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