈ M(D). It is clear that if

u∈ M(D) thenTuCB is bounded onD, since bothTu andCB are bounded.

For the converse, note that without loss of generality we may take B(0) = 0, since if for someawe have B(a) = 0, then, setting

ϕa(z) =

a−z

1−az

we haveB◦ϕa(0) = 0 and

CϕaTuCB =Tu◦ϕaCB◦ϕa.

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Now, given thatTuCB is bounded, letf ∈ D. By Theorem 2.1 we may write

f =

∞

X

k=0

gkBk=

∞ X k=0 n X ℓ=1

akℓeℓBk,

where each gk ∈ KB , {e1, . . . , en} is a basis of KB, and for each ℓ the scalars (akℓ) satisfy

∞

X

k=0

(k+ 1)|akℓ|2.

∞

X

k=0

(k+ 1)kgkk2.kfk2D.

Writefℓ(z) =P∞k=0akℓzk so thatkfℓkD .kfkD. Then

u ∞ X k=0 akℓBk

D

=kTuCBfℓkD .kTuCBkkfkD.

Nowe1, . . . , en lie inM(D) and we conclude that

kufkD=

n X ℓ=1

eℓTuCBfℓ

D

.kTuCBkkfkD,

so thatu∈ M(D) and the Theorem is proved.

Next example shows that the assumption aboutϕbeing inner cannot be relaxed; even ifkϕk∞= 1 and Cϕ is bounded inD.

Example 2.1. We can haveM(ϕ)6=M(D) even ifkϕk∞= 1 andCϕis bounded inD. Let us consider

ϕ(z) = 1−z

2 and u(z) =

∞

X

k=2 zk

k(logk)3/4.

Note thatu∈ D \ M(D) (see [8, Thm. 5.1.6]). Nonetheless, TuCϕ is bounded; that is,

u∈ M(ϕ).

In order to show that, letf ∈ D. We have

(u(f ◦ϕ))′₌_u′₍_f_◦_ϕ_{) +}_u₍_f′_◦_ϕ₎_ϕ′_.

It suffices if we prove that each of these terms lies in the Bergman spaceA2.

Let us split the disc into disjoint measurable setsD₌_D₁_∪_D₂_{, where}_D₁ _{is a small}

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D2 D1

A partition of the unit disc.

OnD1 bothuand u′ are square-integrable with respect to Lebesgue measure, while f◦ϕand f′_◦_ϕ_{are uniformly bounded by constants depending only on the norm of}_f_.

Likewise, onD2,uandu′ are bounded, whilef◦ϕandf′◦ϕare square-integrable. Our

conclusion is thatTuCϕ is bounded on the Dirichlet spaceD.

At the other extreme, we have the following result.

Theorem 2.3. Let ϕbe an analytic self-map ofD_{. Then} _M₍_ϕ_{) =}_D_{if and only if}

(1) kϕk∞<1, and

(2) ϕ∈ M(D).

Proof. Suppose first thatϕsatisfies conditions (1) and (2). Letube a Dirichlet function. In order to prove thatTuCϕ is a bounded operator onD, letf ∈ Dand consider

(u(f ◦ϕ))′=u′(f◦ϕ) +u(f′◦ϕ)ϕ′.

It suffices to show that each of these terms lies in the Bergman spaceA2_.

We have thatkf◦ϕk∞.kfkD since, forw∈D,

(f◦ϕ)(w) =hf, kϕ(w)iD,

where kϕ(w) denotes the reproducing kernel at ϕ(w) in D, which is bounded in norm

independently ofwsincekϕk∞<1. Hence, sinceu∈ D, we have

(1) ku′₍_f_◦_ϕ₎_k

A2 .kfk_D.

Also,kf′_◦_ϕk

∞.kfkD, since the derivative kernelskϕ′(w)satisfying

f′₍_ϕ₍_w_{)) =}_{hf, k}′

ϕ(w)iD

are also uniformly bounded in norm whenkϕk∞ <1. Now by means of condition (2),

ϕ∈ M(D), and therefore the measure|ϕ′₍_z₎_|2_dA₍_z_{) is a Carleson measure for}_D_{. Thus,}

it follows that

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From (1) and (2), one gets thatTuCϕ is a bounded operator for allu∈ D.

Conversely, ifTuCϕ is bounded for allu∈ D, then by the Closed Graph TheoremCϕ maps D boundedly into M(D) and hence into H∞_{. Suppose that}_kϕk

∞ = 1; then we

may find an unboundedf ∈ D, and by considering functionsfn(z) =f(eiθnz) for suitable anglesθn, obtain a sequence of normalized functionsfninDand (zn)⊂Dwith|zn| →1 such that|fn(ϕ(zn))| → ∞. This is a contradiction, so we conclude thatkϕk∞<1.

We now see that, for f ∈ D fixed, we haveku(f′_◦_ϕ₎_ϕ′_k

A2 .kuk_D. So letf(z) =z. We conclude thatTϕ′ :D → A2 is bounded, or equivalently |ϕ′(z)|2dA(z) is a Carleson measure for D. Given that ϕ is bounded, this condition implies that ϕ ∈ M(D), as mentioned in the introduction. This concludes the proof of the Theorem.

3. Spectral properties

In this section, we are interested in describing the spectra of invertible weighted com-position operators in the Dirichlet space. As we pointed out in the introduction, the techniques in [13] depends strongly on the fact that the algebra of the multipliers con-tainsH∞_.

The next result identifies invertible weighted composition operators in the Dirichlet space. It can be found, for example, in [3, Thm 3.3] and [23, Cor. 11].

Proposition 3.1. LetWh,ϕ be a bounded weighted composition operator in the Dirichlet

space D. Then Wh,ϕ is invertible in D if and only if h ∈ M(D), bounded away from

zero in D _and _ϕ _{is an automorphism of} D_{. In such a case, the inverse operator of}

Wh,ϕ:D → Dis also a weighted composition operator and

(Wh,ϕ)−1= 1

h◦ϕ−1Cϕ−1.

Recall that an automorphismϕofD_{can be expressed in the form}

ϕ(z) =eiθ p−z

1−pz (z∈D),

wherep∈D_and_{−π < θ}_≤_π_{. Recall that} _ϕ_{is called}_hyperbolic _if_|p|_>_cos(_θ/_{2) (thus,}

ϕfixes two points on∂D_);_parabolic_if_|p|_{= cos(}_θ/_{2) (so,}_ϕ_{fixes just one point, located}

on∂D_{) and}_elliptic_if_|p|_<_cos(_θ/_{2) (therefore,}_ϕ_{fixes two points, one of them in}D_and

the other in the exterior ofD_{). See [19, Chapter 0], for instance.}

Notation. Assumeϕis a self-map of the unit discD_{. In what follows,}_ϕ_n _{will denote the}

n-th iterate of the mapϕ, that is,

ϕn=ϕ◦ϕ◦ · · ·ϕ (ntimes),

for anyn≥0, whereϕ0is the identity function. It is clear thatCϕn=Cϕn for anyn≥0. If Wh,ϕ is a bounded weighted composition operator in D, it is rather straightforward that

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for anyf ∈ Dandz∈D_{. Following [13], we will denote}

h(n)=

n_Y−1

k=0 h◦ϕk;

whereh(0)= 1 for convenience.

In what follows, we restrict our attention to weighted composition operators Wh,ϕ acting onDinduced by disc automorphismsϕ. By Theorem 2.2, this implies thathis a multiplier ofD.

3.1. Elliptic case. In [13, Section 4.3], the authors describe the spectrum ofWh,ϕacting on a large class of spaces of analytic functions whenever his in the disc algebra A(D₎

and ϕis an elliptic automorphism. Our hypotheses on hin the context of Dis rather more general, sinceWh,ϕis bounded if and only ifh∈ M(D) (and the spacesM(D) and

A(D_{) are not contained in each other). Nevertheless, it is possible to take a bit further}

some of the ideas developed in [13] and show the following result in a similar way.

Theorem 3.1. Suppose that ϕis an elliptic automorphism ofD _{with fixed point}_a_∈D

andWh,ϕ a weighted composition operator onD. Then

(1) either there exists a positive integerj such thatϕj(z) =zfor allz∈D, in which

case, ifm is the smallest such integer, then

σ(Wh,ϕ) ={λ: λm=h(m)(z), z∈D},

(2) orϕn6= Id for every nand, ifWh,ϕ is invertible, then

σ(Wh,ϕ) ={λ: |λ|=|h(a)|}.

Proof. The proof of (1) goes as in [13, Theorem 4.11]. The only minor change concerns the inclusion

σ(Wh,ϕ)⊂ {λ: λm=h(m)(z), z∈D},

where a similar argument applies taking into account the fact that ifg ∈ M(D) and it is bounded away from zero, then 1/gis also inM(D). With respect to (2), just observe that W_h,ϕ−1 = 1

h◦ϕ−1Cϕ−1 is bounded, and hence 1/(h◦ϕ−1) is inM(D) sinceϕ−1 is a disc automorphism (see Theorem 2.2). Therefore, we refer the reader to [13] once more.

3.2. Parabolic case. Now, let us assume that Wh,ϕ is an invertible weighted composi-tion operator onD, whereϕis a parabolic disc automorphism. The previous ideas in [13] to determine the spectrum ofWh,ϕ made an extensive use of the fact that the sequence orbit{ϕn(z0)} of a point z0∈Dis an interpolating sequence for H∞, a space which is

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M(D) and therefore, our proof completely differs from the previous ones. Likewise, the work in [11] made use of inner functions, which are inappropriate in the context ofD.

Theorem 3.2. Suppose that ϕis a parabolic automorphism of D_{with fixed point} _a_∈T

and Wh,ϕ a weighted composition operator on D, determined by an h∈ M(D) that is

continuous ata. IfWh,ϕ is invertible, then

σ(Wh,ϕ) ={λ∈C:|λ|=|h(a)|}.

Proof. We begin by showing that the spectrum of Wh,ϕ is contained in the circle {λ :

|λ|=|h(a)|}. Recall thath∈ M(D) implies thath∈ H∞_{and that}_|h′_|2_dA_{is a Carleson}

measure forD; that is, that there exists a constantK >0 such that

Z

D

|h′₍_z₎_|2_|f_|2_dA₍_z₎_≤_K2_kf_k2

D

for allf ∈ D. We write kh′_k

C for the least suchK. Moreover, we see that 1/h∈ M(D)

sinceWh,ϕ is invertible, which implies thath(a)6= 0.

Recalling that (Wh,ϕ)n = Wh(n),ϕn = Th(n)Cϕn, we estimate its norm. Since the spectral radius ofCϕis 1 (see [12]) so that for eachε >0 we havekCϕnk ≤(1 +ε)

n _for

nsufficiently large, it will be sufficient to consider the operator of multiplication byh(n).

Forf ∈ Dwe have

(h(n)f)′ =h(n)f′+h′(n)f.

TheA2_{norm of the first term can be estimated using the fact that for each}_{ε >}_{0 there}

is anmsuch that

(3) kh(n)k∞≤ khkm∞[(1 +ε)|h(a)|]n−m

for alln≥m, which is given in the proof of [13, Lem. 4.2]. Also

kh′(n)kC ≤

n_X−1

j=0

kh(n),j(h◦ϕj)′kC,

whereh(n),j =h(n)/(h◦ϕj).

Hence, as in (3) we have forj < m

kh(n),jk∞≤ khkm∞−1[(1 +ε)|h(a)|]n−m,

while forj≥mwe have

kh(n),jk∞≤ khkm∞[(1 +ε)|h(a)|]n−m−1.

Also

k(h◦ϕj)′kC =kϕ′j(h′◦ϕj)kC ≤ kϕ′jk∞kh′kC.

Now, we estimatekϕ′

jk∞for anyj. Let us suppose without loss of generality thata=−1.

Hence

ϕn(z) =

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where y∈R_{\ {}₀_} _{(see, for instance, [10] for a similar computation when}_a_{= 1). For a}

M¨obius mapψ(z) = z−α

1−αz we have

|ψ′(z)|= 1− |α|

2

|1−αz|2,

and in the case ofϕn we haveα= 1 +O(1/n), so thatkϕ′nk∞=O(n).

Putting all this together we conclude that

kh′(n)kC ≤Cn2[(1 +ε)|h(a)|]n,

whereC does not depend onn, and hence (since we have this for allε >0)

lim sup n→∞ kTh(n)k

1/n_{≤ |h}₍_a₎_|.

Having bounded the spectral radius of Wh,ϕ by |h(a)|, we may similarly bound the spectral radius of its inverse by 1/|h(a)|, using Proposition 3.1. Thus the spectrum of

Wh,ϕ is contained in the circle of radius|h(a)|.

We now prove that the spectrum ofWh,ϕ is the entire circle, by showing that for each

λwith|λ|=|h(a)|the spectral radiusρ(Wh,ϕ−λI) is at least 2|h(a)|. This technique was also used in [13], but there are extra complications in our case. As we already assumed, we may choose the fixed pointaofϕto be−1.

There is a unitary mappingJ fromD=D(D_{) onto}_D₍C₊_{), the Dirichlet space of the}

right half-plane, induced by the conformal involution

M :z7→ 1−z

1 +z;

an easy calculation shows thatJWh,ϕJ−1=Wfh◦M,τ, the weighted composition operator onD(C₊_{) induced by a parabolic automorphism}_τ ₌_M _◦_ϕ_◦_M_{, which takes the form}

τ:s→s+icfor somec >0 since it fixes∞.

The following is a straightforward Corollary of [4, Thm. 3.2], given that the operator

Z := (Wfh◦M,τ−λI)n is causal (which may be most easily stated by saying that if the inverse Laplace transform of a functionuis supported on (T,∞) for someT > 0, then so is the inverse Laplace transform of its imageZu).

• Let ˜h: C₊ _→C _{be holomorphic and} _τ _: C₊ _→C₊ _{a causal holomorphic function}

such that the weighted compositionWf˜h,τ is bounded on D(C+). Then for all λ∈Cand n≥1 the inequality

k(fW˜_h,τ−λI)nkH2₍C₊)≤ k(Wf˜_h,τ−λI)nkD(C₊)

holds, and henceρ(fW˜_h,τ−λI)H2₍C+)≤ρ(fW˜h,τ−λI)D(C+).

Now consider the operatorWfh◦M,τ onH2(C+), and note that it is unitarily equivalent

to the weighted composition operator W1+ϕ 1+zh,ϕ

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then have

ρ(Wh,ϕ−λI)D = ρ(fW˜_h,τ−λI)D(C+)

≥ ρ(fW˜_h,τ−λI)H2₍C₊)

= ρW1+ϕ 1+zh,ϕ

−λI

H2₍D)

≥ 2

1 +_{1 +}ϕ(_zz)

z=−1

|h(−1)|= 2|h(−1)|,

where the last assertion is a direct consequence of an observation made in the proof of [13, Thm. 4.3] and the fact that

1 +ϕ(z)

1 +z →1 as z→ −1,

sinceϕis parabolic. Sinceρ(Wh,ϕ−λI)D≥2|h(−1)|, we have

σ(Wh,ϕ) ={λ∈C:|λ|=|h(−1)|},

as required.

3.3. Hyperbolic case. The same method as we adopted for the parabolic case can be used to show that the spectrum ofWh,ϕ is contained in an annulus, in the case thatϕis a hyperbolic automorphism.

Theorem 3.3. Suppose that ϕis a hyperbolic automorphism of D_{with attractive fixed}

point a∈T_{and repelling fixed point}_b_∈T_{. Let}_W_h,ϕ _{be a weighted composition operator}

on D, determined by anh∈ M(D) that is continuous ata andb. If Wh,ϕ is invertible,

then

ρ(Wh,ϕ)≤max{|h(a)|,|h(b)|}/µ,

whereϕis conjugate to the automorphism

ψ(z) =(1 +µ)z+ (1−µ) (1−µ)z+ (1 +µ),

with0< µ <1. Henceσ(Wh,ϕ)is contained in the annulus with radiimax{|h(a)|,|h(b)|}/µ

andmin{|h(a)|,|h(b)|}µ.

Proof. For eachε >0 there is an msuch that the estimate

kh(n)k∞≤ khkm∞[(1 +ε) max{|h(a)|,|h(b)|}]n−m

holds, as in [13].

Similarly, for the derivative, we have

kh′(n)kC ≤

n_X−1

j=0

kh(n),j(h◦ϕj)′kC,

whereh(n),j =h(n)/(h◦ϕj), and wherek·kChas been defined in the proof of Theorem 3.2.

Now, however,

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wherekϕ′

jk∞=O(µ−j), using the fact that

ψj(z) =

(1 +µj₎_z_{+ (1}₋_µj₎ (1−µj₎_z_{+ (1 +}_µj₎,

as in [10, p. 38]. Also

kh(n),jk∞≤ khkm∞−1[(1 +ε) max{|h(a)|,|h(b)|}]n−m

forj < mand

kh(n),jk∞≤ khkm∞[(1 +ε) max{|h(a)|,|h(b)|}]n−m−1

forj≥m.

By similar arguments to those used in the proof of Theorem 3.2, we conclude that

lim sup n→∞ kTh(n)k

1/n_≤_max_{|h₍_a₎_|,_|h₍_b₎_|}/µ.

The final assertion follows on considering the spectral radius ofW_h,ϕ−1.

A final remark. Finally, regarding Theorem 3.3, we would like to pose the following open question:

Ifϕis a hyperbolic automorphism ofD_{with attractive fixed point}_a_∈T_{and repelling fixed}

point b ∈T _{conjugated to the automorphism} _ψ₍_z_{) =} (1+µ)z+(1−µ)

(1−µ)z+(1+µ), with 0 < µ <1 and Wh,ϕ is a weighted composition operator onD, determined by anh∈ M(D)continuous

at both aandb, does it follow that

σ(Wh,ϕ) ={z∈C: min{|h(a)|,|h(b)|}µ≤ |z| ≤max{|h(a)|,|h(b)|}/µ}

wheneverWh,ϕ is invertible?

Acknowledgements

This work was initiated during a research visit of the first and the third authors to the Departamento de An´alisis Matem´atico at Universidad Complutense de Madrid. They are grateful for the hospitality and the support of research grant MTM2010-16679. The first author also acknowledges support from the London Mathematical Society (under Scheme 2).

In the first version of this manuscript, we proved Proposition 3.1 ourselves: we thank the referee for pointing out references [3] and [23], where the result is already proved by different methods.

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Universit´e de Lyon; CNRS;

Universit´e Lyon 1; INSA de Lyon; Ecole Centrale de Lyon, CNRS, UMR 5208, Institut Camille Jordan

43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

E-mail address:chalendar@math.univ-lyon1.fr

Universidad Complutense de Madrid e ICMAT Departamento de An´alisis Matem´atico,

Facultad de Ciencias Matem´aticas, Plaza de Ciencias 3

28040, Madrid (SPAIN)

E-mail address:eva.gallardo@mat.ucm.es

School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.