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Density of continuous functions in deBranges-Rovnyak spaces
Bartosz MalmanJoint work with Alexandru Aleman
Lund University
March 17, 2017
33rd Southeastern Analysis MeetingKnoxville, Tennessee
de Branges-Rovnyak space H(b)
For b ∈ H∞ = H∞(D) with ‖b‖∞ ≤ 1 define
H(b) = (1− TbTb)1/2H2
with norm‖f ‖b = inf
g∈H2,f=(1−TbTb)
1/2g
‖g‖2.
Reproducing kernel of H(b)
kb(z , λ) =1− b(λ)b(z)
1− λz.
Dichotomy extreme/non-extreme
If b is non-extreme, then P, set of polynomials, is contained inH(b).
If b is extreme, then P 6⊂ H(b).
Theorem 1 (Sarason, 1986)
If b is non-extreme, then P is dense in H(b).
Special case
If b inner function, then H(b) = H2 bH2 isometrically.
Theorem 2 (Aleksandrov, 1981)
Let b be an inner function. Then the intersection A ∩H(b) isdense in H(b), where A is the disc algebra.
Theorem
Natural question: does this extend to other b?
Answer is yes.
Theorem 3
Let A be the disc algebra. The intersection A ∩H(b) is dense inH(b) for all b in the unit ball of H∞.
Theorem
Natural question: does this extend to other b? Answer is yes.
Theorem 3
Let A be the disc algebra. The intersection A ∩H(b) is dense inH(b) for all b in the unit ball of H∞.
Few words about proof: representation of H(b)
Proposition 4
Let b be an extreme point of the unit ball of H∞ andE = {ζ ∈ T : |b(ζ)| < 1}. Then there exists an isometry J
H(b) 3 f 7→ Jf = (f , g) ∈ H2 ⊕ L2(E )
satisfying
J(H(b))⊥ ={
(bh,√
1− |b|2h) : h ∈ H2}.
Few words about proof: Duality argument
We study the annihilator of J(A ∩H(b)) ⊂ A⊕ L2(E ):
J(A ∩H(b))⊥ ⊂ C ⊕ L2(E ),
C = A′ is the space of Cauchy transforms of finite measures on T.
J(A ∩H(b)) = ∩h∈H2 ker φh,
where
φh =(hb, h
√1− |b|2
)∈ (A⊕ L2(E ))′ = C ⊕ L2(E ).
J(A ∩H(b))⊥ is the weak-star closure of {φh}h∈H2 (Hahn-Banachtheorem).
Few words about proof: Duality argument
We study the annihilator of J(A ∩H(b)) ⊂ A⊕ L2(E ):
J(A ∩H(b))⊥ ⊂ C ⊕ L2(E ),
C = A′ is the space of Cauchy transforms of finite measures on T.
J(A ∩H(b)) = ∩h∈H2 ker φh,
where
φh =(hb, h
√1− |b|2
)∈ (A⊕ L2(E ))′ = C ⊕ L2(E ).
J(A ∩H(b))⊥ is the weak-star closure of {φh}h∈H2 (Hahn-Banachtheorem).
Few words about proof: Duality argument
We study the annihilator of J(A ∩H(b)) ⊂ A⊕ L2(E ):
J(A ∩H(b))⊥ ⊂ C ⊕ L2(E ),
C = A′ is the space of Cauchy transforms of finite measures on T.
J(A ∩H(b)) = ∩h∈H2 ker φh,
where
φh =(hb, h
√1− |b|2
)∈ (A⊕ L2(E ))′ = C ⊕ L2(E ).
J(A ∩H(b))⊥ is the weak-star closure of {φh}h∈H2 (Hahn-Banachtheorem).
Duality argument: the set S
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
Lemma 5
The set S is weak-star closed in C ⊕ L2(E ). Consequently, theannihilator J(A ∩H(b))⊥ is contained in S .
Given Lemma 5, we can prove Theorem 3.
Proof of Theorem 3.
For Jf = (f , g) ∈ H2 ⊕ L2(E ) ⊂ C ⊕ L2(E ) we have
J(A ∩H(b)) ⊥ Jf ⇒ Jf = (f , g) ∈ S ,
i.e.,∃h ∈ H2 s.t. Jf = (bh,
√1− |b|2h) ∈ J(H(b))⊥ ⇒ Jf = 0.
Duality argument: the set S
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
Lemma 5
The set S is weak-star closed in C ⊕ L2(E ). Consequently, theannihilator J(A ∩H(b))⊥ is contained in S .
Given Lemma 5, we can prove Theorem 3.
Proof of Theorem 3.
For Jf = (f , g) ∈ H2 ⊕ L2(E ) ⊂ C ⊕ L2(E ) we have
J(A ∩H(b)) ⊥ Jf ⇒ Jf = (f , g) ∈ S ,
i.e.,∃h ∈ H2 s.t. Jf = (bh,
√1− |b|2h) ∈ J(H(b))⊥ ⇒ Jf = 0.
Duality argument: the set S
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
Lemma 5
The set S is weak-star closed in C ⊕ L2(E ). Consequently, theannihilator J(A ∩H(b))⊥ is contained in S .
Given Lemma 5, we can prove Theorem 3.
Proof of Theorem 3.
For Jf = (f , g) ∈ H2 ⊕ L2(E ) ⊂ C ⊕ L2(E ) we have
J(A ∩H(b)) ⊥ Jf ⇒ Jf = (f , g) ∈ S ,
i.e.,∃h ∈ H2 s.t. Jf = (bh,
√1− |b|2h) ∈ J(H(b))⊥ ⇒ Jf = 0.
Sketch of proof of Lemma 5
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
(Sketch of) proof of Lemma 5
Enough to check for converging sequences.
Let
S 3 (fn, hn)weak-star−−−−−→ (f , h).
Can assume hn → h pointwise a.e on E . Then limn fn = φ existsa.e on E . By a theorem of Khintchin and Ostrowski, f = φ on E .
Sketch of proof of Lemma 5
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
(Sketch of) proof of Lemma 5
Enough to check for converging sequences. Let
S 3 (fn, hn)weak-star−−−−−→ (f , h).
Can assume hn → h pointwise a.e on E .
Then limn fn = φ existsa.e on E . By a theorem of Khintchin and Ostrowski, f = φ on E .
Sketch of proof of Lemma 5
S ={
(f , h) ∈ C ⊕ L2(E ) :f
b∈ N+,
f
b=
h√1− |b|2
on E}.
(Sketch of) proof of Lemma 5
Enough to check for converging sequences. Let
S 3 (fn, hn)weak-star−−−−−→ (f , h).
Can assume hn → h pointwise a.e on E . Then limn fn = φ existsa.e on E . By a theorem of Khintchin and Ostrowski, f = φ on E .
Theorem of Khintchin and Ostrowski
Theorem 6 (Khintchin, Ostrowski [3])
Let fn be a sequence of analytic functions which satisfy
supr∈(0,1)
∫T
log+ |fn(re it)|dt ≤ C .
If the boundary values fn(ζ) converge on a set of positive measureE , then fn converge uniformly on compact subsets of D to aholomorphic function f , and
limn
fn(ζ) = f (ζ)
almost everywhere on E .
Sketch of proof of Lemma 5
(Sketch of) proof of Lemma 5.
Then, for almost every ζ ∈ E
f (ζ)
b(ζ)= lim
n
fn(ζ)
b(ζ)= lim
n
hn(ζ)√1− |b(ζ)|2
=h(ζ)√
1− |b(ζ)|2.
By a theorem of Vinogradov, C has the (contractive) F -property.If fn, f ∈ C and I is the inner factor of b, then
fnweak-star−−−−−→ f ⇒ fn/I
weak-star−−−−−→ f /I ∈ C ⊂ N+.
So f /b ∈ N+, and S is weak-star closed.
Sketch of proof of Lemma 5
(Sketch of) proof of Lemma 5.
Then, for almost every ζ ∈ E
f (ζ)
b(ζ)= lim
n
fn(ζ)
b(ζ)= lim
n
hn(ζ)√1− |b(ζ)|2
=h(ζ)√
1− |b(ζ)|2.
By a theorem of Vinogradov, C has the (contractive) F -property.If fn, f ∈ C and I is the inner factor of b, then
fnweak-star−−−−−→ f ⇒ fn/I
weak-star−−−−−→ f /I ∈ C ⊂ N+.
So f /b ∈ N+, and S is weak-star closed.
Sketch of proof of Lemma 5
(Sketch of) proof of Lemma 5.
Then, for almost every ζ ∈ E
f (ζ)
b(ζ)= lim
n
fn(ζ)
b(ζ)= lim
n
hn(ζ)√1− |b(ζ)|2
=h(ζ)√
1− |b(ζ)|2.
By a theorem of Vinogradov, C has the (contractive) F -property.If fn, f ∈ C and I is the inner factor of b, then
fnweak-star−−−−−→ f ⇒ fn/I
weak-star−−−−−→ f /I ∈ C ⊂ N+.
So f /b ∈ N+, and S is weak-star closed.
References
[1] C. Beneteau, A. A. Condori, C. Liaw, W. T. Ross, and A.A. Sola, Some open problems in complex and harmonicanalysis: Report on problem session held during theconference Completeness problems, Carleson measures, andspace of analytic functions, Contemporary Mathematics,Volume 679
[2] Donald Sarason, Sub-Hardy Hilbert Spaces in the UnitDisk, J. Wiley & Sons, 1994
[3] Victor Havin and Burglind Joricke, The uncertaintyprinciple in harmonic analysis, Springer-Verlag, Berlin, 1994