aa r X i v : . [ m a t h . C V ] S e p A REFINED AGLER DECOMPOSITION ANDGEOMETRIC APPLICATIONS
GREG KNESE
Abstract.
We prove a refined Agler decomposition for boundedanalytic functions on the bidisk and show how it can be used toreprove an interesting result of Guo et al. related to extendingholomorphic functions without increasing their norm. In addition,we give a new treatment of Heath and Suffridge’s characterizationof holomorphic retracts on the polydisk. Introduction
Let D denote the unit disk in C and D = D × D the unit bidisk.[Agler, 1988] proved that a holomorphic function f : D → D satisfiesa decomposition (later called an Agler decomposition ) of the form1 − f ( z ) f ( ζ ) = (1 − z ¯ ζ ) K ( z, ζ ) + (1 − z ¯ ζ ) K ( z, ζ )where K , K are positive semi-definite kernel functions. A kernel func-tion K : Ω × Ω → C is positive semi-definite if for every finite subset F ⊂ Ω the matrix ( K ( z, ζ )) z,ζ ∈ F is positive semi-definite. (In this article, Ω will be either D or D .) TheAgler decomposition generalizes the Pick interpolation theorem fromone-variable complex analysis, which implies that for any f : D → D ,holomorphic, 1 − f ( z ) f ( ζ )1 − z ¯ ζ is a positive semi-definite kernel.In recent years, more refined versions of the Agler decompositionhave been found for rational inner functions. See [Cole and Wermer, 1999],[Geronimo and Woerdeman, 2004], [Knese, 2008], or [Knese, 2010]. (Un-related to rational inner functions, in specific, but still relevant are[Ball et al., 2005] and [Lata et al., 2009]). It has not been clear which Date : July 1, 2018.1991
Mathematics Subject Classification.
Primary 47A57; Secondary 32D15.This research was supported by NSF grant DMS-1001791. of the “refined” aspects of these decompositions for rational inner func-tions would extend to more general bounded analytic functions (andwhich would actually be useful). The following theorem represents anoffering in this direction. Our hope is that others may find it usefulwithout having to learn any of the underlying theory required to proveit.
Theorem 1.1.
Let f : D → D be holomorphic. Then, there existpositive semidefinite kernels K , K , and holomorphic kernels L , L such that − f ( z ) f ( ζ ) = (1 − z ¯ ζ ) K ( z, ζ ) + (1 − z ¯ ζ ) K ( z, ζ ) and f ( z ) − f ( ζ ) = ( z − ζ ) L ( z, ζ ) + ( z − ζ ) L ( z, ζ ) , where ( z, ζ ) = (( z , z ) , ( ζ , ζ )) . In addition, the following (pointwise)inequalities hold | L j ( z, ζ ) | ≤ K j ( z, z ) K j ( ζ , ζ ) for j = 1 , . Notice L j ( z, z ) = ∂f∂z j ( z ). So, estimates on the positive semi-definitekernels in this decomposition provide estimates on the derivatives of f .The analogous inequalities in one variable are (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − f ( ζ ) z − ζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − f ( z ) f ( ζ )1 − z ¯ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − | f ( z ) | − | z | − | f ( ζ ) | − | ζ | which are consequences of the Schwarz-Pick lemma.As an application of this theorem, we are able to reprove a usefultheorem of [Guo et al., 2008] related to norm preserving extensions ofholomorphic functions on the polydisk and holomorphic retracts of thepolydisk. When working in D n +1 we will typically denote points by( z, w ) where z ∈ D n and w ∈ D . Theorem 1.2 ([Guo et al., 2008]) . Let V ⊂ D n +1 , and suppose w | V has a nontrivial norm 1 holomorphic extension to D n . Then, V is asubset of the graph of a holomorphic function of z . Here nontrivial norm 1 extension refers to a function on D n +1 otherthan w which agrees with w on V and whose modulus has supremumnorm at most 1. Guo et al.’s proof involved an interesting use of theone-variable Denjoy-Wolff theorem. Guo et al. used this result to con-tinue some of the work initiated in the paper [Agler and McCarthy, 2003].Additionally, they reproved Heath and Suffridge’s characterization ofholomorphic retracts of the polydisk, which we now define. REFINED AGLER DECOMPOSITION AND GEOMETRIC APPLICATIONS 3
Definition 1.3.
A subset V ⊂ D n is a holomorphic retract if thereexists a holomorphic function ( a retraction ) ρ : D n → D n such that ρ ◦ ρ = ρ and ρ ( D n ) = V Heath and Suffridge characterized all holomorphic retracts of thepolydisk as graphs.
Theorem 1.4 ([Heath and Suffridge, 1981]) . Suppose V ⊂ D n is aholomorphic retract. Then, after applying an automorphism of D n , V can be put into the form { ( z, f ( z )) : z ∈ D k } where f : D k → D n − k is holomorphic. The proof of Heath and Suffridge involves an impressive and technicalstudy of properties of Taylor series of retracts. Guo et al. gave a newproof by rehashing some of their proof of Theorem 1.2. Although it issomething of an aside, we think it is worth it to show a slightly differentapproach in Section 4. While our proofs are different from both Heathand Suffridge and Guo et al., the general roadmap of our approachowes a great deal to Guo et al.2.
Proof of Theorem 1.1
Let us first explain the result for rational inner functions and thenuse an approximation argument to prove it for all analytic functionsbounded by one on D .As shown in [Rudin, 1969] (Theorem 5.2.5), every rational innerfunction on D can be represented as f ( z ) = ˜ p ( z , z ) p ( z , z )where p ∈ C [ z , z ] has no zeros in D , ˜ p ( z , z ) = z n z m p (1 / ¯ z , / ¯ z )for appropriate powers n, m , and ˜ p and p have no common factor.Necessarily p and ˜ p have bidegree at most ( n, m ) (i.e. degree at most n in z and m in z ).[Geronimo and Woerdeman, 2004] proved a detailed version of a two-variable Christoffel-Darboux formula (see their Proposition 2.3.3 andalso [Cole and Wermer, 1999] and [Knese, 2008]), which can be statedas follows: there exist polynomials A , . . . , A n ∈ C [ z , z ] of bidegreeat most ( n − , m ) and polynomials B , . . . , B m ∈ C [ z , z ] of bidegree GREG KNESE at most ( n, m −
1) such that(2.1) p ( z ) p ( ζ ) − ˜ p ( z )˜ p ( ζ ) = (1 − z ¯ ζ ) n X j =1 A j ( z ) A j ( ζ )+(1 − z ¯ ζ ) m X j =1 B j ( z ) B j ( ζ )Let ˜ A j ( z ) := z n − z m A j (1 / ¯ z , / ¯ z ), ˜ B j ( z ) := z n z m − B j (1 / ¯ z , / ¯ z ). Ifwe perform a similar reflection operation to (2.1) (i.e. replace ( z, ζ )with ((1 / ¯ z , / ¯ z ) , (1 / ¯ ζ , / ¯ ζ )), take complex conjugates and multiplythrough by z n z m ¯ ζ n ¯ ζ m ) we get(2.2) p ( z ) p ( ζ ) − ˜ p ( z )˜ p ( ζ ) = (1 − z ¯ ζ ) n X j =1 ˜ A j ( z ) ˜ A j ( ζ )+(1 − z ¯ ζ ) m X j =1 ˜ B j ( z ) ˜ B j ( ζ )If we average (2.1) and (2.2) and rewrite using vector notation, we get(2.3) p ( z ) p ( ζ ) − ˜ p ( z )˜ p ( ζ ) = (1 − z ¯ ζ ) h A ( z ) , A ( ζ ) i + (1 − z ¯ ζ ) h B ( z ) , B ( ζ ) i where A = 1 √ A , . . . , A n , ˜ A , . . . , ˜ A n ] t B = 1 √ B , . . . , B m , ˜ B , . . . , ˜ B m ] t and h v, w i = w ∗ v denotes the standard complex euclidean inner prod-uct (with dimension taken from context).If we reflect (2.3) in z alone we get(2.4) ˜ p ( z ) p ( ζ ) − p ( z )˜ p ( ζ ) = ( z − ζ ) ˜ A ( z ) · A ( ζ ) + ( z − ζ ) ˜ B ( z ) · B ( ζ )where “ · ” denotes the dot product: v · w = w t v .Now, if we divide (2.3) by p ( z ) p ( ζ ) and divide (2.4) by p ( z ) p ( ζ ), weget equations of the form1 − f ( z ) f ( ζ ) = X j =1 (1 − z j ¯ ζ j ) K j ( z, ζ ) f ( z ) − f ( ζ ) = X j =1 ( z j − ζ j ) L j ( z, ζ )where K , K are positive semidefinite kernels given explicitly by K ( z, ζ ) = h A ( z ) , A ( ζ ) i p ( z ) p ( ζ ) K ( z, ζ ) = h B ( z ) , B ( ζ ) i p ( z ) p ( ζ ) REFINED AGLER DECOMPOSITION AND GEOMETRIC APPLICATIONS 5 and L , L are holomorphic kernels given explicitly by L ( z, ζ ) = ˜ A ( z ) · A ( ζ ) p ( z ) p ( ζ ) L ( z, ζ ) = ˜ B ( z ) · B ( ζ ) p ( z ) p ( ζ ) . The inequality | L j ( z, ζ ) | ≤ K j ( z, z ) K j ( ζ , ζ )follows from Cauchy-Schwarz and the fact that | A | = | ˜ A | and | B | = | ˜ B | .This proves the theorem for rational inner functions.It is proven in [Rudin, 1969] (Theorem 5.5.1) that holomorphic func-tions f : D → D can be approximated locally uniformly by ra-tional inner functions. So, let { f ( i ) } i be a sequence of rational in-ner functions converging locally uniformly to f with corresponding K ( i )1 , K ( i )2 , L ( i ) i , L ( i )2 satisfying the above formulas/inequalities. Becauseof the inequalities | L ( i ) j ( z, ζ ) | , | K ( i ) j ( z, ζ ) | ≤ K ( i ) j ( z, z ) K ( i ) j ( ζ , ζ ) ≤ − | z | )(1 − | z | )(1 − | ζ | )(1 − | ζ | )the kernel functions are locally bounded and hence form a normalfamily. We can select subsequences so that K ( i )1 → K , K ( i )2 → K , L ( i )1 → L , L ( i )2 → L locally uniformly. Positive semi-definiteness andpointwise inequalities are preserved under this limit and therefore thestatement of the theorem holds.3. Guo et al.’s extension theorem
As an application we prove Theorem 1.2 in the following slightly moredetailed form. Except for uniqueness, this is contained in [Guo et al., 2008].
Theorem 3.1.
Let V ⊂ D n +1 be a set with more than one w -valueand let πV be the projection of V onto the first n coordinates. Suppose w | V has a nontrivial norm 1 holomorphic extension F . Then, there isa unique holomorphic f : D n → D such that F ( z, f ( z )) = f ( z ) and V = { ( z, f ( z )) : z ∈ πV } . In this section we generally follow the convention of denoting pointsin D n +1 by ( z, w ) with z ∈ D n and w ∈ D . Lemma 3.2. If f : D n +1 → D is holomorphic, φ is an automorphismof D , and there exists a z ∈ D n such that f ( z , w ) = φ ( w ) for all w ,then f ( z, w ) = φ ( w ) for all ( z, w ) . GREG KNESE
Proof.
We may assume φ ( w ) = w and z = (0 , . . . , G ( z, w ) = f ( z, w ) − w − ¯ wf ( z, w )is holomorphic in z , G (0 , w ) = 0, and | G | ≤ | z | ∞ for the maximum modulus of the components of z . Bythe Schwarz lemma, | G ( z, w ) | ≤ | z | ∞ and1 −| z | ∞ ≤ −| G ( z, w ) | = (1 − | w | )(1 − | f ( z, w ) | ) | − ¯ wf ( z, w ) | ≤ − | w | | − ¯ wf ( z, w ) | and so | w − f ( z, w ) | ≤ | − ¯ wf ( z, w ) | ≤ − | w | − | z | ∞ . Then, by the maximum principlesup w ∈ r D | w − f ( z, w ) | ≤ − r − | z | ∞ which implies f ( z, w ) ≡ w after letting r ր (cid:3) Lemma 3.3.
Let F : D n +1 → D be holomorphic and suppose F ( z , w ) = w at some point. Necessarily, (3.1) | ∂F∂w ( z , w ) | ≤ . If equality holds in (3.1) , then F ( z, w ) = φ ( w ) for some automorphism φ . If strict inequality holds in (3.1) , then there exists a unique holo-morphic function f : Ω → D defined in a neighborhood Ω of z suchthat f ( z ) = w and F ( z, f ( z )) = f ( z ) where defined.Proof. By the Schwarz lemma1 − | F ( z , w ) | − | w | = 1 ≥ | ∂F∂w ( z , w ) | . If equality occurs then F ( z , w ) is an automorphism of D and by theprevious lemma F ( z, w ) = φ ( w ) identically.If equality does not hold, then setting G ( z, w ) = F ( z, w ) − w wesee that ∂G∂w ( z , w ) = ∂F∂w ( z , w ) − = 0. By the implicit functiontheorem, there exists a function of z in a neighborhood of z such that G ( z, f ( z )) = 0; i.e. F ( z, f ( z )) = f ( z ).To see that f is unique, we note that if F ( z , w ) = w , there cannotbe a different w = w such that F ( z , w ) = w , for then F ( z , w ) ≡ w and hence F ( z, w ) ≡ w . By assumption this cannot occur, so any REFINED AGLER DECOMPOSITION AND GEOMETRIC APPLICATIONS 7 point ( z , w ) satisfying F ( z , w ) = w is uniquely determined by the z component. In particular, F ( z, f ( z )) = f ( z ) cannot hold for twodifferent choices of f : Ω → D . (cid:3) The final lemma is the most important and it utilizes the main the-orem.
Lemma 3.4.
Let F : D n +1 → D be holomorphic. Suppose F ( z , w ) = w , F ( z , w ) = w , F ( z , w ) = w , where w = w . Then there existsa unique f : D n → D such that F ( z, f ( z )) = f ( z ) . In particular, if F ( z , w ) = w , then f ( z ) = w .Proof. By the three assumptions, F cannot be an automorphism as afunction of w . So, we are in the second case of the previous lemmaand there locally (say on a domain Ω ⊂ D n ) exists a unique f : Ω → D satisfying F ( z, f ( z )) = f ( z ). We need to extend f to all of D n .We will show f can be extended one variable at a time. Letting ζ = ( ζ , ζ , . . . , ζ n ) = ( ζ , ζ ′ ) ∈ Ω, we plan to show f can be ex-tended to D × { ζ ′ } in such a way that the identity F ( z, f ( z )) = f ( z ) is preserved. By Lemma 3.3, the identity will then extend toa unique function on an open neighborhood of D × { ζ ′ } . So, given anyother point η = ( η , . . . , η n ), we will be able to successively extend f to ( η , ζ , . . . , ζ n ) , ( η , η , ζ , . . . ) , . . . , ( η , . . . , η n ). This will imply f isholomorphic on all of D n and F ( z, f ( z )) = f ( z ).For this argument we will use t for the first coordinate of z and write z = ( t, z ′ ) (we are avoiding “ z ” since we have used this in a differentway in the lemma statement).Let g ( t ) = f ( t, ζ ′ ) and G ( t, w ) = F ( t, ζ ′ , w ). Now g is holomorphicin some neighborhood of ζ and G ( t, g ( t )) = g ( t ) holds in said neigh-borhood. If g is constant, then clearly g extends to be holomorphic on D and G ( t, g ( t )) = g ( t ) holds on all of D . So, suppose g is nonconstant.Perturb ζ if necessary to make g ′ ( ζ ) = 0, and let ∂ t , ∂ w denote thepartial derivatives with respect to t, w , respectively.Then, ∂ t G ( t, g ( t )) + ∂ w G ( t, g ( t )) ∂ t g ( t ) = ∂ t g ( t ), so ∂ t G ( t, g ( t )) = ∂ t g ( t )(1 − ∂ w G ( t, g ( t ))). Now, ∂ t G ( ζ , g ( ζ )) = 0 by the previous lemma(i.e. ∂ w G ( t, g ( t )) cannot equal 1, since this would imply G is an auto-morphism as a function of w ) and since ∂ t g ( ζ ) = 0.We apply the main theorem to G ( t, w ). Theorem 1.1 implies1 − G ( t, w ) G ( τ, η ) = (1 − t ¯ τ ) K + (1 − w ¯ η ) K with K , K positive semi-definite, where K , K should be evaluatedat (( t, w ) , ( τ, η )). GREG KNESE
Substituting w = g ( t ) , η = g ( τ ) for t, τ in a neighborhood of ζ , andwriting v ( t, τ ) = (( t, g ( t )) , ( τ, g ( τ ))) for short, we get1 − g ( t ) g ( τ ) = (1 − t ¯ τ ) K ( v ( t, τ )) + (1 − g ( t ) g ( τ )) K ( v ( t, τ ))or(3.2) (1 − g ( t ) g ( τ ))(1 − K ( v ( t, τ ))) = (1 − t ¯ τ ) K ( v ( t, τ ))We cannot have K ( v ( ζ , ζ )) = 1 for then K ( v ( ζ , ζ )) = 0, whichby the main theorem implies a contradiction. Specifically, | ∂ t G ( ζ , g ( ζ )) | = | L ( v ( ζ , ζ )) | ≤ | K ( v ( ζ , ζ )) | = 0which is not the case as ∂ t G ( ζ , g ( ζ )) = 0.So, | K ( v ( t, τ )) | < t, τ in some open set around ζ . By (3.2),for such t, τ − g ( t ) g ( τ )1 − t ¯ τ = K ( v ( t, τ ))1 − K ( v ( t, τ )) = K ( v ( t, τ )) ∞ X j =0 K ( v ( t, τ )) j is positive semi-definite. By the Pick interpolation theorem, g extendsto be holomorphic on all of D . (See [Agler and McCarthy, 2002] for thePick interpolation theorem from this point of view.) Also, G ( t, g ( t )) = g ( t ) then automatically holds on all of D by analyticity. This completesthe proof. (cid:3) Theorem 3.1 is just a rephrasal of this lemma.4.
Retracts and Theorem 1.4
We prove the following refinement of Theorem 1.4 (which can alsobe found in [Guo et al., 2008]).
Theorem 4.1.
Suppose V ⊂ D n is a holomorphic retract. Then, afterapplying an automorphism of D n , V can be put into the form { ( z, e ( z ) , f ( z )) : z ∈ D k } where e : D k → D m is a coordinate function in each component and f : D k → D n − m − k is holomorphic with no components equal to anautomorphism as a function of a single variable. An example for e might be e ( z , z ) = ( z , z , z , z , z ). Lemma 4.2.
Suppose V ⊂ D n +1 is a holomorphic retract, with retrac-tion ρ ( z, w ) = ( ρ , . . . , ρ n , ρ n +1 ) = ( ρ ′ , ρ n +1 ) where we assume ρ n +1 isnot an automorphism as a function of one variable. Then, there exists REFINED AGLER DECOMPOSITION AND GEOMETRIC APPLICATIONS 9 f : D n → D , holomorphic, such that ( z, w ) ( ρ ′ ( z, f ( z )) , f ( z )) is aretraction of V , (4.1) V = { ( z, f ( z )) : z ∈ πV } and πV is a retract with retraction z ρ ′ ( z, f ( z )) .Proof. If ρ n +1 is constant, there is nothing to prove, so assume other-wise. Then, ρ n +1 ( z, w ) = w for two distinct values of w (and necessarilydifferent values of z ) since ρ n +1 ( ρ ′ , ρ n +1 ) = ρ n +1 . Therefore, Theorem3.1 applies. There exists f : D n → D holomorphic satisfying(4.2) ρ n +1 ( z, f ( z )) = f ( z )and(4.3) { ( z, w ) : ρ n +1 ( z, w ) = w } = { ( z, f ( z )) : z ∈ D n } ⊃ V. This proves (4.1).As ρ ( z, f ( z )) ∈ V we see that by (4.3) and (4.2), f ( ρ ′ ( z, f ( z ))) = ρ n +1 ( z, f ( z )) = f ( z ), which shows ( z, w ) ( ρ ′ ( z, f ( z )) , f ( z )) agreeswith the map ( z, w ) ρ ( z, f ( z )). This is a retraction since its compo-sition with itself is ρ ( ρ ′ ( z, f ( z )) , f ( ρ ′ ( z, f ( z )))) = ρ ( ρ ′ ( z, f ( z )) , ρ n +1 ( z, f ( z )))= ρ ( ρ ( z, f ( z )) = ρ ( z, f ( z ))as desired.We need to show that the range of ( z, w ) ρ ( z, f ( z )) contains V (it certainly is contained in V ). If ( z, w ) ∈ V , then w = f ( z ) and ρ ( z, f ( z )) = ( z, f ( z )) = ( z, w ). So, this map is a retraction of V .Finally, we must show πV is a retract with retraction z ρ ′ ( z, f ( z )).This map is indeed a retraction since ( z, w ) ( ρ ′ ( z, f ( z )) , f ( z )) is, andthe first n components necessarily trace out πV . (cid:3) Lemma 4.3.
Let ρ = ( ρ , . . . , ρ n ) : D n → D n be a retraction of V . If ρ ( z , . . . , z n ) is an automorphism as a function of z , then ρ ( z ) ≡ z .If ρ is an automorphism as a function of z then ρ ( z ) ≡ z and ρ ( z ) ≡ φ ( z ) for some φ .Proof. If ρ is an automorphism, say φ , as a function of z , then φ ◦ φ = φ , which means φ = id. This means ρ ( z ) = z . If ρ is anautomorphism, say φ , as a function of z , then φ ( ρ ( z )) = ρ ( ρ ( z )) = ρ ( z ) = φ ( z ). This implies ρ ( z ) = z . (cid:3) Lemma 4.4.
Let ρ = ( ρ , . . . , ρ n ) : D n → D n be a retraction of V withall components equal to an automorphism as a function of a single variable. After conjugating by automorphisms of D n we may put ρ intothe form ρ ( z, w ) = ( z, e ( z )) where z ∈ D k , w ∈ D n − k , e : D k → D n − k , where each component of e is a coordinate function.Proof. Let us just illustrate. If ρ ( z ) = z and ρ ( z ) = φ ( z ) for someone variable automorphism φ , we can conjugate by the automorphismof D n given by ψ ( z ) = ( φ − ( z ) , z , . . . , z n ) to get ψ ◦ ρ ◦ ψ − ( z ) = ( z , z , ρ ◦ ψ − ( z ) , . . . , ρ n ◦ ψ − ( z ))The lemma then follows from the previous lemma after reordering andconjugating by analogous automorphisms as necessary. (cid:3) Proof of Theorem 4.1.
There is no harm in assuming V is not a Carte-sian product of a point and a retract (this is equivalent to assumingour retractions do not possess a constant component).We proceed by induction. Let n = 1 and let ρ : D → V be a retrac-tion. One variable retractions are either constant (which by assumptionis ruled out) or equal to the identity (by the Schwarz-Pick lemma, aself-map of the disk with two fixed points equals the identity).Suppose the theorem holds for n dimensional retracts. Let ρ : D n +1 → V be a retraction onto V . If all components of ρ are au-tomorphisms (in a single variable) then we are finished by Lemma 4.4.So, we assume some component is not an automorphism in a singlevariable and relabel to make ρ n +1 such a component. By Lemma 4.2,we can replace ρ with a retraction r of the form r ( z, w ) = ( r ′ ( z ) , f ( z ))where z ∈ D n and w ∈ D and the projection of V onto the first n co-ordinates, denoted πV , is a retract with retraction r ′ ( z ). By induction(after possibly applying automorphisms of D n ) we may put πV intothe form πV = { ( z, e ( z ) , g ( z )) : z ∈ D k } where e : D k → D m consists of coordinate functions, g : D k → D n − m − k is holomorphic with no components equal to an automorphism.Then, V = { ( z, e ( z ) , g ( z ) , f ( z, e ( z ) , g ( z ))) : z ∈ D k } which is of the desired form. (cid:3) REFINED AGLER DECOMPOSITION AND GEOMETRIC APPLICATIONS 11
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University of Alabama, Tuscaloosa, AL, 35487-0350
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