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Toeplitz operators on the space of analytic functions with logarithmic growth
J. Math. Anal. Appl. 353 (2009) 428–435
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Toeplitz operators on the space of analytic functions with logarithmic growth José Bonet a,∗ , Jari Taskinen b a b
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universidad Politécnica de Valencia, E-46071 Valencia, Spain Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FIN-00014 Helsinki, Finland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 August 2008 Available online 11 December 2008 Submitted by T. Ransford Keywords: Toeplitz operators Berezin transform Weighted Banach spaces of analytic functions Weighted inductive limits
Continuous and compact Toeplitz operators for positive symbols are characterized on the space H ∞ V of analytic functions with logarithmic growth on the open unit disc of the complex plane. The characterizations are in terms of the behaviour of the Berezin transform of the symbol. The space H ∞ V was introduced and studied by Taskinen. The Bergman projection is continuous on this space in a natural way, which permits to define Toeplitz operators. Sufficient conditions for general symbols are also presented. © 2008 Elsevier Inc. All rights reserved.
1. Introduction The purpose of this paper is to study Toeplitz operators T ϕ in the space H ∞ V of analytic functions on the unit disc which is quite far from the Bergman Hilbert space A 2 , our results grow logarithmically at the boundary. Though the space H ∞ V will be astonishingly similar to that basic case. For example, in the case of positive symbols ϕ we obtain a necessary and sufficient condition for the continuity in terms of a growth condition of the Berezin transform ϕ˜ of ϕ , see Theorem 3.1. ˜ Moreover, the difference between the A 2 and H ∞ V -cases is most typical: in the former, ϕ must be bounded on D whereas in the latter a growth like a power of the logarithm of the boundary distance is allowed (see [22], Chapter 6, for the basic results for Toeplitz operators on A 2 ). In Theorem 3.2 we also characterize compact Toeplitz operators (with positive ϕ ) on H∞ V again in terms of a Berezin-type condition. It is remarkable how beautifully the abstract definition of continuous and compact operators in countable inductive limit of Banach spaces is transformed into concrete conditions on logarithmic weight families, see (3.1) and (3.2). For general not necessarily positive ϕ we are able to provide sufficient conditions for continuity and compactness, see Remark 3.4, Proposition 4.1 and Theorem 4.2. ∞ The space H ∞ V and the related space L V were introduced and studied by the second author in [15] as substitutes of the usual sup-normed spaces H ∞ and L ∞ , to avoid the well-known discontinuity problem of the Bergman projection P with respect to the sup-norm. These spaces are topological algebras and they have the properties that H ∞ V is a closed subspace ∞ ∞ ∞ of L ∞ V and that P is a continuous projection from L V onto H V . In the article [15] the space H V was introduced using weighted sup-seminorms and it was proved, as a consequence of abstract results due to Bierstedt, Meise and Summers [3], that it is in fact a countable inductive limit (in fact a union) of weighted Banach spaces of analytic functions on the disc with compact linking maps. This description is better adapted to the study of operators defined on the space, and is the one we outline in Section 2. The research of Taskinen [15] in this direction was continued by several authors [6,8,9,16].
*
Corresponding author. E-mail addresses: [email protected] (J. Bonet), jari.taskinen@helsinki.fi (J. Taskinen).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.12.009
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J. Bonet, J. Taskinen / J. Math. Anal. Appl. 353 (2009) 428–435
429
From the harmonic analysis point of view, the advantage of using the space H ∞ V is that very simple pointwise estimates can all the time be applied. The emerging logarithmic singularities are absorbed by the inductive limit. A reader who is not familiar with inductive limits may be satisfied by the observation that their presence only amounts to statements on weight functions like “for all powers of log” or “for some power of log” instead of the usual unique weight functions in the Banach space case. Properties of operators in inductive limits are explicitely formulated in terms of containment of sets of analytic functions satisfying certain weight estimates. Toeplitz operators on spaces of analytic functions have been investigated since long. We refer the reader e.g. to the books by Böttcher and Silbermann [5] and by Zhu [22]. For Toeplitz operators on Bergman spaces, in particular on A 2 of various domains, see [1,10,12,14,17,18,20,21]. As for other spaces, let us also mention the recent papers [7,19]. 2. Notation All function spaces are defined on the open unit disc D of the complex plane C, unless otherwise stated. By d A we denote the normalized two-dimensional Lebesgue measure on D, and by L p := L p (d A ) the space of p-integrable functions on the disc D with respect to the measure d A. Here 1 p ∞. Moreover, A p stands for the Bergman space, which is the closed subspace of L p consisting of analytic functions. The Bergman projection P is the integral operator
P f ( z) := D
f (ζ )
(1 − zζ¯ )2
d A (ζ ),
z ∈ D,
(2.1)
defined at least for all f ∈ L 1 (d A ). Given a function (symbol) ϕ ∈ L 1loc (d A ) we denote by M ϕ the pointwise multiplication with ϕ , and by T ϕ we denote the corresponding Toeplitz-operator
T ϕ f ( z) := P M ϕ f ( z) := D
ϕ (ζ ) f (ζ ) (1 − zζ¯ )2
The Berezin transform of a function
ϕ˜ (z) =
ϕ (ζ )(1 − |z|2 )2 |1 − zζ¯ |4
D
d A (ζ ).
(2.2)
ϕ ∈ L 1 is defined by
d A (ζ ).
(2.3)
See more details in [22]. ∞ We recall the definitions of the spaces H ∞ V and L V from [15]; notice that we use here slightly different notations. First, denote
w ( z) := 1 + log(1 − z).
(2.4)
This function is mainly used to define the radial weight w (| z|). Sometimes it is however necessary to define the logarithm as an analytic function on D: then the argument of 1 − z is understood to belong to the interval ]−π , π [ for all z ∈ D, and the definition becomes unambiguous. ∞ The space H ∞ V (respectively, L V ) consists of analytic (respectively, measurable) functions f : D → C such that for some n ∈ N and constant C n > 0
f (z) C n w | z| n for (almost) all z ∈ D.
(2.5)
The space H ∞ V is an
(LB)-space, i.e., countable inductive limits of Banach spaces. In fact, it is a complete (LB)-space. We refer the reader to [11,13] for locally convex inductive limits. A convenient way to describe the topology of H ∞ V is obtained with a projective description of the inductive limit topology. This can be done as a consequence of the work of Bierstedt, Meise and Summers [3]. More precisely, it allows us to define the topology by means of the following family of weighted sup-seminorms: f v := sup f (z) v (z), z∈D
v ∈ V,
(2.6)
where V consists of all continuous, positive, radial functions v : D → R such that for all n ∈ N, v ( z) C n w ( z)−n .
(2.7)
More details, results, examples and motivation about weighted inductive limits of Banach spaces of analytic functions can be seen in [2] and the references therein. We still denote B n :=
sup f (z) w | z| f ∈ H∞ V z∈D
n
;
(2.8)
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∞ in the same way (using “ess sup”) we define the subsets B nL of L ∞ V . The sets B n are bounded and even precompact in H V . ∞ ∞ L Every bounded subset of H V is contained in a multiple of some B n . The same holds for the sets B n in L V , except that bounded sets need not be precompact in this case. We collect in the following proposition the results of [15] that are needed later (see also [4]).
Proposition 2.1. (1) If v ∈ V , then the pointwise product w k v also belongs to V , for every k. ∞ (2) The mapping P is a continuous projection from L ∞ V onto H V . L (3) In addition, P ( B n ) ⊂ C n B n+1 for all n. A linear operator from a complete locally convex space into itself is called compact (respectively, bounded), if it maps a neighbourhood of zero into a precompact (respectively, bounded) set. Recall that in a complete space a subset is precompact if and only if it is relatively compact (see [11]). Remark 2.2. (1) A linear operator T between two (LB)-spaces is continuous, if and only if it maps bounded sets into bounded sets. In ∞ our case this means that T ϕ : H ∞ V → H V is continuous if and only if for every n ∈ N one can find C n > 0 and m ∈ N such that T ϕ ( B n ) ⊂ C n B m . ∞ (2) One can also show that T ϕ : H ∞ V → H V is compact, if and only if there exists m ∈ N such that for every n ∈ N one can find a constant C n with T ϕ ( B n ) ⊂ C n B m . The symbols C , c (respectively, C n ) denote strictly positive absolute (respectively, depending on n) constants, which may vary from place to place but not in the same sequence of inequalities. Definition 2.3. Denote for all L ∈ N and λ ∈ ]−2 L , 2 L ] ∩ Z =: I L
D L ,λ := z := re i θ 1 − 2 L −1 < r 1 − 2− L and and for each L and λ denote z L ,λ
π (λ − 1)2−L < θ π λ2−L ,
−L := (1 − 2−L )e i2π λ2
(2.9)
∈ D L ,λ .
Remark 2.4. For positive real valued functions F and G (defined for example on D or on some other domain) we denote F∼ = G, if there exist absolute, positive constants C > c > 0 such that cF G C F
(2.10)
on the whole domain of definition. The following simple estimate will be needed: 1 − | z| + |arg z| ∼ = |1 − z|,
(2.11)
where z ∈ D with arg z ∈ [−π , π ]. Namely, it is obviously enough to prove the claim on a domain where, say, |1 − z| 1/4. (For other z both sides of (2.11) are bounded from above and below by positive constants.) But for such z =: re i θ we have |1 − z | ∼ = |1 − r cos θ| + |sin θ| ∼ = |1 − r | + |θ|, which is the statement. 3. Characterization of continuity and compactness for positive symbols In the following main results of our paper we characterize the continuity and compactness of T ϕ with positive ϕ in terms of the growth properties of the Berezin transform of ϕ . For general, not necessarily positive ϕ , we show that these growth criterions are necessary conditions for continuity and compactness, respectively. ∞ Theorem 3.1. Assume ϕ 0. The Toeplitz operator T ϕ is continuous H ∞ V → H V if and only if there exist k ∈ N and C > 0 such that the Berezin transform ϕ˜ satisfies
k k ϕ˜ (z) C w |z| = C 1 + log 1 − |z| , z ∈ D.
(3.1)
∞ Theorem 3.2. Assume ϕ 0. The operator T ϕ : H ∞ V → H V is compact, if and only there exists k ∈ N such that for every q ∈ N there exists C q > 0 with
D
ϕ (ζ )(1 − |z|2 )2 |1 −
z ζ |4
k
w (ζ z¯ )q d A (ζ ) C q w | z| ,
z ∈ D.
(3.2)
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431
Proof of Theorems 3.1 and 3.2. I. Sufficiency. We prove the sufficiency of the condition (3.1) in Theorem 3.1 (we call this the case (A); of course, k ∈ N will be fixed all the time). During the course of the proof we describe the changes needed to prove the sufficiency of (3.2) in Theorem 3.2; this is called the case (B). We first claim that in the case (A)
ϕ d A C k Lk 2−2L
(3.3)
D L ,λ
for all L and λ. In the case (B) and the assumption (3.2) we get
ϕ d A C q L −q+k 2−2L for all q.
(3.4)
D L ,λ
In the case (A), fixing L and λ we use the positivity of the symbol, the definition of the Berezin transform and the assumption (3.1) to obtain
D L ,λ
ϕ (ζ )(1 − |z L ,λ |2 )2 |1 − z L ,λ ζ¯ |4
d A (ζ ) D
ϕ (ζ )(1 − |z L ,λ |2 )2 |1 − z L ,λ ζ¯ |4
k d A (ζ ) C log 1 − | z L ,λ | .
We have 1 − | z L ,λ | ∼ = 2−L and also |1 − z L ,λ ζ¯ | C 2−L for ζ ∈ D L ,λ . The inequality (3.3) follows already from this, since In the case (B) we in addition make the observation that
log(1 − ζ z¯ L ,λ ) ∼ = − log |1 − ζ z¯ L ,λ | + arg(1 − ζ z¯ L ,λ ) ∼ =L
(3.5)
ϕ 0. (3.6)
ζ z¯ L ,λ is of order 2−L and the argument of it is only between
−π and π . Otherfor ζ ∈ D L ,λ , since the modulus of 1 − wise (3.4) follows in the same way as in (3.5). In the case (A), for the continuity of T ϕ it is sufficient to show that for an arbitrary n ∈ N one can find a constant C n,k > 0 such that T ϕ ( B n ) ⊂ C n,k B n+k+1 (see Remark 2.2(1)). For the compactness, case (B), we show that for every n ∈ N there exists cn > 0 such that T ϕ ( B n ) ⊂ cn B 2+k (Remark 2.2(2)). To prove these facts, we let n ∈ N and f ∈ B n be arbitrary and evaluate the expression T ϕ f (z)
ϕ (ζ )| f (ζ )| |1 − zζ¯ |2
D
d A (ζ ),
(3.7)
z ∈ D. We may assume that z > 0 (by the rotation symmetry) and that z actually of the form z = z N ,ν =: z N for some N ∈ N and ν = 2 N (since a positive z anyway belongs to some set D N ,ν with ν = 2 N ; we have then |1 − z N ζ¯ | ∼ = |1 − zζ¯ | for ζ ∈ D). We get ∞
ϕ (ζ )| f (ζ )| T ϕ f (z N ) d A (ζ ). |1 − z N ζ¯ |2 L =1 λ∈ I L
(3.8)
D L ,λ
For ζ ∈ D L ,λ one has 1 − | z N ζ¯ | ∼ = 1 − (1 − 2−N )(1 − 2−L ) ∼ = 2−N + 2−L and, since z N is real, we obtain from (2.11) of Remark 2.4,
|1 − z N ζ¯ | ∼ = 2−N + 2−L + |λ|2−L .
(3.9)
Moreover, since f ∈ B n , we have
f (ζ ) C n Ln .
(3.10)
On account these estimates and (3.3), we find that in the case (A) that (3.8) can be bounded by a constant times
L ,λ
C n Ln
(2−N
+ 2−L
+ |λ|2−L )2
ϕ (ζ ) d A (ζ ) D L ,λ
L ,λ
(2−N
C n L n+k 2−2L
+ 2−L + |λ|2−L )2
.
(3.11)
In case (B) we get an estimate where C L n+k is replaced by C q L n−q+k . We are thus left with combinatorial estimates. In the case (A) (3.11) will be shown to have a bound C N n+k+1 . This proves the continuity of T ϕ , since this shows
T ϕ f (z N ) C N n+k+1 C log 1 − | z N | n+k+1 ,
(3.12)
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i.e., T ϕ maps B n into C n B n+k+1 (see the discussion about z and z N above). In the case (B) we aim to the bound
T ϕ f (z N ) C q n! N n−q+1+k
(3.13)
for every q n. Then we choose q = n − 1, and obtain
T ϕ f (z N ) C N n−q+1+k C log 1 − | z N | 2+k , and thus T ϕ maps B n into cn B 2+k . We need to estimate
λ∈ I L
2−2L
(2−N + 2−L + |λ|2−L )2
=
Comparing with the integral (with
2L
1
(1 + α +
1
x)2
dx
λ∈ I L
1
(1 + 2L −N + |λ|)2
(3.14)
.
α = 2L−N )
C
(3.15)
1+α
we find that the right-hand side of (3.14) is bounded by C /(1 + 2 L − N ). From this we find that (3.11) has in the case (A) the bound
C n Ln+k . 1 + 2L−N
(3.16)
L
In the case (B) the bound is the same with L n+k replaced by L n−q+k . The inequality
LN
L n+k 1 + 2L−N
C (n + k)! N n+k+1
(3.17)
can easily be derived e.g. by integrating by parts the expression
LN
L n+k 1 + 2L−N
∞ N
xn−k e −x dx. On the other hand,
L n+k N n+k+1 .
(3.18)
LN
This proves the case (A). Obviously, in the case (B) we also get the desired estimate N n−q+1+k . II. Necessity for Theorem 3.1. We assume T ϕ is continuous and find k ∈ N such that T ϕ maps B 1 into C B k (see the remark after (2.8) and Remark 2.2(1)). Considering for a moment the number z ∈ D as a parameter, we define the analytic function K z (ζ ) :=
1
(1 − ζ z¯ )2
,
ζ ∈ D.
(3.19)
Trivially the estimate
1 − | z |2 2 K z
∞
4
(3.20)
holds for every z. Hence T ϕ ((1 − | z|2 )2 K z ) ∈ 4C B k for every z, and
T ϕ 1 − | z|2 2 K z (ω) 4C w |ω| k for all
(3.21)
ω ∈ D. But taking ω = z, this means that
2 2 ϕ (ζ ) (1 − |z|2 )2 ϕ˜ (z) = ϕ (ζ )(1 − | z| ) d A (ζ ) = d A (ζ ) (1 − zζ¯ )2 (1 − ζ z¯ )2 |1 − zζ¯ |4 D
D
2 k = T ϕ 1 − | z|2 K z (z) C w | z| .
(3.22)
This is the condition (3.1). III. Necessity for Theorem 3.2. Our assumption is that there exists a k ∈ N such that for every q ∈ N, T ( B q ) ⊂ C q B k for some constant C q > 0. We define for this q and each z ∈ D the analytic function
q
1 − ζ z¯ Λq,z (ζ ) := log 10q , e
ζ ∈ D.
(3.23)
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433
Combining (3.20) and (3.25) of Lemma 3.3 (below) we find that the function (1 − | z|2 )2 K z Λq,z belongs to the set C q B q for all z. Hence, it is mapped into the set C B k by T ϕ , i.e.
C w | z|
k
2 T ϕ 1 − | z|2 K z Λq,z (z) ϕ (ζ ) (1 − |z|2 )2 Λq,z (ζ ) d A (ζ ) = 2 2 (1 − ζ z¯ ) (1 − zζ¯ ) D
ϕ (ζ )(1 − | z|2 )2 Λq,z (ζ ) = d A (ζ ). |1 − ζ z¯ |4
(3.24)
D
Since all the factors of the integrand are positive, except the function Λq,z , we can conclude from (3.26) of Lemma 3.3 that (3.24) is bounded from below by a constant times
ϕ (ζ )(1 − | z|2 )2 w (ζ z¯ )q . d A (ζ ) |1 − ζ z¯ |4
2
D
To complete the proof above, we still have to show the following lemma. Lemma 3.3. For each q ∈ N there exists C q > 0 such that the function Λq,z of (3.23) satisfies
Λq,z (ζ ) C q w |ζ | q
(3.25)
and
(−1)q Re Λq,z (ζ )
1 Cq
w (ζ z¯ )q
(3.26)
for all z, ζ ∈ D. Proof. We may assume that | z| 1 − e −10q ; otherwise Λq,z is bounded from below and above by positive constants. To prove (3.25), we have
|I | := Im log e −10q (1 − ζ z¯ ) = arg e −10q (1 − ζ z¯ ) π ,
(3.27)
since ζ z¯ ∈ D. Hence, denoting
R := − Re log e −10q (1 − ζ z¯ ) = − log e −10q (1 − ζ z¯ ) > 0
(3.28)
we can estimate
Λq,z (ζ ) C q (1 + R)q = C q 1 − log e −10q (1 − ζ z¯ ) q q C q 1 + log 1 − |ζ | ,
(3.29)
since C |1 − ζ z¯ | 1 − |ζ |. This proves (3.25). Now we deal with (3.26). To estimate the real part of Λq,z we first observe that R 10q, since |e −10q (1 − ζ z¯ )| 3e −10q . By (3.27) we thus have R 2q|I |. But for the number R + i I one thus has (by elementary geometry in the complex plane, based on the fact that the modulus of the argument of R + i I is small, i.e., about 1/(4π q)) Re(R + i I )q
1 2
|R + i I |q
(3.30)
and, furthermore,
q = cq (1 + log e −10q (1 − ζ z¯ ) q C q 1 + log(1 − ζ z¯ ) = C q w (¯zζ )q .
Re(R + i I )q cq 1 + |R + i I |
This is our statement (3.26).
q
(3.31)
2
Remark 3.4. If ϕ is not positive, condition (3.1) is still necessary for the continuity. Sufficient conditions for continuity and compactness are obtained by replacing ϕ by |ϕ | in (3.1) and (3.2), respectively. These claims will not be given separate proofs. A careful inspection of our proof above will reveal their validity.
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4. Sufficient conditions for general symbols In the case of the reflexive Bergman spaces A p it is clear that a Toeplitz operator is bounded, if its symbol ϕ is bounded on D. This follows from the boundedness of the Bergman projection P : L p (d A ) → A p for 1 < p < ∞. In case of H ∞ V there is an analogous, quite straightforward sufficient condition based on a pointwise estimate of ϕ . It is however of some interest that many unbounded symbols satisfy this condition. Moreover, there is a similar condition for the compactness of T ϕ . The analogy of these pointwise conditions with those in Theorems 3.1 and 3.2 should be observed. Proposition 4.1. If there exists a k ∈ N such that
ϕ (z) C w | z| k for all z ∈ D,
(4.1)
∞ then T ϕ : H ∞ V → H V is continuous.
Proof. If v ∈ V , then the weight that
ω := w k v still belongs to V , by Proposition 2.1(1). The assumption (4.1) moreover implies
f ϕ v C f ω
(4.2)
∞ ∞ for all f ∈ H ∞ V , i.e., the multiplication operator M ϕ is continuous H V → L V . By Proposition 2.1(2), the Bergman projection ∞ , hence, T = P M is continuous on H ∞ . 2 is continuous L ∞ → H ϕ ϕ V V V
Theorem 4.2. Assume that for all q ∈ N there exist C q ∈ N such that
ϕ (z) C q w | z| −q for all z ∈ D. Then T ϕ :
H∞ V
→
(4.3)
H∞ V is compact.
∞ Proof. We first show that under the condition (4.3) the operator M ϕ : H ∞ V → L V is bounded, i.e. it maps a neighbourhood U of 0 into a bounded set. We first define
v ( z) := inf C q w | z| q∈N
−q
(4.4)
.
Clearly, v ∈ V ; see the definition of the weight family V in (2.7). Let U be a neighbourhood of 0 which is the closed unit ball of · v , i.e. U := { f ∈ H ∞ v | f v 1}. Every f ∈ U satisfies | f ( z)| 1/ v ( z), hence, using (4.3), we obtain
M ϕ f (z) = ϕ (z) f (z) ϕ (z)sup C −1 w | z| q 1.
(4.5)
q
q∈N
L ∞ ∞ Hence, M ϕ f ∈ B 1L ⊂ L ∞ V . Proposition 2.1(3), shows that P maps B 1 into B 2 ⊂ H V , which is a precompact subset of H V . Hence, T ϕ = P M ϕ is compact. 2
It is not so difficult to see that there are positive symbols which do not satisfy (4.1) but nevertheless determine continuous and even compact Toeplitz operators on H ∞ V . We actually have the following result. ∞ Proposition 4.3. For any ϕ ∈ L 1 (D) such that the support of ϕ is a compact subset of D, the operator T ϕ : H ∞ V → H V is compact.
Proof. Let 0 < r < 1 be such that supp(ϕ ) is contained in the closed disc D (0, r ) (with center 0 and radius r). There exists a constant C > 0 such that |1 − zζ¯ | = |1 − ζ z¯ | C for all ζ ∈ D (0, r ) and z ∈ D. Hence, for every q we have w (ζ z¯ )q C q for such ζ and z, and we can estimate in (3.2)
D
ϕ (ζ )(1 − |z|2 )2 |1 − zζ¯ |4
ϕ (ζ )(1 − |z|2 )2
w (ζ z¯ )q d A (ζ ) = D (0,r )
|1 − zζ¯ |4
w (ζ z¯ )q d A (ζ )
ϕ (ζ ) d A (ζ ) 1 − |z|2
Cq
2
C q .
(4.6)
D (0,r )
The result follows from Theorem 3.2.
2
Obviously, all functions ϕ which satisfy the assumption of Proposition 4.3, and which in addition are essentially unbounded, violate the condition (4.1).
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Acknowledgments The research of Bonet was partially supported by FEDER and MEC Project MTM2007-62643 and by GV Project Prometeo/2008/101. The research of Taskinen was partially supported by a grant of the Väisälä Foundation of the Finnish Academy of Science and Letters.
References [1] S. Axler, D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (2) (1998) 387–400. [2] K.D. Bierstedt, A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions, Hommage à Pascal Laubin, Bull. Soc. Roy. Sci. Liège 70 (2001) 167–182. [3] K.D. Bierstedt, R. Meise, W.H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982) 107–160. [4] J. Bonet, Weighted spaces of holomorphic functions and operators between them, in: Seminar of Mathematical Analysis, Malaga/Seville, 2002/2003, in: Colecc. Abierta, vol. 64, Univ. Sevilla Secr. Publ., Seville, 2003, pp. 117–138. [5] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, second ed., Springer Monogr. Math., Springer-Verlag, Berlin, 2006. [6] M. Engliš, T.T. Hänninen, J. Taskinen, Minimal L ∞ -type spaces on strictly pseudoconvex domains on which the Bergman projection is continuous, Houston J. Math. 32 (2006) 253–275. [7] A. Harutyunyan, W. Lusky, Toeplitz operators on weighted spaces of holomorphic functions, Math. Scand. 103 (1) (2008) 40–52. [8] M. Jasiczak, On locally convex extension of H ∞ in the unit ball and continuity of the Bergman projection, Studia Math. 156 (2003) 261–275. [9] M. Jasiczak, Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains, Arch. Math. 87 (5) (2006) 436–448. p [10] A.N. Karapetyants, The space BMOλ (D), compact Toeplitz operators with BMO1λ (D) symbols on weighted Bergman spaces, and the Berezin transform, Izv. Vyssh. Uchebn. Zaved. Mat. 50 (8) (2006) 76–79 (in Russian); translation in: Russian Math. (Iz. VUZ) 50 (8) (2006) 71–74 (2007). [11] G. Köthe, Topological Vector Spaces I and II, Springer-Verlag, Berlin, 1969 and 1979. [12] D.H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (2) (1987) 345–368. [13] R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon, Oxford, 1997. [14] D. Suarez, The essential norm of operators in the Toeplitz algebra on A p (Bn ), Indiana Univ. Math. J. 56 (5) (2007) 2185–2232. [15] J. Taskinen, On the continuity of the Bergman and Szegö projections, Houston J. Math. 30 (2004) 171–190. [16] J. Taskinen, Atomic decomposition of a weighted inductive limit, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 97 (2003) 325–337. [17] J. Taskinen, J.A. Virtanen, Spectral theory of Toeplitz and Hankel operators on the Bergman space A 1 , New York J. Math. 14 (2008) 1–19. [18] N. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Oper. Theory Adv. Appl., vol. 185, Birkhäuser Verlag, 2008. [19] Z. Wu, R. Zhao, N. Zorboska, Toeplitz operators on Bloch-type spaces, Proc. Amer. Math. Soc. 134 (2006) 3531–3542. [20] K. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (2) (1988) 329–357. [21] K. Zhu, BMO and Hankel operators on Bergman spaces, Pacific J. Math. 155 (2) (1992) 377–395. [22] K. Zhu, Operator Theory in Function Spaces, second ed., Math. Surveys Monogr., vol. 138, American Mathematical Society, Providence, RI, 2007.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Toeplitz operators on the space of analytic functions with logarithmic growth José Bonet a,∗ , Jari Taskinen b a b
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universidad Politécnica de Valencia, E-46071 Valencia, Spain Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FIN-00014 Helsinki, Finland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 August 2008 Available online 11 December 2008 Submitted by T. Ransford Keywords: Toeplitz operators Berezin transform Weighted Banach spaces of analytic functions Weighted inductive limits
Continuous and compact Toeplitz operators for positive symbols are characterized on the space H ∞ V of analytic functions with logarithmic growth on the open unit disc of the complex plane. The characterizations are in terms of the behaviour of the Berezin transform of the symbol. The space H ∞ V was introduced and studied by Taskinen. The Bergman projection is continuous on this space in a natural way, which permits to define Toeplitz operators. Sufficient conditions for general symbols are also presented. © 2008 Elsevier Inc. All rights reserved.
1. Introduction The purpose of this paper is to study Toeplitz operators T ϕ in the space H ∞ V of analytic functions on the unit disc which is quite far from the Bergman Hilbert space A 2 , our results grow logarithmically at the boundary. Though the space H ∞ V will be astonishingly similar to that basic case. For example, in the case of positive symbols ϕ we obtain a necessary and sufficient condition for the continuity in terms of a growth condition of the Berezin transform ϕ˜ of ϕ , see Theorem 3.1. ˜ Moreover, the difference between the A 2 and H ∞ V -cases is most typical: in the former, ϕ must be bounded on D whereas in the latter a growth like a power of the logarithm of the boundary distance is allowed (see [22], Chapter 6, for the basic results for Toeplitz operators on A 2 ). In Theorem 3.2 we also characterize compact Toeplitz operators (with positive ϕ ) on H∞ V again in terms of a Berezin-type condition. It is remarkable how beautifully the abstract definition of continuous and compact operators in countable inductive limit of Banach spaces is transformed into concrete conditions on logarithmic weight families, see (3.1) and (3.2). For general not necessarily positive ϕ we are able to provide sufficient conditions for continuity and compactness, see Remark 3.4, Proposition 4.1 and Theorem 4.2. ∞ The space H ∞ V and the related space L V were introduced and studied by the second author in [15] as substitutes of the usual sup-normed spaces H ∞ and L ∞ , to avoid the well-known discontinuity problem of the Bergman projection P with respect to the sup-norm. These spaces are topological algebras and they have the properties that H ∞ V is a closed subspace ∞ ∞ ∞ of L ∞ V and that P is a continuous projection from L V onto H V . In the article [15] the space H V was introduced using weighted sup-seminorms and it was proved, as a consequence of abstract results due to Bierstedt, Meise and Summers [3], that it is in fact a countable inductive limit (in fact a union) of weighted Banach spaces of analytic functions on the disc with compact linking maps. This description is better adapted to the study of operators defined on the space, and is the one we outline in Section 2. The research of Taskinen [15] in this direction was continued by several authors [6,8,9,16].
*
Corresponding author. E-mail addresses: [email protected] (J. Bonet), jari.taskinen@helsinki.fi (J. Taskinen).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.12.009
© 2008 Elsevier Inc.
All rights reserved.
J. Bonet, J. Taskinen / J. Math. Anal. Appl. 353 (2009) 428–435
429
From the harmonic analysis point of view, the advantage of using the space H ∞ V is that very simple pointwise estimates can all the time be applied. The emerging logarithmic singularities are absorbed by the inductive limit. A reader who is not familiar with inductive limits may be satisfied by the observation that their presence only amounts to statements on weight functions like “for all powers of log” or “for some power of log” instead of the usual unique weight functions in the Banach space case. Properties of operators in inductive limits are explicitely formulated in terms of containment of sets of analytic functions satisfying certain weight estimates. Toeplitz operators on spaces of analytic functions have been investigated since long. We refer the reader e.g. to the books by Böttcher and Silbermann [5] and by Zhu [22]. For Toeplitz operators on Bergman spaces, in particular on A 2 of various domains, see [1,10,12,14,17,18,20,21]. As for other spaces, let us also mention the recent papers [7,19]. 2. Notation All function spaces are defined on the open unit disc D of the complex plane C, unless otherwise stated. By d A we denote the normalized two-dimensional Lebesgue measure on D, and by L p := L p (d A ) the space of p-integrable functions on the disc D with respect to the measure d A. Here 1 p ∞. Moreover, A p stands for the Bergman space, which is the closed subspace of L p consisting of analytic functions. The Bergman projection P is the integral operator
P f ( z) := D
f (ζ )
(1 − zζ¯ )2
d A (ζ ),
z ∈ D,
(2.1)
defined at least for all f ∈ L 1 (d A ). Given a function (symbol) ϕ ∈ L 1loc (d A ) we denote by M ϕ the pointwise multiplication with ϕ , and by T ϕ we denote the corresponding Toeplitz-operator
T ϕ f ( z) := P M ϕ f ( z) := D
ϕ (ζ ) f (ζ ) (1 − zζ¯ )2
The Berezin transform of a function
ϕ˜ (z) =
ϕ (ζ )(1 − |z|2 )2 |1 − zζ¯ |4
D
d A (ζ ).
(2.2)
ϕ ∈ L 1 is defined by
d A (ζ ).
(2.3)
See more details in [22]. ∞ We recall the definitions of the spaces H ∞ V and L V from [15]; notice that we use here slightly different notations. First, denote
w ( z) := 1 + log(1 − z).
(2.4)
This function is mainly used to define the radial weight w (| z|). Sometimes it is however necessary to define the logarithm as an analytic function on D: then the argument of 1 − z is understood to belong to the interval ]−π , π [ for all z ∈ D, and the definition becomes unambiguous. ∞ The space H ∞ V (respectively, L V ) consists of analytic (respectively, measurable) functions f : D → C such that for some n ∈ N and constant C n > 0
f (z) C n w | z| n for (almost) all z ∈ D.
(2.5)
The space H ∞ V is an
(LB)-space, i.e., countable inductive limits of Banach spaces. In fact, it is a complete (LB)-space. We refer the reader to [11,13] for locally convex inductive limits. A convenient way to describe the topology of H ∞ V is obtained with a projective description of the inductive limit topology. This can be done as a consequence of the work of Bierstedt, Meise and Summers [3]. More precisely, it allows us to define the topology by means of the following family of weighted sup-seminorms: f v := sup f (z) v (z), z∈D
v ∈ V,
(2.6)
where V consists of all continuous, positive, radial functions v : D → R such that for all n ∈ N, v ( z) C n w ( z)−n .
(2.7)
More details, results, examples and motivation about weighted inductive limits of Banach spaces of analytic functions can be seen in [2] and the references therein. We still denote B n :=
sup f (z) w | z| f ∈ H∞ V z∈D
n
;
(2.8)
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∞ in the same way (using “ess sup”) we define the subsets B nL of L ∞ V . The sets B n are bounded and even precompact in H V . ∞ ∞ L Every bounded subset of H V is contained in a multiple of some B n . The same holds for the sets B n in L V , except that bounded sets need not be precompact in this case. We collect in the following proposition the results of [15] that are needed later (see also [4]).
Proposition 2.1. (1) If v ∈ V , then the pointwise product w k v also belongs to V , for every k. ∞ (2) The mapping P is a continuous projection from L ∞ V onto H V . L (3) In addition, P ( B n ) ⊂ C n B n+1 for all n. A linear operator from a complete locally convex space into itself is called compact (respectively, bounded), if it maps a neighbourhood of zero into a precompact (respectively, bounded) set. Recall that in a complete space a subset is precompact if and only if it is relatively compact (see [11]). Remark 2.2. (1) A linear operator T between two (LB)-spaces is continuous, if and only if it maps bounded sets into bounded sets. In ∞ our case this means that T ϕ : H ∞ V → H V is continuous if and only if for every n ∈ N one can find C n > 0 and m ∈ N such that T ϕ ( B n ) ⊂ C n B m . ∞ (2) One can also show that T ϕ : H ∞ V → H V is compact, if and only if there exists m ∈ N such that for every n ∈ N one can find a constant C n with T ϕ ( B n ) ⊂ C n B m . The symbols C , c (respectively, C n ) denote strictly positive absolute (respectively, depending on n) constants, which may vary from place to place but not in the same sequence of inequalities. Definition 2.3. Denote for all L ∈ N and λ ∈ ]−2 L , 2 L ] ∩ Z =: I L
D L ,λ := z := re i θ 1 − 2 L −1 < r 1 − 2− L and and for each L and λ denote z L ,λ
π (λ − 1)2−L < θ π λ2−L ,
−L := (1 − 2−L )e i2π λ2
(2.9)
∈ D L ,λ .
Remark 2.4. For positive real valued functions F and G (defined for example on D or on some other domain) we denote F∼ = G, if there exist absolute, positive constants C > c > 0 such that cF G C F
(2.10)
on the whole domain of definition. The following simple estimate will be needed: 1 − | z| + |arg z| ∼ = |1 − z|,
(2.11)
where z ∈ D with arg z ∈ [−π , π ]. Namely, it is obviously enough to prove the claim on a domain where, say, |1 − z| 1/4. (For other z both sides of (2.11) are bounded from above and below by positive constants.) But for such z =: re i θ we have |1 − z | ∼ = |1 − r cos θ| + |sin θ| ∼ = |1 − r | + |θ|, which is the statement. 3. Characterization of continuity and compactness for positive symbols In the following main results of our paper we characterize the continuity and compactness of T ϕ with positive ϕ in terms of the growth properties of the Berezin transform of ϕ . For general, not necessarily positive ϕ , we show that these growth criterions are necessary conditions for continuity and compactness, respectively. ∞ Theorem 3.1. Assume ϕ 0. The Toeplitz operator T ϕ is continuous H ∞ V → H V if and only if there exist k ∈ N and C > 0 such that the Berezin transform ϕ˜ satisfies
k k ϕ˜ (z) C w |z| = C 1 + log 1 − |z| , z ∈ D.
(3.1)
∞ Theorem 3.2. Assume ϕ 0. The operator T ϕ : H ∞ V → H V is compact, if and only there exists k ∈ N such that for every q ∈ N there exists C q > 0 with
D
ϕ (ζ )(1 − |z|2 )2 |1 −
z ζ |4
k
w (ζ z¯ )q d A (ζ ) C q w | z| ,
z ∈ D.
(3.2)
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431
Proof of Theorems 3.1 and 3.2. I. Sufficiency. We prove the sufficiency of the condition (3.1) in Theorem 3.1 (we call this the case (A); of course, k ∈ N will be fixed all the time). During the course of the proof we describe the changes needed to prove the sufficiency of (3.2) in Theorem 3.2; this is called the case (B). We first claim that in the case (A)
ϕ d A C k Lk 2−2L
(3.3)
D L ,λ
for all L and λ. In the case (B) and the assumption (3.2) we get
ϕ d A C q L −q+k 2−2L for all q.
(3.4)
D L ,λ
In the case (A), fixing L and λ we use the positivity of the symbol, the definition of the Berezin transform and the assumption (3.1) to obtain
D L ,λ
ϕ (ζ )(1 − |z L ,λ |2 )2 |1 − z L ,λ ζ¯ |4
d A (ζ ) D
ϕ (ζ )(1 − |z L ,λ |2 )2 |1 − z L ,λ ζ¯ |4
k d A (ζ ) C log 1 − | z L ,λ | .
We have 1 − | z L ,λ | ∼ = 2−L and also |1 − z L ,λ ζ¯ | C 2−L for ζ ∈ D L ,λ . The inequality (3.3) follows already from this, since In the case (B) we in addition make the observation that
log(1 − ζ z¯ L ,λ ) ∼ = − log |1 − ζ z¯ L ,λ | + arg(1 − ζ z¯ L ,λ ) ∼ =L
(3.5)
ϕ 0. (3.6)
ζ z¯ L ,λ is of order 2−L and the argument of it is only between
−π and π . Otherfor ζ ∈ D L ,λ , since the modulus of 1 − wise (3.4) follows in the same way as in (3.5). In the case (A), for the continuity of T ϕ it is sufficient to show that for an arbitrary n ∈ N one can find a constant C n,k > 0 such that T ϕ ( B n ) ⊂ C n,k B n+k+1 (see Remark 2.2(1)). For the compactness, case (B), we show that for every n ∈ N there exists cn > 0 such that T ϕ ( B n ) ⊂ cn B 2+k (Remark 2.2(2)). To prove these facts, we let n ∈ N and f ∈ B n be arbitrary and evaluate the expression T ϕ f (z)
ϕ (ζ )| f (ζ )| |1 − zζ¯ |2
D
d A (ζ ),
(3.7)
z ∈ D. We may assume that z > 0 (by the rotation symmetry) and that z actually of the form z = z N ,ν =: z N for some N ∈ N and ν = 2 N (since a positive z anyway belongs to some set D N ,ν with ν = 2 N ; we have then |1 − z N ζ¯ | ∼ = |1 − zζ¯ | for ζ ∈ D). We get ∞
ϕ (ζ )| f (ζ )| T ϕ f (z N ) d A (ζ ). |1 − z N ζ¯ |2 L =1 λ∈ I L
(3.8)
D L ,λ
For ζ ∈ D L ,λ one has 1 − | z N ζ¯ | ∼ = 1 − (1 − 2−N )(1 − 2−L ) ∼ = 2−N + 2−L and, since z N is real, we obtain from (2.11) of Remark 2.4,
|1 − z N ζ¯ | ∼ = 2−N + 2−L + |λ|2−L .
(3.9)
Moreover, since f ∈ B n , we have
f (ζ ) C n Ln .
(3.10)
On account these estimates and (3.3), we find that in the case (A) that (3.8) can be bounded by a constant times
L ,λ
C n Ln
(2−N
+ 2−L
+ |λ|2−L )2
ϕ (ζ ) d A (ζ ) D L ,λ
L ,λ
(2−N
C n L n+k 2−2L
+ 2−L + |λ|2−L )2
.
(3.11)
In case (B) we get an estimate where C L n+k is replaced by C q L n−q+k . We are thus left with combinatorial estimates. In the case (A) (3.11) will be shown to have a bound C N n+k+1 . This proves the continuity of T ϕ , since this shows
T ϕ f (z N ) C N n+k+1 C log 1 − | z N | n+k+1 ,
(3.12)
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i.e., T ϕ maps B n into C n B n+k+1 (see the discussion about z and z N above). In the case (B) we aim to the bound
T ϕ f (z N ) C q n! N n−q+1+k
(3.13)
for every q n. Then we choose q = n − 1, and obtain
T ϕ f (z N ) C N n−q+1+k C log 1 − | z N | 2+k , and thus T ϕ maps B n into cn B 2+k . We need to estimate
λ∈ I L
2−2L
(2−N + 2−L + |λ|2−L )2
=
Comparing with the integral (with
2L
1
(1 + α +
1
x)2
dx
λ∈ I L
1
(1 + 2L −N + |λ|)2
(3.14)
.
α = 2L−N )
C
(3.15)
1+α
we find that the right-hand side of (3.14) is bounded by C /(1 + 2 L − N ). From this we find that (3.11) has in the case (A) the bound
C n Ln+k . 1 + 2L−N
(3.16)
L
In the case (B) the bound is the same with L n+k replaced by L n−q+k . The inequality
LN
L n+k 1 + 2L−N
C (n + k)! N n+k+1
(3.17)
can easily be derived e.g. by integrating by parts the expression
LN
L n+k 1 + 2L−N
∞ N
xn−k e −x dx. On the other hand,
L n+k N n+k+1 .
(3.18)
LN
This proves the case (A). Obviously, in the case (B) we also get the desired estimate N n−q+1+k . II. Necessity for Theorem 3.1. We assume T ϕ is continuous and find k ∈ N such that T ϕ maps B 1 into C B k (see the remark after (2.8) and Remark 2.2(1)). Considering for a moment the number z ∈ D as a parameter, we define the analytic function K z (ζ ) :=
1
(1 − ζ z¯ )2
,
ζ ∈ D.
(3.19)
Trivially the estimate
1 − | z |2 2 K z
∞
4
(3.20)
holds for every z. Hence T ϕ ((1 − | z|2 )2 K z ) ∈ 4C B k for every z, and
T ϕ 1 − | z|2 2 K z (ω) 4C w |ω| k for all
(3.21)
ω ∈ D. But taking ω = z, this means that
2 2 ϕ (ζ ) (1 − |z|2 )2 ϕ˜ (z) = ϕ (ζ )(1 − | z| ) d A (ζ ) = d A (ζ ) (1 − zζ¯ )2 (1 − ζ z¯ )2 |1 − zζ¯ |4 D
D
2 k = T ϕ 1 − | z|2 K z (z) C w | z| .
(3.22)
This is the condition (3.1). III. Necessity for Theorem 3.2. Our assumption is that there exists a k ∈ N such that for every q ∈ N, T ( B q ) ⊂ C q B k for some constant C q > 0. We define for this q and each z ∈ D the analytic function
q
1 − ζ z¯ Λq,z (ζ ) := log 10q , e
ζ ∈ D.
(3.23)
J. Bonet, J. Taskinen / J. Math. Anal. Appl. 353 (2009) 428–435
433
Combining (3.20) and (3.25) of Lemma 3.3 (below) we find that the function (1 − | z|2 )2 K z Λq,z belongs to the set C q B q for all z. Hence, it is mapped into the set C B k by T ϕ , i.e.
C w | z|
k
2 T ϕ 1 − | z|2 K z Λq,z (z) ϕ (ζ ) (1 − |z|2 )2 Λq,z (ζ ) d A (ζ ) = 2 2 (1 − ζ z¯ ) (1 − zζ¯ ) D
ϕ (ζ )(1 − | z|2 )2 Λq,z (ζ ) = d A (ζ ). |1 − ζ z¯ |4
(3.24)
D
Since all the factors of the integrand are positive, except the function Λq,z , we can conclude from (3.26) of Lemma 3.3 that (3.24) is bounded from below by a constant times
ϕ (ζ )(1 − | z|2 )2 w (ζ z¯ )q . d A (ζ ) |1 − ζ z¯ |4
2
D
To complete the proof above, we still have to show the following lemma. Lemma 3.3. For each q ∈ N there exists C q > 0 such that the function Λq,z of (3.23) satisfies
Λq,z (ζ ) C q w |ζ | q
(3.25)
and
(−1)q Re Λq,z (ζ )
1 Cq
w (ζ z¯ )q
(3.26)
for all z, ζ ∈ D. Proof. We may assume that | z| 1 − e −10q ; otherwise Λq,z is bounded from below and above by positive constants. To prove (3.25), we have
|I | := Im log e −10q (1 − ζ z¯ ) = arg e −10q (1 − ζ z¯ ) π ,
(3.27)
since ζ z¯ ∈ D. Hence, denoting
R := − Re log e −10q (1 − ζ z¯ ) = − log e −10q (1 − ζ z¯ ) > 0
(3.28)
we can estimate
Λq,z (ζ ) C q (1 + R)q = C q 1 − log e −10q (1 − ζ z¯ ) q q C q 1 + log 1 − |ζ | ,
(3.29)
since C |1 − ζ z¯ | 1 − |ζ |. This proves (3.25). Now we deal with (3.26). To estimate the real part of Λq,z we first observe that R 10q, since |e −10q (1 − ζ z¯ )| 3e −10q . By (3.27) we thus have R 2q|I |. But for the number R + i I one thus has (by elementary geometry in the complex plane, based on the fact that the modulus of the argument of R + i I is small, i.e., about 1/(4π q)) Re(R + i I )q
1 2
|R + i I |q
(3.30)
and, furthermore,
q = cq (1 + log e −10q (1 − ζ z¯ ) q C q 1 + log(1 − ζ z¯ ) = C q w (¯zζ )q .
Re(R + i I )q cq 1 + |R + i I |
This is our statement (3.26).
q
(3.31)
2
Remark 3.4. If ϕ is not positive, condition (3.1) is still necessary for the continuity. Sufficient conditions for continuity and compactness are obtained by replacing ϕ by |ϕ | in (3.1) and (3.2), respectively. These claims will not be given separate proofs. A careful inspection of our proof above will reveal their validity.
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4. Sufficient conditions for general symbols In the case of the reflexive Bergman spaces A p it is clear that a Toeplitz operator is bounded, if its symbol ϕ is bounded on D. This follows from the boundedness of the Bergman projection P : L p (d A ) → A p for 1 < p < ∞. In case of H ∞ V there is an analogous, quite straightforward sufficient condition based on a pointwise estimate of ϕ . It is however of some interest that many unbounded symbols satisfy this condition. Moreover, there is a similar condition for the compactness of T ϕ . The analogy of these pointwise conditions with those in Theorems 3.1 and 3.2 should be observed. Proposition 4.1. If there exists a k ∈ N such that
ϕ (z) C w | z| k for all z ∈ D,
(4.1)
∞ then T ϕ : H ∞ V → H V is continuous.
Proof. If v ∈ V , then the weight that
ω := w k v still belongs to V , by Proposition 2.1(1). The assumption (4.1) moreover implies
f ϕ v C f ω
(4.2)
∞ ∞ for all f ∈ H ∞ V , i.e., the multiplication operator M ϕ is continuous H V → L V . By Proposition 2.1(2), the Bergman projection ∞ , hence, T = P M is continuous on H ∞ . 2 is continuous L ∞ → H ϕ ϕ V V V
Theorem 4.2. Assume that for all q ∈ N there exist C q ∈ N such that
ϕ (z) C q w | z| −q for all z ∈ D. Then T ϕ :
H∞ V
→
(4.3)
H∞ V is compact.
∞ Proof. We first show that under the condition (4.3) the operator M ϕ : H ∞ V → L V is bounded, i.e. it maps a neighbourhood U of 0 into a bounded set. We first define
v ( z) := inf C q w | z| q∈N
−q
(4.4)
.
Clearly, v ∈ V ; see the definition of the weight family V in (2.7). Let U be a neighbourhood of 0 which is the closed unit ball of · v , i.e. U := { f ∈ H ∞ v | f v 1}. Every f ∈ U satisfies | f ( z)| 1/ v ( z), hence, using (4.3), we obtain
M ϕ f (z) = ϕ (z) f (z) ϕ (z)sup C −1 w | z| q 1.
(4.5)
q
q∈N
L ∞ ∞ Hence, M ϕ f ∈ B 1L ⊂ L ∞ V . Proposition 2.1(3), shows that P maps B 1 into B 2 ⊂ H V , which is a precompact subset of H V . Hence, T ϕ = P M ϕ is compact. 2
It is not so difficult to see that there are positive symbols which do not satisfy (4.1) but nevertheless determine continuous and even compact Toeplitz operators on H ∞ V . We actually have the following result. ∞ Proposition 4.3. For any ϕ ∈ L 1 (D) such that the support of ϕ is a compact subset of D, the operator T ϕ : H ∞ V → H V is compact.
Proof. Let 0 < r < 1 be such that supp(ϕ ) is contained in the closed disc D (0, r ) (with center 0 and radius r). There exists a constant C > 0 such that |1 − zζ¯ | = |1 − ζ z¯ | C for all ζ ∈ D (0, r ) and z ∈ D. Hence, for every q we have w (ζ z¯ )q C q for such ζ and z, and we can estimate in (3.2)
D
ϕ (ζ )(1 − |z|2 )2 |1 − zζ¯ |4
ϕ (ζ )(1 − |z|2 )2
w (ζ z¯ )q d A (ζ ) = D (0,r )
|1 − zζ¯ |4
w (ζ z¯ )q d A (ζ )
ϕ (ζ ) d A (ζ ) 1 − |z|2
Cq
2
C q .
(4.6)
D (0,r )
The result follows from Theorem 3.2.
2
Obviously, all functions ϕ which satisfy the assumption of Proposition 4.3, and which in addition are essentially unbounded, violate the condition (4.1).
J. Bonet, J. Taskinen / J. Math. Anal. Appl. 353 (2009) 428–435
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Acknowledgments The research of Bonet was partially supported by FEDER and MEC Project MTM2007-62643 and by GV Project Prometeo/2008/101. The research of Taskinen was partially supported by a grant of the Väisälä Foundation of the Finnish Academy of Science and Letters.
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