Two classes of Blaschke products and their applications to operator - - PowerPoint PPT Presentation

Two classes of Blaschke products and their applications to operator theory

Pamela Gorkin

Bucknell University joint work with

• J. Akeroyd, University of Arkansas

Kent State, 2014

Blaschke products and inner functions

A Blaschke product is a function of the form B(z) = eiθzn

j

|aj| aj aj − z 1 − ajz . Blaschke products are analytic on the open unit disk, D, map D to itself, and have radial limits of modulus one almost everywhere on the unit circle: |B∗| = 1 a.e. A function I analytic on the disk, mapping D to D, with |I ∗| = 1 a.e. is an inner function. Inner functions with no zeros in D are singular inner functions, given by a singular measure.

Why study Blaschke products

Every bounded analytic function f = IG where I is inner and G has no zeros on D. Every inner function I is a product of such functions: I = BS where B is Blaschke and S is singular inner. The Blaschke products are uniformly dense in the set of all inner functions. There are universal Blaschke products.

Universal Blaschke products

Ln(z) = (z + zn)/(1 + znz) (zn) in D is universally admissible for the set of Blaschke products if there is a Blaschke product B such that {B((z + zn)/(1 + znz)) : n ∈ N} is locally uniformly dense in the unit ball of the space of bounded analytic functions. The corresponding Blaschke product is a universal Blaschke product. The existence of universal Blaschke products is due to Maurice Heins (1955) and every sequence that tends to the boundary is universally admissible (G., Mortini, 2004).

Today’s questions

Blaschke products and composition

Definition An inner function I is prime (indecomposable) if whenever I = U ◦ V with U, V inner, either U or V is an automorphism of D. Question 1: When is an inner function prime? Definition An inner function I is semi-prime if whenever I = U ◦ V with U and V inner, either U or V is a finite Blaschke product. Question 2: When is I semi-prime? Examples? Non-examples?

Some simple examples

Let B(z) =

n

• j=1

z − aj 1 − ajz have prime degree; i.e. n is a prime. B = C ◦ D, degree of C = m, degree of D = k

Some simple examples

Let B(z) =

n

• j=1

z − aj 1 − ajz have prime degree; i.e. n is a prime. B = C ◦ D, degree of C = m, degree of D = k = ⇒ n = mk. So any such B is prime. Of course, any such B is also finite.

Breaking up is hard to do!

Breaking up is hard to do!

Are there “more interesting” examples?

Critical Points (B degree n, B(0) = 0)

Critical point: B′(z) = 0; critical value w = B(z), B′(z) = 0. Theorem (Heins, 1942; Zakeri, BLMS 1998) Let z1, . . . , zd ∈ D. There exists a unique Blaschke B, degree d + 1, B(0) = 0, B(1) = 1, and B′(zj) = 0, all j. Corollary (Nehari, 1947; Zakeri) Blaschke pdts. B1, B2 have the same critical pts. iff B1 = ϕa ◦ B2 for some automorphism ϕa.

• Remark. B with distinct zeros has 2n − 2 critical points,

Critical Points (B degree n, B(0) = 0)

Critical point: B′(z) = 0; critical value w = B(z), B′(z) = 0. Theorem (Heins, 1942; Zakeri, BLMS 1998) Let z1, . . . , zd ∈ D. There exists a unique Blaschke B, degree d + 1, B(0) = 0, B(1) = 1, and B′(zj) = 0, all j. Corollary (Nehari, 1947; Zakeri) Blaschke pdts. B1, B2 have the same critical pts. iff B1 = ϕa ◦ B2 for some automorphism ϕa.

• Remark. B with distinct zeros has 2n − 2 critical points, only

n − 1 are in D: {z1, . . . , zn−1, 1/z1, . . . , 1/zn−1}: B has ≤ n − 1 critical values in D.

Counting critical values

Degree of D = k, degree of C = m. B = C ◦ D = ⇒ B′(z) = C ′(D(z))D′(z); D has k − 1 critical points, D partitions the others into m − 1 sets. Theorem B = C ◦ D iff there exists a subproduct D of B sharing k − 1 critical pts. with B that partitions the others into m − 1 sets. B can have at most (k − 1)

Counting critical values

Degree of D = k, degree of C = m. B = C ◦ D = ⇒ B′(z) = C ′(D(z))D′(z); D has k − 1 critical points, D partitions the others into m − 1 sets. Theorem B = C ◦ D iff there exists a subproduct D of B sharing k − 1 critical pts. with B that partitions the others into m − 1 sets. B can have at most (k − 1) + (m − 1) critical values.

Which one is a composition?

Figure : Blaschke products of degree 16

Which one is a composition?

Figure : Blaschke products of degree 16

First: Finite Blaschke products

• 1922-3, J. Ritt reduced to result about groups (Trans. AMS)
• 1974: Carl Cowen gave result for rational functions. (ArXiv)
• The group: Associated with the set of covering transformations
• f the Riemann surface of the inverse of the Blaschke product;

Compositions correspond to (proper) normal subgroups.

First: Finite Blaschke products

• 1922-3, J. Ritt reduced to result about groups (Trans. AMS)
• 1974: Carl Cowen gave result for rational functions. (ArXiv)
• The group: Associated with the set of covering transformations
• f the Riemann surface of the inverse of the Blaschke product;

Compositions correspond to (proper) normal subgroups.

• 2000, JLMS Beardon, Ng simplified Ritt’s work,
• 2011 Tsang and Ng, Extended to finite mappings between

Riemann surfaces

• 2013, Daepp, G., Schaffer, Sokolowsky, Voss: applets to check

when finite BP’s are prime. What else the applet demonstrates: Prime finite BP’s are dense in the set of all finite BP’s.

What about infinite Blaschke products?

What about infinite Blaschke products? Let’s look at famous classes of Blaschke products

Introducing

Indestructible Blaschke products

Blaschke Product: B(z) = eiθ

j |aj| aj aj−z 1−ajz .

ϕa(z) = a−z

1−az denotes a conformal automorphism.

Theorem (Frostman’s Theorem) Let I be an inner function. Then for all a ∈ D, except possibly a set of capacity zero, ϕa ◦ I is a Blaschke product. Definition A Blaschke product is indestructible if ϕa ◦ B is a Blaschke product for all a ∈ D. Which BP’s are indestructible?

Figure : The atomic singular inner function S(z) = exp

• − 1+z

1−z

• Constant Modulus on circles

Discontinuity at 1 Radial limit 0 at 1

S and indestructibility

Figure : ϕa ◦ S(−iz), for a = .727

Indestructibility: ϕa ◦ I Blaschke for all a

I is indestructible if ϕa ◦ I Blaschke for all a ∈ D. S(z) = exp

• − 1+z

1−z

• ; ϕα ◦ S is a BP for all α = 0, so we have

examples of destructible Blaschke products. Clever name due to Renate McLaughlin (1972): gave necessary and sufficient conditions for indestructibility. Morse (1980): Example of a destructible Blaschke product that becomes indestructible when you delete a single zero.

S and indecomposability

I is prime if whenever I = U ◦ V either U or V is an automorphism. Sensitive Question. S(z) = exp

• − 1+z

1−z

• is decomposable.

S and indecomposability

I is prime if whenever I = U ◦ V either U or V is an automorphism. Sensitive Question. S(z) = exp

• − 1+z

1−z

• is decomposable.

You can take roots of singular inner functions. So zn ◦ S1/n = S.

S and indecomposability

I is prime if whenever I = U ◦ V either U or V is an automorphism. Sensitive Question. S(z) = exp

• − 1+z

1−z

• is decomposable.

You can take roots of singular inner functions. So zn ◦ S1/n = S. But zS(z) is prime! (K. Stephenson, 1982)

S and indecomposability

I is prime if whenever I = U ◦ V either U or V is an automorphism. Sensitive Question. S(z) = exp

• − 1+z

1−z

• is decomposable.

You can take roots of singular inner functions. So zn ◦ S1/n = S. But zS(z) is prime! (K. Stephenson, 1982) Why?

Something you should do with an automorphism ϕa first

To show I prime: ϕI(0) ◦ I(0) = 0. I = U ◦V with U or V non-automorphisms (non-prime) iff ϕI ◦I is. So assume I(0) = 0.

Something you should do with an automorphism ϕa first

To show I prime: ϕI(0) ◦ I(0) = 0. I = U ◦V with U or V non-automorphisms (non-prime) iff ϕI ◦I is. So assume I(0) = 0. I(0) = (U ◦ ϕV (0)) ◦ (ϕV (0) ◦ V ), so assume V (0) = 0. 0 = I(0) = U(V (0)) = U(0), so U(z) = z(U1(z)). And so...

Something you should do with an automorphism ϕa first

To show I prime: ϕI(0) ◦ I(0) = 0. I = U ◦V with U or V non-automorphisms (non-prime) iff ϕI ◦I is. So assume I(0) = 0. I(0) = (U ◦ ϕV (0)) ◦ (ϕV (0) ◦ V ), so assume V (0) = 0. 0 = I(0) = U(V (0)) = U(0), so U(z) = z(U1(z)). And so...I(z) = V (z) (U1(V (z)) and V divides I.

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z).

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))).

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0.

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0. Not much divides S.

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0. Not much divides S. So V = Sα (α ≥ 0) & S1−α(z) = U(zV (z)).

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0. Not much divides S. So V = Sα (α ≥ 0) & S1−α(z) = U(zV (z)). α = 0 = ⇒ V (r) = Sα(r) → 0 as r → 1 and

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0. Not much divides S. So V = Sα (α ≥ 0) & S1−α(z) = U(zV (z)). α = 0 = ⇒ V (r) = Sα(r) → 0 as r → 1 and α = 1 = ⇒ 0 = limr→1 S1−α(r) = limr→1 U(rV (r)) = U(0).

Proof that zS(z) is prime, S atomic singular inner function

zS(z) = (U0 ◦ V0)(z) already vanishes at zero. WOLOG U0(z) = zU(z) and V0(z) = zV (z). zS(z) = (zU(z)) ◦ (zV (z)) = ⇒ zS(z) = zV (z)(U(zV (z))). S(z) = V (z)(U(zV (z))) and S(0) = V (0) · U(0). So, U(0) = 0. Not much divides S. So V = Sα (α ≥ 0) & S1−α(z) = U(zV (z)). α = 0 = ⇒ V (r) = Sα(r) → 0 as r → 1 and α = 1 = ⇒ 0 = limr→1 S1−α(r) = limr→1 U(rV (r)) = U(0). So, α = 0, zV (z) = z or α = 1 and U(zS(z)) = 1, so zU(z) = z.

Other singular inner functions

Theorem (Stephenson, 1983) Let Sµ be a singular inner function. There exists α > 0 and an inner function I such that Sµ = Sα ◦ I, where S is the atomic singular inner function.

Other singular inner functions

Theorem (Stephenson, 1983) Let Sµ be a singular inner function. There exists α > 0 and an inner function I such that Sµ = Sα ◦ I, where S is the atomic singular inner function. Consequence: Theorem (G., Laroco, Mortini, Rupp, 1994.) A singular inner function Sµ is semi-prime iff Sµ has only finitely many singularities.

How about Blaschke products?

How about Blaschke products?

A Blaschke product is: Definition interpolating if ∃δ > 0 such that inf

k

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

Definition Thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.

Definition Uniform Frostman Blaschke Product if sup

λ∈T ∞

• j=1

1 − |zj|2 |1 − zjλ| < ∞.

Interpolating Blaschke products

Definition A BP with zeros (zn) is interpolating if ∃δ > 0 such that inf

k

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

(Zero sequence is interpolating for H∞: For all bounded (wn), there exists f ∈ H∞ with f (zn) = wn) for all n.) S the atomic singular inner function. ϕa ◦ S is a Blaschke product for all a = 0.

Interpolating Blaschke products

Definition A BP with zeros (zn) is interpolating if ∃δ > 0 such that inf

k

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

(Zero sequence is interpolating for H∞: For all bounded (wn), there exists f ∈ H∞ with f (zn) = wn) for all n.) S the atomic singular inner function. ϕa ◦ S is an interpolating Blaschke product for all a = 0. So interpolating Blaschke products can be destructible and non-prime (decomposable).

Thin Blaschke products

Definition A Blaschke sequence (zn) is thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.
• 1. Every thin BP is a finite product of interpolating BPs.

Thin Blaschke products

Definition A Blaschke sequence (zn) is thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.
• 1. Every thin BP is a finite product of interpolating BPs.
• 2. The zero sequence of a thin interpolating BP is interpolating

for VMO ∩ H∞ = QA = H∞ ∩ H∞ + C. (Wolff, 1982)

Thin Blaschke products

Definition A Blaschke sequence (zn) is thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.
• 1. Every thin BP is a finite product of interpolating BPs.
• 2. The zero sequence of a thin interpolating BP is interpolating

for VMO ∩ H∞ = QA = H∞ ∩ H∞ + C. (Wolff, 1982)

• 3. Every thin BP is very indestructible; i.e. if ϕα is a conformal

automorphism of D, then ϕα is thin for every ϕα ∈ D.

Thin Blaschke Products, continued

• 4. Can rotate the zeros of every BP to get a thin BP.

(Chalendar, Fricain, Timotin 2003)

• 5. Every thin product is semi-prime; i.e. if B = C ◦ D then C or

D is a finite BP. Why?

Thin: lim

n→∞(1 − |zn|2)|B′(zn)| = lim n→∞

• j=n
• zj − zn

1 − zjzn

• = 1.

To show semi-prime: if B = U ◦ V then U or V is a finite BP. Fact: If B is thin with zeros (zn), then for Ln(z) = (z + zn)/(1 + znz) we have B ◦ Ln(z) → z.

Thin: lim

n→∞(1 − |zn|2)|B′(zn)| = lim n→∞

• j=n
• zj − zn

1 − zjzn

• = 1.

To show semi-prime: if B = U ◦ V then U or V is a finite BP. Fact: If B is thin with zeros (zn), then for Ln(z) = (z + zn)/(1 + znz) we have B ◦ Ln(z) → z. If B = U ◦ V , then ϕa ◦ B thin for all a = ⇒ we can assume B(0) = U(0) = V (0) = 0 and B and V are thin. Suppose V has infinitely many zeros.

Thin: lim

n→∞(1 − |zn|2)|B′(zn)| = lim n→∞

• j=n
• zj − zn

1 − zjzn

• = 1.

To show semi-prime: if B = U ◦ V then U or V is a finite BP. Fact: If B is thin with zeros (zn), then for Ln(z) = (z + zn)/(1 + znz) we have B ◦ Ln(z) → z. If B = U ◦ V , then ϕa ◦ B thin for all a = ⇒ we can assume B(0) = U(0) = V (0) = 0 and B and V are thin. Suppose V has infinitely many zeros. So B ◦ Ln(z) → z and (B ◦ Ln)(z) = U(V ◦ Ln)(z) → U(z).

Thin: lim

n→∞(1 − |zn|2)|B′(zn)| = lim n→∞

• j=n
• zj − zn

1 − zjzn

• = 1.

To show semi-prime: if B = U ◦ V then U or V is a finite BP. Fact: If B is thin with zeros (zn), then for Ln(z) = (z + zn)/(1 + znz) we have B ◦ Ln(z) → z. If B = U ◦ V , then ϕa ◦ B thin for all a = ⇒ we can assume B(0) = U(0) = V (0) = 0 and B and V are thin. Suppose V has infinitely many zeros. So B ◦ Ln(z) → z and (B ◦ Ln)(z) = U(V ◦ Ln)(z) → U(z). So U(z) = z or V is finite. (!) These don’t have to be prime, because V really can have degree > 1.

Summary of Results on Prime BP up to 2013

Semi-prime: I = U ◦ V = ⇒ U or V finite. Prime: I = U ◦ V = ⇒ U or V a disk automorphism. Type of function Semi-prime Prime Singular iff finitely-many never singularities Omits > 1 point of D never never Omits exactly 1 point of D iff finitely-many never singularities surjective inner thin Blaschke ϕaB, B thin zSµ a / ∈ E, E a countable zSµ

Three classes of Blaschke products. A BP is:

Definition interpolating if ∃δ > 0 such that

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

Three classes of Blaschke products. A BP is:

Definition interpolating if ∃δ > 0 such that

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

destructible or indestructible, prime, semi-prime, or neither

Three classes of Blaschke products. A BP is:

Definition interpolating if ∃δ > 0 such that

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

destructible or indestructible, prime, semi-prime, or neither Definition thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.

Three classes of Blaschke products. A BP is:

Definition interpolating if ∃δ > 0 such that

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

destructible or indestructible, prime, semi-prime, or neither Definition thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.

indestructible, prime or semi-prime

Three classes of Blaschke products. A BP is:

Definition interpolating if ∃δ > 0 such that

• j=k
• zj − zk

1 − zjzk

• ≥ δ > 0.

destructible or indestructible, prime, semi-prime, or neither Definition thin if lim

k→∞

• j=k
• zj − zk

1 − zjzk

• = 1.

indestructible, prime or semi-prime Definition uniform Frostman Blaschke product if sup

λ∈T ∞

• j=1

1 − |zj|2 |1 − zjλ| < ∞.

Why we care about UFBP, function theory

Theorem (Frostman’s theorem) Let B be an (infinite) Blaschke product with zeros (an). Then B and all of B’s subproducts have radial limit of modulus one at λ ∈ T iff

∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞,

Why we care about UFBP, function theory

Theorem (Frostman’s theorem) Let B be an (infinite) Blaschke product with zeros (an). Then B and all of B’s subproducts have radial limit of modulus one at λ ∈ T iff

∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞, Definition A Blaschke product is a Frostman Blaschke product if

∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞. Easier to construct! But not as nice (not a finite product of IBP’s).

Why we care about UFBP, function theory

Theorem (Frostman’s theorem) Let B be an (infinite) Blaschke product with zeros (an). Then B and all of B’s subproducts have radial limit of modulus one at λ ∈ T iff

∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞, Definition A Blaschke product is a uniform Frostman Blaschke product if sup

λ∈T ∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞.

Why we care about UFBP; function/operator theory

Definition µ finite Borel measure (∈ M), the Cauchy transform of µ is (1) (Kµ)(z) =

• T

1 1 − ξz dµ(ξ), z ∈ D K = {Kµ : µ finite Borel measure} space of Cauchy transforms. f K = inf{µ : µ ∈ M and (1) holds}. Definition φ analytic on D is a multiplier if f ∈ K = ⇒ φf ∈ K. Theorem (Hruˇ sˇ cev, Vinagradov, 1980) UFBP is the set of inner functions that are multipliers of K.

UFBP if sup

λ∈T ∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞.

UFBP if sup

λ∈T ∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞.

“Specific examples of Blaschke products in UFB are somewhat difficult to come by.” –Cima, Matheson, Ross

UFBP if sup

λ∈T ∞

• j=1

1 − |aj|2 |1 − ajλ| < ∞.

“Specific examples of Blaschke products in UFB are somewhat difficult to come by.” –Cima, Matheson, Ross (Specific Example): 0 < rn < 1, 0 < θn < 1, sup θn+1 θn

• < 1 and

∞

• n=1

1 − rn θn < ∞ then (rneiθn) is the zero sequence of a UFBP.

First step ϕB(0) ◦ B

Is every UFBP indestructible?

First step ϕB(0) ◦ B

Is every UFBP indestructible? Is ϕa ◦ B ∈ UFBP?

First step ϕB(0) ◦ B

Is every UFBP indestructible? Is ϕa ◦ B ∈ UFBP? Is every B ∈ UFBP prime or semi-prime? What we know about UFBPs that can help us:

Cima’s question

How big can the closure of the zero set in T of B, a UFBP (i.e. the spectrum of B = σB) be?

Cima’s question

How big can the closure of the zero set in T of B, a UFBP (i.e. the spectrum of B = σB) be? Ans: Not very big.

Cima’s question

How big can the closure of the zero set in T of B, a UFBP (i.e. the spectrum of B = σB) be? Ans: Not very big. Theorem (Matheson, 2007) If B ∈ UFBP, then σB is a closed nowhere dense subset of T.

UFBP Examples and conditions?

Theorem (Vasyunin, 1979) B ∈ UFBP with zeros (zn) = ⇒

n(1 − |zn|) log(1/(1 − |zn|)) < ∞.

UFBP Examples and conditions?

Theorem (Vasyunin, 1979) B ∈ UFBP with zeros (zn) = ⇒

n(1 − |zn|) log(1/(1 − |zn|)) < ∞.

Theorem Let (rn)∞

n=1 increasing in [0, 1). For there to exist a B ∈ UFBP

with zeros (zn), |zn| = rn, it is sufficient that ∃ε > 0 such that

1 (Vasyunin) ∞

• n=1

(1 − rn) log(e/(1 − rn))[log(log(3/(1 − rn)))]1+ε < ∞;

UFBP Examples and conditions?

Theorem (Vasyunin, 1979) B ∈ UFBP with zeros (zn) = ⇒

n(1 − |zn|) log(1/(1 − |zn|)) < ∞.

Theorem Let (rn)∞

n=1 increasing in [0, 1). For there to exist a B ∈ UFBP

with zeros (zn), |zn| = rn, it is sufficient that ∃ε > 0 such that

1 (Vasyunin) ∞

• n=1

(1 − rn) log(e/(1 − rn))[log(log(3/(1 − rn)))]1+ε < ∞;

2 (Akeroyd, G.) ∞

• n=1

(1 − rn) log(e/(1 − rn))[log(log(3/(1 − rn)))]ε converges

UFBP Examples and conditions?

Theorem (Vasyunin) B ∈ UFB with zeros (zn) = ⇒

n(1−|zn|) log(1/(1−|zn|)) < ∞.

Theorem Let (rn)∞

n=1 increasing in [0, 1). For there to exist a B ∈ UFB with

zeros (zn), |zn| = rn, it is sufficient that ∃ε > 0 such that

1 (Vasyunin) ∞

• n=1

(1 − rn) log(e/(1 − rn))[log(log(3/(1 − rn)))]1+ε < ∞;

2 (Akeroyd, G.) ∞

• n=1

(1 − rn) log(e/(1 − rn))[log(log(3/(1 − rn)))]ε converges

• OK. We have more info, now we need evidence

Operator theory viewpoint Function theory viewpoint All signs point to B ∈ UFBP implies B is semi-prime.

Operator viewpoint: Composition operators and UFBPs

Cϕ(f ) = f ◦ ϕ (Akeroyd, Ghatage 2008) On the classical Bergman space: (1) Let B be a BP. If B∗(T \ σB) = T, the composition operator CB will have closed range. (2) But B ∈ UFBP maps components of T \ σB onto arcs of length 2π − δ for arbitrary δ (with 0 < δ < 2π). It’s hard to do (2) without doing (1). Conjecture: CB has closed range for all B ∈ UFBP.

Function theory viewpoint: More evidence

To get more evidence, we need to know what happens to ϕa ◦ B when B ∈ UFBP.

Function theory viewpoint: More evidence

To get more evidence, we need to know what happens to ϕa ◦ B when B ∈ UFBP. Theorem (Matheson and Ross, CMFT 2007) If B ∈ UFBP, then ϕa ◦ B ∈ UFBP for all a ∈ D. “You can’t Frostman shift your way into (or out of) the class UFBP”

Function theory viewpoint: More evidence

To get more evidence, we need to know what happens to ϕa ◦ B when B ∈ UFBP. Theorem (Matheson and Ross, CMFT 2007) If B ∈ UFBP, then ϕa ◦ B ∈ UFBP for all a ∈ D. “You can’t Frostman shift your way into (or out of) the class UFBP” Theorem (Kraus, Roth (2013)) If B and C are indestructible BP’s, then B ◦ C indestructible. In fact, for every UFBP, C, and every BP, B, B ◦ C ∈ BP.

Back to our question

Q: Is every B ∈ UFBP prime? semi-prime? Write B = C ◦ D. Then ϕa ◦ B ∈ UFBP, wlog, B(0) = 0;

Back to our question

Q: Is every B ∈ UFBP prime? semi-prime? Write B = C ◦ D. Then ϕa ◦ B ∈ UFBP, wlog, B(0) = 0; wlogging again, D(0) = 0. So C(0) = 0 and C(z) = zC1(z). Thus, B(z) = D(z) C1(D(z)) and D ∈ UFBP.

Back to our question

Q: Is every B ∈ UFBP prime? semi-prime? Write B = C ◦ D. Then ϕa ◦ B ∈ UFBP, wlog, B(0) = 0; wlogging again, D(0) = 0. So C(0) = 0 and C(z) = zC1(z). Thus, B(z) = D(z) C1(D(z)) and D ∈ UFBP. Matheson = ⇒ σB and σD are closed, nowhere dense sets.

Back to our question

Q: Is every B ∈ UFBP prime? semi-prime? Write B = C ◦ D. Then ϕa ◦ B ∈ UFBP, wlog, B(0) = 0; wlogging again, D(0) = 0. So C(0) = 0 and C(z) = zC1(z). Thus, B(z) = D(z) C1(D(z)) and D ∈ UFBP. Matheson = ⇒ σB and σD are closed, nowhere dense sets. If D wraps a component of T \ σB around T, since B and D are both continuous, C is too.

Back to our question

Q: Is every B ∈ UFBP prime? semi-prime? Write B = C ◦ D. Then ϕa ◦ B ∈ UFBP, wlog, B(0) = 0; wlogging again, D(0) = 0. So C(0) = 0 and C(z) = zC1(z). Thus, B(z) = D(z) C1(D(z)) and D ∈ UFBP. Matheson = ⇒ σB and σD are closed, nowhere dense sets. If D wraps a component of T \ σB around T, since B and D are both continuous, C is too. So we need a UFBP that doesn’t wrap a component around T. Conclusion: This is hard; maybe all B ∈ UFBP’s are semi-prime.

Another reason why all B ∈ UFBP are probably semi-prime

Theorem (Chalendar, G., Partington, 2013) Given B ∈ UFBP with distinct zeros, there exists a ∈ D such that ϕ2

aB is semi-prime.

Actually a more general fact about Blaschke products that are continuous on an arc.

Surprising answer, at least for us

Example (Akeroyd, G.) There exists B ∈ UFBP such that B ◦ B ∈ UFBP. How do you do it?

Surprising answer, at least for us

Example (Akeroyd, G.) There exists B ∈ UFBP such that B ◦ B ∈ UFBP. How do you do it? Fact: If you’re an inner function close (uniformly) to a UFBP, you’re a UFBP.

Surprising answer, at least for us

Example (Akeroyd, G.) There exists B ∈ UFBP such that B ◦ B ∈ UFBP. How do you do it? Fact: If you’re an inner function close (uniformly) to a UFBP, you’re a UFBP. Create B so that on a “hot spot” B ◦ B =

• j

B − aj 1 − ajB ∼ λk B − ak 1 − akB , a Frostman shift of a UFBP.

Surprising answer, at least for us

Example (Akeroyd, G.) There exists B ∈ UFBP such that B ◦ B ∈ UFBP. How do you do it? Fact: If you’re an inner function close (uniformly) to a UFBP, you’re a UFBP. Create B so that on a “hot spot” B ◦ B =

• j

B − aj 1 − ajB ∼ λk B − ak 1 − akB , a Frostman shift of a UFBP. Make B very thin, so that the zeros of B ◦ B again lie in the “hot spots.”

Surprising answer, at least for us

Example (Akeroyd, G.) There exists B ∈ UFBP such that B ◦ B ∈ UFBP. How do you do it? Fact: If you’re an inner function close (uniformly) to a UFBP, you’re a UFBP. Create B so that on a “hot spot” B ◦ B =

• j

B − aj 1 − ajB ∼ λk B − ak 1 − akB , a Frostman shift of a UFBP. Make B very thin, so that the zeros of B ◦ B again lie in the “hot spots.” Then control all the Frostman constants.

Question

Are prime Blaschke products dense in the set of all Blaschke products in the uniform norm?

• Prime finite Blaschke products are uniformly dense in finite

Blaschke products.

Question

Are prime Blaschke products dense in the set of all Blaschke products in the uniform norm?

• Prime finite Blaschke products are uniformly dense in finite

Blaschke products.

• Prime thin BP’s are dense in thin BP’s. (G., Laroco, Mortini,

Rupp, 1994)

Question

Are prime Blaschke products dense in the set of all Blaschke products in the uniform norm?

• Prime finite Blaschke products are uniformly dense in finite

Blaschke products.

• Prime thin BP’s are dense in thin BP’s. (G., Laroco, Mortini,

Rupp, 1994) Why? If B = C ◦ D and D is infinite, C is an automorphism. So D is finite and (wolog) a subproduct of B. So we can count the possible D.

Question

Are prime Blaschke products dense in the set of all Blaschke products in the uniform norm?

• Prime finite Blaschke products are uniformly dense in finite

Blaschke products.

• Prime thin BP’s are dense in thin BP’s. (G., Laroco, Mortini,

Rupp, 1994) Why? If B = C ◦ D and D is infinite, C is an automorphism. So D is finite and (wolog) a subproduct of B. So we can count the possible D.

• Prime finite products of thin BP’s are dense in finite products
• f thin BP’s. (Chalendar, G., Partington, 2013)

Much harder question because you need to move infinitely many zeros. Still can be done.

Pictures by A. Beurling (June) & O. Frostman (October)

Figure : U. Daepp, G, G. Semmler, E. Wegert