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 | a j | a j − z B ( z ) = e i θ z n � 1 − a j z . a j j 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 L n ( z ) = ( z + z n ) / (1 + z n z ) ( z n ) in D is universally admissible for the set of Blaschke products if there is a Blaschke product B such that { B (( z + z n ) / (1 + z n z )) : 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 n z − a j � Let B ( z ) = 1 − a j z have prime degree; i.e. n is a prime. j =1 B = C ◦ D , degree of C = m , degree of D = k

Some simple examples n z − a j � Let B ( z ) = 1 − a j z have prime degree; i.e. n is a prime. j =1 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 z 1 , . . . , z d ∈ D . There exists a unique Blaschke B, degree d + 1 , B (0) = 0 , B (1) = 1 , and B ′ ( z j ) = 0 , all j. Corollary (Nehari, 1947; Zakeri) Blaschke pdts. B 1 , B 2 have the same critical pts. iff B 1 = ϕ a ◦ B 2 for some automorphism ϕ a . Remark. B with distinct zeros has 2 n − 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 z 1 , . . . , z d ∈ D . There exists a unique Blaschke B, degree d + 1 , B (0) = 0 , B (1) = 1 , and B ′ ( z j ) = 0 , all j. Corollary (Nehari, 1947; Zakeri) Blaschke pdts. B 1 , B 2 have the same critical pts. iff B 1 = ϕ a ◦ B 2 for some automorphism ϕ a . Remark. B with distinct zeros has 2 n − 2 critical points, only n − 1 are in D : { z 1 , . . . , z n − 1 , 1 / z 1 , . . . , 1 / z n − 1 } : B has ≤ n − 1 critical values in D .

Counting critical values Degree of D = k , degree of C = m . ⇒ B ′ ( z ) = C ′ ( D ( z )) D ′ ( z ); D has k − 1 critical B = C ◦ D = 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 ′ ( z ) = C ′ ( D ( z )) D ′ ( z ); D has k − 1 critical B = C ◦ D = 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 of 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 of 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 | a j | a j − z Blaschke Product: B ( z ) = e i θ � 1 − a j z . j a j ϕ a ( 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, ϕ a ◦ I is a Blaschke product. Definition A Blaschke product is indestructible if ϕ a ◦ B is a Blaschke product for all a ∈ D . Which BP’s are indestructible ?

� � − 1+ z Figure : The atomic singular inner function S ( z ) = exp 1 − z Constant Modulus on circles Discontinuity at 1 Radial limit 0 at 1

S and indestructibility Figure : ϕ a ◦ S ( − iz ), for a = . 727

Indestructibility: ϕ a ◦ I Blaschke for all a I is indestructible if ϕ a ◦ I Blaschke for all a ∈ D . � � − 1+ z S ( z ) = exp ; ϕ α ◦ S is a BP for all α � = 0, so we have 1 − z 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. � � − 1+ z Sensitive Question . S ( z ) = exp is decomposable. 1 − z

S and indecomposability I is prime if whenever I = U ◦ V either U or V is an automorphism. � � − 1+ z Sensitive Question . S ( z ) = exp is decomposable. 1 − z You can take roots of singular inner functions. So z n ◦ S 1 / n = S .

S and indecomposability I is prime if whenever I = U ◦ V either U or V is an automorphism. � � − 1+ z Sensitive Question . S ( z ) = exp is decomposable. 1 − z You can take roots of singular inner functions. So z n ◦ S 1 / 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. � � − 1+ z Sensitive Question . S ( z ) = exp is decomposable. 1 − z You can take roots of singular inner functions. So z n ◦ S 1 / n = S . But zS ( z ) is prime! (K. Stephenson, 1982) Why?

Something you should do with an automorphism ϕ a first To show I prime: ϕ I (0) ◦ I (0) = 0. I = U ◦ V with U or V non-automorphisms (non-prime) iff ϕ I ◦ I is. So assume I (0) = 0.

Something you should do with an automorphism ϕ a first To show I prime: ϕ I (0) ◦ I (0) = 0. I = U ◦ V with U or V non-automorphisms (non-prime) iff ϕ I ◦ I is. So assume I (0) = 0. I (0) = ( U ◦ ϕ V (0) ) ◦ ( ϕ V (0) ◦ V ), so assume V (0) = 0. 0 = I (0) = U ( V (0)) = U (0), so U ( z ) = z ( U 1 ( z )). And so...

Something you should do with an automorphism ϕ a first To show I prime: ϕ I (0) ◦ I (0) = 0. I = U ◦ V with U or V non-automorphisms (non-prime) iff ϕ I ◦ I is. So assume I (0) = 0. I (0) = ( U ◦ ϕ V (0) ) ◦ ( ϕ V (0) ◦ V ), so assume V (0) = 0. 0 = I (0) = U ( V (0)) = U (0), so U ( z ) = z ( U 1 ( z )). And so... I ( z ) = V ( z ) ( U 1 ( V ( z )) and V divides I .

Proof that zS ( z ) is prime, S atomic singular inner function zS ( z ) = ( U 0 ◦ V 0 )( z ) already vanishes at zero. WOLOG U 0 ( z ) = zU ( z ) and V 0 ( z ) = zV ( z ).

Proof that zS ( z ) is prime, S atomic singular inner function zS ( z ) = ( U 0 ◦ V 0 )( z ) already vanishes at zero. WOLOG U 0 ( z ) = zU ( z ) and V 0 ( z ) = zV ( z ). zS ( z ) = ( zU ( z )) ◦ ( zV ( z )) = ⇒ zS ( z ) = zV ( z )( U ( zV ( z ))).