Constructive approximation in de Branges-Rovnyak spaces

0.16 MB·English

by  O. El-Fallah

CONSTRUCTIVE APPROXIMATION IN DE BRANGES–ROVNYAK SPACES
O. EL-FALLAH,E. FRICAIN,K.KELLAY,J. MASHREGHI,ANDT. RANSFORD
5 Abstract. In most classical holomorphic function spaces on the unit disk, a function f can be
1 approximated in thenorm of thespace by its dilates fr(z):=f(rz) (r<1). Weshow that this is
0 not thecaseforthedeBranges–RovnyakspacesH(b). Moreprecisely,wegiveanexampleofanon-
n 2 extIrtemisekpnoowinnt bthoaft,thifebuinsitabnaolnl-oefxHtre∞maenpdoiantfuonfctthioenufnit∈bHal(lbo)fsHuc∞h,ththaetnlipmorl→yn1o−mkifarlksHa(rbe)d=en∞se.
a in H(b). Wegive the first constructive proof of thisfact.
J

3
1

]
A
1. Introduction
F
h. The de Branges–Rovnyak spaces are a family of subspaces (b) of the Hardy space H2, para-
H
t metrized by elements b of the closed unit ball of H∞. We shall give the precise definition in 2. In
a §
m general (b) is not closed in H2, but it carries its own norm H(b) making it a Hilbert space.
H k·k
[ The spaces (b) were introduced by de Branges and Rovnyak in the appendix of [2] and further
H
studied in their book [3]. The initial motivation was to provide canonical model spaces for certain
1
types ofcontractions on Hilbertspaces. Subsequentlyitwas realized that thesespaces have several
v
0 interesting connections with other topics in complex analysis andoperator theory. For background
1 information we refer to the books of de Branges and Rovnyak [3], Sarason [6], and the forthcoming
9
monograph of Fricain and Mashreghi [5].
2
0 The general theory of (b)-spaces splits into two cases, according to whether b is an extreme
1. point or a non-extreme poHint of the unit ball of H∞. For example, if b is non-extreme, then (b)
0 contains all functions holomorphic in a neighborhood of the closed unit disk D, whereas ifHb is
5
extreme, then (b) contains very few such functions. In particular, (b) contains the polynomials
1 H H
if and only if b is non-extreme, and in this case, the polynomials are dense in (b). Proofs of all
:
v H
these facts can be found in Sarason’s book [6].
i
X Thedensityofpolynomialsisprovedin[6]byshowingthattheirorthogonalcomplementin (b)
H
r is zero. The proof is non-constructive in the sense that it gives no clue how to find polynomial
a
approximants to a given function. We know of no published work describing constructive methods
of polynomial approximation in (b), and it is surely of interest to have such methods available.
H
Perhaps the most natural approach is to try using dilations. Writing f (z) := f(rz), the idea
r
is to approximate f by f for some r < 1, and then f by the partial sums of its Taylor series.
r r
This idea works in many function spaces, but, as we shall see, it fails dismally in (b), at least for
H
certain choices of b. Indeed, it can happen that limr→1− fr H(b) = , even though f (b). We
k k ∞ ∈ H
shall prove this in 3, thereby answering a question posed in [1].
§
Date: 6 January 2015.
Research of OE supported byCNRST (URAC/03) and Acad´emie Hassan II des sciences et techniques.
Research of EF supported by CEMPI.
Research of KK supported by UMI-CRM.
Research of JM supported by NSERC.
Research of TR supported byNSERCand theCanada research chairs program.
1
2 O.EL-FALLAH,E.FRICAIN,K.KELLAY,J.MASHREGHI,ANDT.RANSFORD
This phenomenon has other negative consequences, among them the surprising fact that the
formula for f in terms of the Taylor coefficients of f, previously known to hold for f holo-
H(b)
k k
morphic on a neighborhood of D, actually breaks down for general f (b). It also shows that, in
∈ H
general, neither the Taylor partial sums of f, nor their Cesa`ro means need converge to f in (b).
H
So, to construct polynomial approximants to functions in (b), a different idea is needed. In 4
H §
weshalldescribeaschemethatachieves thisbasedonToeplitzoperators. Themethodpresupposes
that b is non-extreme (as it must), but one of its consequences is an approximation theorem for
Toeplitz operators that extends even to the case when b is extreme. We shall give a proof of this
extension in 5.
§
2. Background on (b)-spaces
H
Given ψ L∞(T), the corresponding Toeplitz operator T : H2 H2 is defined by
ψ
∈ →
T f := P (ψf) (f H2),
ψ +
∈
where P : L2(T) H2 denotes the othogonal projection of L2(T) onto H2. Clearly T is a
+ ψ
→
bounded operator on H2 with Tψ ψ L∞(T) (in fact, by a theorem of Brown and Halmos,
k k ≤ k k
Tψ = ψ L∞(T), but we do not need this). If h H∞, then Th is simply the operator of
k k k k ∈
multiplication by h and its adjoint is T . Consequently, if h,k H∞, then T T = T = T T , a
h ∈ h k hk k h
useful fact that we shall exploit frequently in what follows.
Definition 2.1. Let b H∞ with b H∞ 1. The associated de Branges–Rovnyak space (b) is
∈ k k ≤ H
the image of H2 under the operator (I T T )1/2. We define a norm on (b) making (I T T )1/2
− b b H − b b
a partial isometry from H2 onto (b), namely
H
(I T T )1/2f := f (f H2 ker(I T T )1/2)).
k − b b kH(b) k kH2 ∈ ⊖ − b b
This is the definition of (b) as given in [6]. The original definition of de Branges and Rovnyak,
H
based on the notion of complementary space, is different but equivalent. An explanation of the
equivalence can be found in [6, pp.7–8]. A third approach is to start from the positive kernel
1 b(w)b(z)
kb(z) := − (z,w D),
w 1 wz ∈
−
and to define (b) as the reproducing kernel Hilbert space associated with this kernel.
H
As mentioned in the introduction, the theory of (b)-spaces is pervaded by a fundamental
H
dichotomy, namely whether b is or is not an extreme point of the unit ball of H∞. This dichotomy
is illustrated by following result.
Theorem 2.2. Let b H∞ with b H∞ 1. The following statements are equivalent:
∈ k k ≤
(i) b is a non-extreme point of the unit ball of H∞;
(ii) log(1 b 2) L1(T);
(iii) (b) −con|t|ain∈s all functions holomorphic in a neighborhood of D;
H
(iv) (b) contains all polynomials.
H
Proof. The equivalence between (i) and (ii) is proved in [4, Theorem 7.9]. That (i) implies (iii) is
proved in [6, IV-6]. Clearly (iii) implies (iv). That (iv) implies (i) follows from [6, V-1]. (cid:3)
§ §
Henceforth we shall simply say that b is ‘extreme’ or ‘non-extreme’, it being understood that
this relative to the unit ball of H∞.
From the equivalence between (i) and (ii), it follows that, if b is non-extreme, then there is an
outer function a such that a(0) > 0 and a 2 + b 2 = 1 a.e. on T (see [6, IV-1]). The function
| | | | §
a is uniquely determined by b. We shall call (b,a) a pair. The following result gives a useful
characterization of (b) in this case.
H
CONSTRUCTIVE APPROXIMATION IN DE BRANGES–ROVNYAK SPACES 3
Theorem 2.3 ([6, IV-1]). Let b be non-extreme, let (b,a) be a pair and let f H2. Then
§ ∈
f (b) if and only if T f T (H2). In this case, there exists a unique function f+ H2 such
∈ H b ∈ a ∈
that T f = T f+, and
b a
(1) f 2 = f 2 + f+ 2 .
k kH(b) k kH2 k kH2
We end this section with an example that was studied in [7]. Let
τz
(2) b (z) := ,
0 1 τ2z
−
where τ := (√5 1)/2. The equivalence between (i) and (ii) in Theorem 2.2 shows that b is
0
−
non-extreme, and a calculation shows that the function a making (b ,a ) a pair is given by
0 0 0
τ(1 z)
a (z) = − .
0 1 τ2z
−
It was shown in [7] that b has the special property that f f for all f (b )
0 k rkH(b0) ≤ k kH(b0) ∈ H 0
and all r < 1. Using a standard argument of reproducing kernel Hilbert spaces, it is easy to see
that this implies that limr→1−kfr−fkH(b0) = 0. As we shall see in the next section, this property
is not shared by general b.
3. Dilation in (b)
H
Our principal goal in this section is to prove the following theorem.
Theorem 3.1. Let b := b B2, where b is the function given by (2), and B is the Blaschke product
0 0
with zeros at w := 1 4−n (n 1). Let f(z) := 2−n/(1 w z). Then b is non-extreme,
n − ≥ n≥1 − n
f (b), and we have P
∈H
(3) lim (f )+(0) = and lim f = .
r r H(b)
r→1−| | ∞ r→1−k k ∞
Notice that, by Theorem 2.3, if b is non-extreme and f (b), then
∈H
f f+ f+(0).
H(b) H2
k k ≥ k k ≥ | |
Thus the second conclusion in (3) is actually a consequence of the first. We shall therefore con-
centrate our attention on the first conclusion.
To simplify the notation in what follows, we shall write k (z) := 1/(1 wz), the Cauchy kernel.
w
It is the reproducing kernel for H2 in the sense that f(w) = f,k f−or all f H2 and w D.
w H2
h i ∈ ∈
In particular, k 2 = k ,k = k (w) = 1/(1 w 2). We remark that k has the useful
k wkH2 h w wiH2 w −| | w
property that T (k )= h(w)k for all h H∞. Indeed, given g H2, we have
h w w ∈ ∈
g,T (k ) = hg,k , = h(w)g(w) = h(w) g,k = g,h(w)k , .
h h w iH2 h w iH2 h wiH2 h w iH2
Theproof of Theorem 3.1 dependson two lemmas. Thefirstlemma provides a class of functions
f for which (f )+(0) is readily identifiable.
r
Lemma 3.2. Let b be non-extreme, let (b,a) be a pair and let φ:= b/a. Let
f := c k ,
n wn
X
n≥1
where (w ) are zeros of b and (c ) are complex numbers with c (1 w )−1/2 < .
n n≥1 n n≥1 n| n| −| n| ∞
Then f (b) and P
∈H
(4) (f )+(0) = c φ(rw ) (0 < r < 1).
r n n
X
n≥1
4 O.EL-FALLAH,E.FRICAIN,K.KELLAY,J.MASHREGHI,ANDT.RANSFORD
Proof. The series defining f clearly converges absolutely in H2. Also, since T k = b(w )k = 0
b wn n wn
for all n, we have T f = 0, and consequently f (b) by Theorem 2.3.
b ∈H
Now fix r (0,1) and consider
∈
g := c φ(rw )k .
n n wn
X
n≥1
As (φ(rw )) is a bounded sequence, this series also converges absolutely in H2, and a simple
n n≥1
calculation gives T (f )= T (g ). Thus f (b) and (f )+ = g . In particular (4) holds. (cid:3)
b r a r r ∈H r r
The second lemma is a technical result about Blaschke products.
Lemma 3.3. Let B be an infinite Blaschke product whose zeros (w ) lie in (0,1) and satisfy
n n≥1
1 w 1
n+1
(5) 0 < α − β < (n 1).
≤ 1 w ≤ 2 ≥
n
−
Then there exists a constant C > 0 such that
B(rw ) C (w r w , n 1).
n n n+1
| | ≥ ≤ ≤ ≥
Proof. We have
∞
B(rw ) = ρ(w ,rw ),
n k n
| |
Y
k=1
where ρ denotes the pseudo-hyperbolic metric on D, namely ρ(z,w) := z w /1 wz . The
| − | | − |
condition (5) implies that w < w for all n, and even that w < w2. Indeed, we have
n−1 n n−1 n
(6) 1 w2 2(1 w ) 2β(1 w )< 1 w .
− n ≤ − n ≤ − n−1 − n−1
It follows that, if r [w ,w ], then rw [w2,w w ] (w ,w ), and consequently
∈ n n+1 n ∈ n n n+1 ⊂ n−1 n
n−2 ∞
(7) B(rw ) ρ(w ,w ) ρ(w ,w2) ρ(w ,w w ) ρ(w ,w ) .
| n | ≥ (cid:16)Y k n−1 (cid:17)× n−1 n × n n n+1 ×(cid:16) Y k n (cid:17)
k=1 k=n+1
Thus the lemma will be proved if we can show that each of the four terms on the right-hand side
of (7) is bounded below by a positive constant independent of n.
By [4, Theorem 9.2], the condition (5) implies that the sequence (w ) is uniformly separated,
n
in other words, there exists a constant C′ > 0 such that
ρ(w ,w ) C′ (j 1).
k j
≥ ≥
Y
k6=j
Applying this with j = n 1 and j = n takes care of the first and fourth terms in (7).
−
For the second term in (7), note that (6) gives w2 w (1 2β)(1 w ), and clearly also
n− n−1 ≥ − − n−1
1 w2w 1 w2 2(1 w ), whence
− n n−1 ≤ − n−1 ≤ − n−1
w2 w 1 2β
ρ(w ,w2)= n− n−1 − .
n−1 n 1 w2w ≥ 2
− n n−1
Finally, for the third term in (7), we observe that w w w w (1 w ) and also
n n n+1 1 n+1
− ≥ −
1 w2w = (1 w )+(w w2)+(w2 w2w ) 3(1 w ), whence
− n n+1 − n n − n n− n n+1 ≤ − n
w w w w 1 w w
ρ(w ,w w )= n− n n+1 1 − n+1 1α. (cid:3)
n n n+1 1 w2w ≥ 3 1 w ≥ 3
− n n+1 − n
CONSTRUCTIVE APPROXIMATION IN DE BRANGES–ROVNYAK SPACES 5
Proof of Theorem 3.1. As remarked in 2, the function b is non-extreme and the function a
0 0
§
making (b ,a ) a pair satisfies φ := b /a = z/(1 z). As b and b have the same outer factors, it
0 0 0 0 0 0
−
follows that b is non-extreme and the function a making (b,a) a pair is just a . Hence φ:= b/a=
0
B2b /a = B2φ .
0 0 0
By Lemma 3.2, we have f (b). The lemma also gives that
∈ H
rw
(f )+(0) = 2−nφ(rw )= 2−nB(rw )2 n .
r n n
1 rw
X X n
n≥1 n≥1 −
As the terms in this series are non-negative, each one of them provides a lower bound for the sum.
Given r [w ,1), we choose n so that w r w . By Lemma 3.3 we have B(rw ) C > 0,
1 n n+1 n
∈ ≤ ≤ | | ≥
where C is a constant independent of r and n. Thus
rw w2
(f )+(0) 2−nC2 n 2−nC2 n 2n (1 r)−1/2.
r ≥ 1 rw ≥ 1 w2 ≍ ≍ −
− n − n
In particular (f )+(0) as r 1−, as claimed. Finally, as already remarked, this implies that
r
f as r 1→−.∞ → (cid:3)
r H(b)
k k → ∞ →
We now present some consequences of this result.
Corollary 3.4. Let b,f be as in Theorem 3.1. Then the Taylor partial sums s (f) of f and their
n
Ces`aro means σ (f) satisfy
n
limsup s (f) = and limsup σ (f) = .
n H(b) n H(b)
n→∞ k k ∞ n→∞ k k ∞
Proof. This follows immediately from Theorem 3.1 and the elementary identities
f = (1 r)rns (f) and f = (n+1)(1 r)2rnσ (f). (cid:3)
r n r n
− −
X X
n≥0 n≥0
Letbbenon-extreme, let(b,a)beapairandletφ:= b/a, say φ(z) = φ(j)zj. Itwas shown
j≥0
in [1, Theorem 4.1] that, if f is holomorphic in a neighborhood of D,Psay fb(z) = f(k)zk,
k≥0
P
then the series f(j +k)φ(j) converges absolutely for each k, and b
j≥0
P
b b 2
(8) f 2 = f(k)2 + f(j +k)φ(j) .
k kH(b) X| | X(cid:12)X (cid:12)
k≥0 b k≥0(cid:12)j≥0 b b (cid:12)
(cid:12) (cid:12)
It was left open whether the same formula holds for all f (b). Using Theorem 3.1, we can now
∈ H
show that it does not.
Corollary 3.5. Let b,f be as in Theorem 3.1. Then f(j)φ(j) diverges.
j≥0
P
Proof. For r (0,1), the dilated function f is holomorphbic inb a neighborhood of D, and the
r
∈
argument in [1] that establishes the formula (8) shows that (f )+(0) = rjf(j)φ(j). If b,f are
r j≥0
P
as in Theorem 3.1 then (fr)+(0) → ∞ as r → 1−, in other words, limr→1− j≥b0rjbf(j)φ(j) = ∞.
By Abel’s theorem, it follows that the series f(j)φ(j) diverges. P b b (cid:3)
j≥0
P
In Theorem 3.1, we chose b so as to have a sibmplebconcrete example. With a more astute
0
choice, we can prove more, obtaining examples where f grows ‘fast’. There is a limit on
r H(b)
k k
how fast it can grow: it was shown in [1, Theorem 5.2] that, if b is non-extreme and f (b),
then log+ f = o((1 r)−1) as r 1−. We now prove that this estimate is sharp. ∈ H
r H(b)
k k − →
Theorem 3.6. Let γ : (0,1) (1, ) be a function such that logγ(r) = o((1 r)−1). Then there
→ ∞ −
exist b non-extreme and f (b) such that f γ(r) for all r in some interval (r ,1).
r H(b) 0
∈ H k k ≥
6 O.EL-FALLAH,E.FRICAIN,K.KELLAY,J.MASHREGHI,ANDT.RANSFORD
Proof. Let φ beany function in the Smirnov class N+ that is positive and increasing on (0,1). To
1
say that φ N+ means we can write φ = b /a , where a ,b H∞ and a is outer. Multiplying
1 1 1 1 1 1 1
∈ ∈
a and b by an appropriately chosen outer function, we may furtherensure that a 2+ b 2 = 1 a.e.
1 1
onT andthata (0) > 0, inother words, that(b ,a )isapair. Repeatingtheproo|f|of T|h|eorem3.1
1 1 1
with b replaced by b (but with the same B), we obtain f (b) such that
0 1
∈ H
(f )+(0) C2(1 r)1/2φ (8r 7) (3/4 < r < 1).
r 1
≥ − −
Since logγ(r) = o((1 r)−1), it is possible to choose φ so that right-hand side exceeds γ(r) for
1
all r sufficiently close−to 1. For these r, we therefore have f (f )+(0) γ(r). (cid:3)
r H(b) r
k k ≥ ≥
4. Polynomial approximation in (b)
H
In this section we present a recipefor polynomial approximation in (b) when b is non-extreme.
H
It is based upon three lemmas.
Lemma 4.1. Let h H∞ and let p be a polynomial. Then T p is a polynomial.
∈ h
Proof. If k is strictly larger than the degree of p, then T p,zk = p,zkh = 0. (cid:3)
h h iH2 h iH2
Lemma 4.2. Let h H∞ with h H∞ 1. Then, for all g H∞, we have
∈ k k ≤ ∈
T g g 2 2(1 Reh(0)) g 2 .
k h − kH2 ≤ − k k∞
Proof. Expanding the left-hand side, we obtain
T g g 2 = T g 2 + g 2 2Re T g,g
k h − kH2 k h kH2 k kH2 − h h iH2
2 g 2 2Re g,hg
≤ k kH2 − h iH2
2π dθ
= 2 g(eiθ)2 1 Reh(eiθ)
Z | | − 2π
0 (cid:0) (cid:1)
2π dθ
2 g 2 1 Reh(eiθ)
≤ k kH∞ Z − 2π
0 (cid:0) (cid:1)
= 2 g 2 (1 Reh(0)). (cid:3)
k kH∞ −
Lemma 4.3. Let b be non-extreme and let (b,a) be a pair. If h aH∞, then T is a bounded
∈ h
operator from H2 into H(b) and kThkH2→H(b) ≤ kh/akH∞.
Proof. Let h= ah , where h H∞. For f H2, we have
0 0
∈ ∈
T T f = T T T f = T T T f
b h b a h0 a h0 b
so, by Theorem 2.3, T f (b) and (T f)+ = T T f. Consequently
h ∈ H h h0 b
T f 2 = T f 2 + T T f 2
k h kH(b) k h kH2 k h0 b kH2
= T T f 2 + T T f 2
k h0 a kH2 k h0 b kH2
h 2 ( T f 2 + T f 2 )
≤ k 0kH∞ k a kH2 k b kH2
≤ kh0k2H∞kfk2H2,
the last inequality coming from the fact that a 2 + b 2 = 1 a.e. on T, since (b,a) is a pair. (cid:3)
| | | |
We now put these results together to produce a new, constructive proof of the following result.
Theorem 4.4. If b is non-extreme, then polynomials are dense in (b).
H
CONSTRUCTIVE APPROXIMATION IN DE BRANGES–ROVNYAK SPACES 7
Proof. Letf (b)andletǫ > 0. Pick g ,g H∞ suchthat f g ǫand f+ g ǫ.
1 2 1 H2 2 H2
∈ H ∈ k − k ≤ k − k ≤
Then pick h ∈ aH∞ such that khkH∞ ≤ 1 and 2(1 − Reh(0)) ≤ ǫ2/(kg1k2H∞ + kg2k2H∞). (For
example, let h be the outer function satisfying h(0) > 0 and h = min 1,n a , where n is chosen
| | { | |}
sufficiently large.) Finally, pick a polynomial p such that f p H2 ǫ/ h/a H∞. Then, by
k − k ≤ k k
Lemma 4.1 T p is a polynomial, and we shall show that f T p 6ǫ.
h k − h kH(b) ≤
First of all, using Lemma 4.2 we have
f T f f g + g T g + T g T f
k − h kH2 ≤ k − 1kH2 k 1 − h 1kH2 k h 1− h kH2
f g1 H2 +√2(1 Reh(0))1/2 g1 H∞ + g1 f H2
≤ k − k − k k k − k
3ǫ.
≤
Likewise,
f+ T f+ f+ g + g T g + T g T f+
k − h kH2 ≤ k − 2kH2 k 2 − h 2kH2 k h 2− h kH2
f+ g2 H2 +√2(1 Reh(0))1/2 g2 H∞ + g2 f+ H2
≤ k − k − k k k − k
3ǫ.
≤
Now T (T f)= T (T f)=T (T (f+)) = T (T (f+)), so by Theorem 2.3 we have T f (b) and
b h h b h a a h h ∈ H
(T f)+ = T (f+), and
h h
f T f 2 = f T f 2 + f+ T f+ 2 9ǫ2+9ǫ2 (5ǫ)2.
k − h kH(b) k − h kH2 k − h kH2 ≤ ≤
Finally, using Lemma 4.3, we have
f T p f T f + T f T p
k − h kH(b) ≤ k − h kH(b) k h − h kH(b)
≤ kf −ThfkH(b)+kh/akH∞kf −pkH2
5ǫ+ǫ = 6ǫ,
≤
as claimed. (cid:3)
5. Toeplitz approximation in (b)
H
Recapitulating the proof of Theorem 4.4, given f (b), we can approximate it in (b) by
∈ H H
a polynomial of the form T p, where h H∞ and p is a polynomial. Indeed, by the triangle
h ∈
inequality,
f T p f T f + T f T p .
k − h kH(b) ≤ k − h kH(b) k h − h kH(b)
The first term on the right-hand can be made small by choosing h with h H∞ 1 and h(0)
k k ≤
sufficiently close to 1. If, in addition, h aH∞, then the second term can be made small by
∈
choosing p sufficiently close to f in H2, for example a Taylor partial sum of f.
This construction presupposes that b is non-extreme. Indeed it must, since otherwise (b) may
H
well contain no non-zero polynomials. However, the fact that f T f can be made small
k − h kH(b)
remainstrueeveninthecasewhenbisextreme. Weisolate theideainthefollowingapproximation
theorem, valid for all b, extreme and non-extreme.
Theorem 5.1. Let (hn)n≥1 be a sequence in H∞ such that hn H∞ 1 and limn→∞hn(0) = 1.
k k ≤
Then, given b in the unit ball of H∞ and f (b), we have T f (b) for all n and
∈ H hn ∈ H
lim T f f = 0.
n→∞k hn − kH(b)
Theproof of this result requires a little more background on de Branges–Rovnyak spaces, which
we now briefly summarize.
8 O.EL-FALLAH,E.FRICAIN,K.KELLAY,J.MASHREGHI,ANDT.RANSFORD
Let b H∞ with b H∞ 1. We define the space (b) in the same way as (b), but with the
∈ k k ≤ H H
roles of b and b interchanged. Thus (b) is the image of H2 under the operator (I T T )1/2, with
H − b b
norm defined by
(I T T )1/2f := f (f H2 ker(I T T )1/2)).
k − b b kH(b) k kH2 ∈ ⊖ − b b
The spaces (b) and (b) are related through the following theorem.
H H
Theorem 5.2 ([6, II-4]). Let b be an element of the unit ball of H∞ and let f H2. Then
§ ∈
f (b) if and only T f (b), and in this case
∈H b ∈H
f 2 = f 2 + T f 2 .
k kH(b) k kH2 k b kH(b)
The advantage of (b) over (b), for our purposes at least, is that it has another description
H H
making it a little more amenable.
Theorem 5.3 ([6, III-2]). Let b H∞ with b H∞ 1. Let ρ := 1 b 2 on T, let H2(ρ) be
the closure of the pol§ynomials in L2∈(T, ρdθ/2π)kaknd let≤J : H2 H2(ρ)−b|e|the natural inclusion.
ρ
→
Then J∗ is an isometry of H2(ρ) onto (b).
ρ H
Using this result, we can prove a version of Theorem 5.1 for (b).
H
Theorem 5.4. Let (hn)n≥1 be a sequence in H∞ such that hn H∞ 1 and limn→∞hn(0) = 1.
k k ≤
Then, given b in the unit ball of H∞ and f (b), we have T f (b) for all n and
∈ H hn ∈ H
lim T f f = 0.
n→∞k hn − kH(b)
Proof. Let ρ:= 1 b 2 on T and defineH2(ρ) and J as in the precedingtheorem. Given h H∞,
ρ
−| | ∈
let M : H2(ρ) H2(ρ) be the operator of multiplication by h, namely M g := hg (g H2(ρ)).
h h
→ ∈
Note that M J = J T , so, taking adjoints, we have
h ρ ρ h
(9) J∗M∗ = T J∗.
ρ h h ρ
Now let f (b). By Theorem 5.3, there exists g H2(ρ) such that f = J∗g. Using (9),
∈ H ∈ ρ
we have T f = T J∗g = J∗M∗ g. As J∗ is an isometry of H2(ρ) onto (b), it follows that
hn hn ρ ρ hn ρ H
T f (b) and that
hn ∈ H
T f f = M∗ g g .
k hn − kH(b) k hn − kH2(ρ)
It therefore remains to show that lim M∗ g g = 0 for all g H2(ρ). It suffices
n→∞k hn − kH2(ρ) ∈
to prove this when g H∞, because H∞ is dense in H2(ρ) and the operators M∗ are uniformly
∈ hn
bounded in norm (by 1). For g H∞, the same proof as that of Lemma 4.2 gives that
∈
M∗ g g 2 2(1 Reh (0)) g 2 ,
k hn − kH2(ρ) ≤ − n k kH∞
and as the right-hand side tends to zero, the proof is complete. (cid:3)
Finally, we deduce the corresponding result for (b).
H
Proof of Theorem 5.1. Let f (b). By Theorem 5.2 we have T f (b). For each n, we have
∈ H b ∈ H
T T f = T T f (b), and hence T f (b). Also, by Theorem 5.2 again,
b hn hn b ∈ H hn ∈ H
T f f 2 = T f f 2 + T T f T f 2 .
k hn − kH(b) k hn − kH2 k hn b − b kH(b)
The second term on the right-hand side tends to zero by Theorem 5.4. The first term tends to
zero by Lemma 4.2 and the density of H∞ in H2. (cid:3)
CONSTRUCTIVE APPROXIMATION IN DE BRANGES–ROVNYAK SPACES 9
Acknowledgement
Part of this research was carried out during a Research-in-Teams meeting at the Banff Interna-
tional Research Station (BIRS). We thank BIRS for its hospitality.
References
[1] N. Chevrot, D. Guillot, T. Ransford, De Branges–Rovnyak spaces and Dirichlet spaces, J. Funct. Anal. 259
(2010), 2366–2383.
[2] L. de Branges, J. Rovnyak, Canonical models in quantum scattering theory, in Perturbation Theory and its
Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst.,
Univ. of Wisconsin, Madison, Wis., 1965), pp.295–392, Wiley, New York,1966.
[3] L. de Branges, J. Rovnyak,Square Summable Power Series, Holt, Rinehart and Winston, New York,1966.
[4] P. Duren,Theory of Hp Spaces, Dover,Mineola NY,2000.
[5] E. Fricain, J. Mashreghi, Theory of H(b) Spaces, New Mathematical Monographs vols 20 and 21, Cambridge
University Press, Cambridge, to appear.
[6] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, John Wiley & SonsInc., New York,1994.
[7] D.Sarason,LocalDirichletspacesasdeBranges-Rovnyakspaces,Proc.Amer.Math.Soc.125(1997),2133–2139.
Laboratoire Analyse et Applications (URAC/03), Universit´e Mohamed V, B.P. 1014 Rabat, Mo-
rocco
E-mail address: [email protected]
Laboratoire Paul Painlev´e, UFR des Math´ematiques, Universit´e des Sciences et Technologies
Lille 1, 59655 Villeneuve d’Ascq Cedex, France.
E-mail address: [email protected]
Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux, 351 cours de la Lib´eration,
33405 Talence Cedex, France
E-mail address: [email protected]
D´epartement de math´ematiques et de statistique, Universit´e Laval, 1045 avenue de la M´edecine,
Qu´ebec (QC), G1V 0A6, Canada
E-mail address: [email protected]
D´epartement de math´ematiques et de statistique, Universit´e Laval, 1045 avenue de la M´edecine,
Qu´ebec (QC), G1V 0A6, Canada
E-mail address: [email protected]