Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels
aa r X i v : . [ m a t h . C V ] J a n COMPOSITION OPERATORS ON HARDY-SOBOLEV SPACESWITH BOUNDED REPRODUCING KERNELS
GUANGFU CAO AND LI HE*A
BSTRACT . For any real β let H β be the Hardy-Sobolev space on theunit disk D . H β is a reproducing kernel Hilbert space and its reproducingkernel is bounded when β > / . In this paper, we study compositionoperators C ϕ on H β for / < β < . Our main result is that, for anon-constant analytic function ϕ : D → D , the operator C ϕ has denserange in H β if and only if the polynomials are dense in a certain Dirichletspace of the domain ϕ ( D ) . It follows that if the range of C ϕ is dense in H β , then ϕ is a weak-star generator of H ∞ . Note that this conclusion isfalse for the classical Dirichlet space D . We also characterize Fredholmcomposition operators on H β .
1. I
NTRODUCTION
Let D be the unit disk in the complex plane C and H ( D ) be the space ofall analytic functions on D . For f ∈ H ( D ) we use R f ( z ) = z ∂f∂z ( z ) to denote the radial derivative of f at z . If f ( z ) = P ∞ k =0 a k z k is the Taylorexpansion of f , it is easy to see that R f ( z ) = ∞ X k =0 ka k z k = ∞ X k =1 ka k z k . More generally, for any real number β and any f ∈ H ( D ) with the Taylorexpansion above, we define R β f ( z ) = ∞ X k =1 k β a k z k Mathematics Subject Classification.
Key words and phrases.
Hardy-Sobolev space, composition operator, Fredholm opera-tor, reproducing kernel, automorphism.G.C. was supported by NNSF of China (Grant No. 12071155), and L.H. was supportedby NNSF of China (Grant No. 11871170).*Corresponding author, email: [email protected]. and call it the radial derivative of f of order β .It is clear that these fractional radial differential operators satisfy R α R β = R α + β . When β < , the effect of R β on f is actually “integration” insteadof “diffferentiation”. For example, radial differentiation of order − is ac-tually radial integration of order .For β ∈ R , the Hardy-Sobolev space H β consists of all analytic functions f on D such that R β f belongs to the classical Hardy space H . It is clearthat H β is a Hilbert space with the inner product h f, g i β = f (0) g (0) + hR β f, R β g i H . The induced norm in H β is then given by k f k β = | f (0) | + kR β f k H . Recall that H is the space of analytic functions f on D such that k f k H = sup
Proposition 1 ([6]) . Suppose β ∈ R and f ∈ H ( D ) . Then the followingconditions are equivalent. (a) f ∈ H β . (b) R β +1 f ∈ A .If N is a nonnegative integer with N > β , then the conditions above arealso equivalent to (c) R N f ∈ A N − β ) − . Hardy-Sobolev spaces contain many classical analytic function spacesas special cases. For example, H − / is the Bergman space A , H is theHardy space H , and H / is the Dirichlet space D consisting of analyticfunctions f on D such that k f k = | f (0) | + Z D | f ′ ( z ) | dA ( z ) < ∞ . More generally, for any domain G ⊂ C and any positive measure dω on G , we will use A ( G, dω ) to denote the weighted Bergman space of analyticfunctions f on G such that Z G | f ( z ) | dω ( z ) < ∞ . Similarly, we use D ( G, dω ) for the weighted Dirichlet space of analyticfunctions f on G with Z G | f ′ ( z ) | dω ( z ) < ∞ . When ω is ordinary area measure, we will simply write A ( G ) and D ( G ) .Let ϕ : D → D be an analytic self-map D . For any Hilbert space H ofanalytic functions on D we consider the composition operator C ϕ : H → H defined by C ϕ f = f ◦ ϕ . For β < / , every composition operator isbounded on H β . However, this is not so for β ≥ / . For example, notevery composition operator is bounded on the Dirichlet space. There areconditions (in terms of Carleson type measures, for example) that tell usexactly when C ϕ is bounded on D . See [12, 21, 30] for example.The density of the range of a composition operator is an interesting prob-lem. Bourdon and Roan studied the problem for the Hardy space (see[2, 22]) and Cima raised the problem for the Dirichlet space in [10]. In[7], we settled Cima’s problem completely: Theorem 2.
Suppose ϕ : D → D is analytic, non-constant, and G = ϕ ( D ) .Then the following two conditions are equivalent. GUANGFU CAO AND LI HE* (i) C ϕ : D → D is bounded and has dense range. (ii) ϕ is univalent and the polynomials are dense in A ( G ) . In [2], Bourdon proved the following result.
Theorem 3. If G = ϕ ( D ) , where ϕ is a weak-star generator of H ∞ , thenthe polynomials are dense in A ( G ) . It is thus natural for us to consider the following problem.
Question 4.
Does the density of polynomials in A ( G ) imply that ϕ is aweak-star generator of H ∞ ? In general, the answer is no. In fact, Sarason gave a condition in [24] for ϕ to be a weak-star generator of H ∞ , which when combined with Corol-lary 2 in [24] yields a bounded simply connected domain G such that thepolynomials are dense in A ( G ) but any Riemann map ϕ : D → G is not aweak-star generator of H ∞ ; see [2] and [18].In this paper, we give a necessary and sufficient condition for compo-sition operators to have dense range on Hardy-Sovolev spaces. Our resultshows that if ϕ is a univalent self-map of D , then the density of polynomialsin the weighted Dirichlet spaces D (cid:0) ϕ ( D ) , (1 − | ϕ − | ) − β dA (cid:1) , < β < , implies that ϕ is a weak-star generator of H ∞ .Another widely-studied problem in this area is to characterize Fredholmcomposition operators on various function spaces. We mention the follow-ing result from [8]. Theorem 5.
Suppose ϕ : G → G is an analytic self-map on a domain G ⊂ C and H is a Hilbert space of analytic functions on G . If the reproducingkernel of H has the property that K ( w, w ) → ∞ as w approaches theboundary of G , then the following conditions are equivalent. (i) C ϕ is a Fredholm operator on H . (ii) C ϕ is an invertible operator on H . (iii) ϕ ∈ Aut (Ω) . This covers several previous results in the literature. For example, Fred-holm composition operators on the Hardy space were characterized in [2,9, 11], Fredholm composition operators on the Bergman space were char-acterized in [1, 2], and Fredholm composition operators on the Dirichletspace were characterized in [10]. For the study of Fredholm compositionoperators on other spaces of analytic functions on domains in C and C n , see[2, 11, 12, 14, 16, 17, 19, 31]. OMPOSITION OPERATORS ON HARDY-SOBOLEV SPACES 5
Although the assumption K ( w, w ) → ∞ as w → ∂ Ω is a mild andnatural condition, we suspect that the result above may still be true withoutthis assumption. It is easy to construct Hilbert spaces of analytic functionswhose reproducing kernel does not satisfy K ( w, w ) → ∞ as w → ∂ Ω . Forexample, if β > / , then H β has a bounded reproducing kernel K ( z, w ) ,so the proof in [8] fails to work in this case. We will give a different proofto show that the result above still holds when β > / .2. P OLYNOMIAL APPROXIMATION AND COMPOSITION OPERATORS
In [18], S. N. Mergeljan and A. P. Talmadjan showed that if sufficientlymany slits are put in the unit disk then we can obtain a domain G such thatthe polynomials are dense in A ( G ) . By the Riemann mapping theorem,there is an analytic homeomorphism ϕ : D → G , so C ϕ has dense range in D by Theorem 2 but ϕ is not a weak-star generator of H ∞ by Corollary 2 of[24]. However, the boundary of the above domain is not a Jordan curve, theRiemann map may not be continuous up to the boundary, and ϕ does notbelong to the disc algebra A ( D ) . Furthermore, ϕ / ∈ D − β for / < β < ,where D − β = (cid:8) f ∈ H ( D ) | f ′ ∈ A − β (cid:9) is the weighted Dirichlet space with the norm k f k D − β = | f (0) | + (cid:20)Z D | f ′ ( z ) | (1 − | z | ) − β dA (cid:21) . The following result is due to P. Bourdon.
Proposition 6 (Corollary 3.7 in [2]) . Let ϕ map D univalently onto G ⊂ D .If the polynomials are dense in A ( G, (1 − | ϕ − | ) dA ) , then C ϕ : H → H has dense range. Proposition 6 extends a result of Roan [22] and supplies additional ex-amples of composition operators with dense range. As a special case of ournext result, we see that the density of polynomials in A ( G, (1 −| ϕ − | ) dA ) is also a necessary condition for the density of the range of C ϕ in H , thatis, the converse of Bourdon’s result above is also true.We will use the notion R ( C ϕ ) to denote the range of a composition oper-ator. The space on which C ϕ acts is usually obvious from the context, or itwill be specified whenever there is a possibility for confusion. Theorem 7.
Suppose β is real and ϕ is a non-constant analytic self-map of D . Then C ϕ has dense range in H β = D − β if and only if ϕ is univalent andthe polynomials are dense in D ( G, (1 − | ϕ − | ) − β dA ) , where G = ϕ ( D ) . GUANGFU CAO AND LI HE*
Proof.
First assume that C ϕ has dense range in D − β . It is easy to see that ϕ must be univalent. In fact, if there are z , z ∈ D , z = z , such that ϕ ( z ) = ϕ ( z ) , then for any f ∈ D − β we have C ϕ f ( z ) = C ϕ f ( z ) , which clearlycontradicts the assumption that the range of C ϕ is dense in D − β . To provethat the polynomials are dense in D ( G, (1 − | ϕ − ( z ) | ) − β dA ) , fix any g ∈ D ( G, (1 − | ϕ − ( z ) | ) − β dA ) . Since C ϕ g ∈ D − β and C ϕ hasdense range in D − β , we can find a sequence { p k } of polynomials suchthat k C ϕ p k − C ϕ g k→ in D − β . This, by a change of variables, isequivalent to k p k − g k → in D ( G, (1 − | ϕ − | ) − β dA ) .Conversely, assume ϕ is univalent and the polynomials are dense in thespace D ( G, (1 − | ϕ − | ) − β ) dA ) . It is clear that C ϕ is an invertible op-erator from D ( G, (1 − | ϕ − | ) − β dA ) onto D − β , with the inverse being C ϕ − . Thus for any g ∈ D − β there is an f ∈ D ( G, (1 − | ϕ − | ) − β dA ) such that C ϕ f = g . Let { p k } be a sequence of polynomials such that p k → f in D ( G, (1 − | ϕ − | ) − β dA ) . Then, by a change of variablesagain, k C ϕ p k − g k = k C ϕ p k − C ϕ f k→ in D − β . This shows that the range of C ϕ is dense in D − β . (cid:3) Choosing β = 0 and β = ± / in Theorem 7, we see that, for univalentfunctions ϕ : D → D , R ( C ϕ ) is dense in A ( D ) if and only if the poly-nomials are dense in A ( ϕ ( D ) , (1 − | ϕ − | ) dA ) , R ( C ϕ ) is dense H ( D ) if and only if the polynomials are dense in A ( ϕ ( D ) , (1 − | ϕ − | ) dA ) (see[2]), and R ( C ϕ ) is dense in D if and only if the polynomials are dense in A ( ϕ ( D )) (see [7]).Closely related to these discussions, we mention the following result ofHedberg’s from [26]. Theorem 8. If f is in the Bergman space A and if f is the derivative of aunivalent function, then f is a cyclic vector for A . Equivalently, if ϕ ∈ D is univalent, then H ∞ ( ϕ ( D )) is dense in A ( ϕ ( D )) . The proof of Theorem 8 in [26] is quite technical. If ϕ is univalent and ( ϕ − ) ′ can be approximated by polynomials on ϕ ( D ) , we will give a simplerproof for the density of H ∞ ( ϕ ( D )) in A ( ϕ ( D )) . The above condition about ( ϕ − ) ′ seems natural because, as the (normalized) area of D = ϕ − ( ϕ ( D )) ,we have Z ϕ ( D ) | ( ϕ − ) ′ | dA = 1 . Thus ( ϕ − ) ′ ∈ A ( ϕ ( D )) . Proposition 9.
Suppose ϕ is an analytic self-map of D and ϕ ∈ D . Thenthe function z belongs to R ( C ϕ ) in D if and only if ϕ is univalent and ( ϕ − ) ′ can be approximated by polynomials in A ( ϕ ( D )) . OMPOSITION OPERATORS ON HARDY-SOBOLEV SPACES 7
Proof. If ϕ is univalent and there is a sequence { p k } of polynomials suchthat Z ϕ ( D ) | ( p k − ( ϕ − ) ′ )( w ) | dA ( w ) → , then Z ϕ ( D ) | ( p k − ( ϕ − ) ′ )( w ) | dA ( w )= Z D | p k ( ϕ ( z )) − ( ϕ − ) ′ ( ϕ ( z )) | | ϕ ′ ( z ) | dA ( z )= Z D | p k ( ϕ ( z )) ϕ ′ ( z ) − ( ϕ − ) ′ ( ϕ ( z )) ϕ ′ ( z ) | dA ( z )= Z D | p k ( ϕ ( z )) ϕ ′ ( z ) − | dA ( z ) → as k → ∞ . Write q k ( z ) = Z z p k ( u ) du, k ≥ . Then q k is also a polynomial for each k and ( C ϕ q k ) ′ ( z ) = Z ϕ ( z )0 p k ( u ) du ! ′ = p k ( ϕ ( z )) ϕ ′ ( z ) . Thus Z D | ( C ϕ q k ) ′ − | dA ( z ) → , k → ∞ , so the function z belongs to R ( C ϕ ) in D .Conversely, if the function z is in the closure of R ( C ϕ ) in D , then ϕ is obviously univalent (see the beginning of the proof of Theorem 7), andreversing the calculations above implies that there is a sequence { p k } ofpolynomials such that Z ϕ ( D ) | ( p k − ( ϕ − ) ′ )( w ) | dA ( w ) → as k → ∞ . (cid:3) The following result gives a simpler proof for Hedberg’s theorem (i.e.Theorem 8) under an additional assumption.
Proposition 10.
Suppose ϕ is an analytic self-map of D and ϕ ∈ D . If thefunction z belongs to R ( C ϕ ) in D , then H ∞ ( ϕ ( D )) is dense in A ( ϕ ( D )) . GUANGFU CAO AND LI HE*
Proof.
Assume ˜ f ∈ A ( ϕ ( D )) . Once again, z ∈ R ( C ϕ ) implies that ϕ isunivalent. Thus there is an f ∈ A ( D ) such that ˜ f ( w ) = f ( ϕ − ( w ))( ϕ − ) ′ ( w ) . Let p k be the k -th partial sum of the Taylor series of f . Then k p k − f k A → , k → ∞ . By the change of variables formula, k ( p k ◦ ϕ − )( ϕ − ) ′ − ˜ f k A ( ϕ ( D )) → , k → ∞ . Since z ∈ R ( C ϕ ) , it follows from Proposition 9 that there is a sequence { q n } of polynomials such that q n converges to ( ϕ − ) ′ in A ( ϕ ( D )) . For any ǫ > choose K such that k ( p k ◦ ϕ − )( ϕ − ) ′ − ˜ f k A ( ϕ ( D )) < ǫ for k ≥ K . Choose a positive integer N such that k ( p K ◦ ϕ − )( q n − ( ϕ − ) ′ ) k A ( ϕ ( D )) < ǫ for n ≥ N. Then for n ≥ N we have k ( p K ◦ ϕ − ) q n − ˜ f k A ( ϕ ( D )) ≤ k ( p K ◦ ϕ − ) q n − p K ◦ ϕ − ( ϕ − ) ′ k A ( ϕ ( D )) + k p K ◦ ϕ − ( ϕ − ) ′ − ˜ f k A ( ϕ ( D )) < ǫ. This shows that H ∞ ( ϕ ( D )) is dense in A ( ϕ ( D )) . (cid:3) If ϕ ∈ A ( D ) and the image ϕ ( D ) has infinite area, then the polynomialsmay not be dense in A ( ϕ ( D )) . Here is an example.Let f ( z ) = 1 / √ z be the principal branch of / √ z on C \ [0 , + ∞ ) . Thenthe function ϕ ( z ) = f (1 + z ) = 1 √ z is analytic function on D . It is obvious that ϕ belongs to A ( D ) and isunivalent in the open unit disc. However, ϕ ′ / ∈ A ( D ) , that is, the region ϕ ( D ) has infinite area. This implies that the polynomials are not dense in A ( ϕ ( D )) . In fact, if g ( w ) = ϕ − ( w ) = 1 w − , then g / ∈ A ( ϕ ( D )) , but g ′ ∈ A ( ϕ ( D )) . However, g ′ cannot be approxi-mated by polynomials in A ( ϕ ( D )) . OMPOSITION OPERATORS ON HARDY-SOBOLEV SPACES 9
This example also implies that the Dirichlet space is not necessarily con-tained in the Bergman space on a general domain in the complex plane. See[7] and additional references there.
Proposition 11.
Suppose β < and ϕ ∈ D − β is univalent. Then C ϕ isan invertible operator from D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) onto D − β withthe inverse being C ϕ − . Moreover, C ϕ preserves the Dirichlet semi-norms.Proof. This follows from an easy change of variables. We leave the routinedetails to the interested reader. (cid:3)
Lemma 12 ([24]) . A sequence { ψ n } ∞ in H ∞ converges weak-star to thefunction ψ if and only if it is uniformly bounded and converges to ψ at everypoint of D . Lemma 13 (Mergelyan’s Theorem [25]) . If K is a compact subset of theplane whose complement is connected, then every complex function that iscontinuous on K and analytic on its (topological) interior can be uniformlyapproximated on K by polynomials. It follows from Proposition 11 that if / < β < and if ϕ ∈ D − β isunivalent then ϕ − ∈ D ( ϕ ( D ) , (1 − | ϕ − ( z ) | ) − β dA ) . A standard argument shows that the operators from Proposition 11 satisfy C ∗ ϕ − ˜ K w = K ϕ − ( w ) , C ∗ ϕ K z = ˜ K ϕ ( z ) , where ˜ K w ( u ) = ˜ K ( u, w ) and K z ( v ) = K ( v, z ) are the reproducing ker-nels of D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) at w ∈ ϕ ( D ) and of D − β at z ∈ D , respectively. Since K ( w, z ) is continuous on D × D , we know that ˜ K ( u, v ) is also continuous on ϕ ( D ) × ϕ ( D ) . Hence each function f in D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) is continuous on ϕ ( D ) by properties of thereproducing kernel. In particular, ϕ − is continuous on ϕ ( D ) . Furthermore,by Lemma 13, ϕ − can be uniformly approximated on ϕ ( D ) by polynomi-als. Proposition 14.
Suppose / < β < and ϕ is a univalent analytic self-map of D with ϕ ∈ D − β . Then Lat ( M βϕ ) = Lat ( M βz ) , where M βϕ and M βz are multiplication operators on the weighted Bergman space A − β and Lat ( M βϕ ) and Lat ( M βz ) are their invariant subspace lattices.Proof. Since ϕ ∈ D − β , it is clear that M βϕ is bounded on A − β . Lemma 12implies that there is a sequence { p k } of polynomials such that p k ( z ) → ϕ − ( z ) uniformly on D , and this implies that p k ( ϕ ( z )) → f ( z ) = z uni-formly on D . Thus Z D | ( p k ( ϕ ) − z ) g | (1 − | z | ) − β dA → , g ∈ A − β . This shows that M βp k ( ϕ ) converges to M βz in the weak operator topology.Hence, Lat ( M βϕ ) ⊂ Lat ( M βz ) . The reversed inclusion is obvious, so wehave Lat ( M βϕ ) = Lat ( M βz ) . (cid:3) Corollary 15.
Suppose / < β < and ϕ is a univalent analytic self-mapof D with ϕ ∈ D − β . Then C ϕ has dense range in D − β if and only if H ∞ ( ϕ ( D )) is dense in A ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) .Proof. This is a direct consequence of Theorem 7, because every boundedanalytic function can be approximated by polynomials in the norm topologyof A ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) . (cid:3) Theorem 16.
Suppose / < β < and ϕ is a univalent analytic self-mapof D with ϕ ∈ D − β . If C ϕ has dense range in D − β , then ϕ ′ is a cyclicvector for both M βz and M βϕ on D − β .Proof. Define E ϕ : A ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) → A − β by E ϕ ( f )( z ) = ( f ◦ ϕ )( z ) ϕ ′ ( z ) . Similarly, define E ϕ − : A − β → A ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) by E ϕ − ( f )( w ) = ( f ◦ ϕ − )( w )( ϕ − ) ′ ( w ) . Direct calculation shows that both E ϕ and E ϕ − are isometric operators and E ϕ E ϕ − = I A − β , E ϕ − E ϕ = I A ( ϕ ( D ) , (1 −| ϕ − | ) − β dA ) , are identity operators. Thus for any function f ∈ A − β there is a function ˜ f ∈ A ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) such that f ( z ) = ˜ f ( ϕ ( z )) ϕ ′ ( z ) .Assume { p k } is a sequence of polynomials such that Z ϕ ( D ) | p k ( w ) − ˜ f ( w ) | (1 − | ϕ − | ) − β dA ( w ) → , k → ∞ . OMPOSITION OPERATORS ON HARDY-SOBOLEV SPACES 11
Then Z D | p k ( ϕ )( z ) ϕ ′ ( z ) − f ( z ) | (1 − | z | ) − β dA ( z )= Z D | p k ( ϕ )( z ) ϕ ′ ( z ) − ˜ f ( ϕ ( z )) ϕ ′ ( z ) | (1 − | z | ) − β dA ( z )= Z D | p k ( ϕ ( z )) − ˜ f ( ϕ ( z )) | | ϕ ′ ( z ) | (1 − | z | ) − β dA ( z )= Z ϕ ( D ) | p k ( w ) − ˜ f ( w ) | (1 − | ϕ − | ) − β dA ( w ) → as k → ∞ . Not p k ( ϕ )( z ) ϕ ′ ( z ) = p k ( M ϕ )( ϕ ′ )( z ) , this shows that ϕ ′ is acyclic vector of M ϕ on D − β . By Proposition 14, ϕ ′ is also a cyclic vectorfor M z on D − β . (cid:3) Theorem 17.
Suppose / < β < and ϕ is an analytic self-map of D such that C ϕ is bounded on D − β = H β . If R ( C ϕ ) is dense in D − β , then ϕ is a weak-star generator of H ∞ .Proof. For any f ∈ D − β there is a sequence { p k } of polynomials suchthat k C ϕ p k − f k D − β → . Note that | p k ( ϕ )( z ) − f ( z ) | = |h p k ( ϕ ) − f, K z i| ≤ k p k ( ϕ ) − f k D − β k K z k D − β , where K z is the reproducing kernel of D − β at z . Since / < β < , thefunction z
7→ k K z k D − β = p K ( z, z ) is bounded on D . Thus p k ( ϕ )( z ) converges uniformly to f ( z ) . Furthermore, k p k ( ϕ ) − f k ∞ → as k → ∞ .If f ∈ H ∞ , then for any < r < , f r ( z ) = f ( rz ) ∈ D − β . Choose r n ∈ (0 , such that r n → , then f r n w ∗ −→ f in H ∞ by the dominatedconvergence theorem. For any n, there is a sequence of polynomials { p ( n ) k } such that k p ( n ) k ( ϕ ) − f r n k ∞ → as k → ∞ . Hence, we may find subse-quence k n such that p ( n ) k n ( ϕ ) w ∗ −→ f in H ∞ . It follows that { C ϕ p k : p k is a polynomial } = { p k ( ϕ ) : p k is a polynomial } is weak-star dense in H ∞ . (cid:3) It is clear that if C ϕ maps H β to itself and / < β < , then ϕ ∈ H β ⊂ A ( D ) , the disk algebra. If β ≥ , then z / ∈ D − β , so the polynomialscannot be dense in D − β . In this case, we need to consider higher orderderivatives. From the discussion above, we see that the density of polynomials in D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) for / < β < implies that ϕ is a weak-star generator of H ∞ . On the other hand, ϕ being a weak-star generator of H ∞ implies that the polynomials are dense in the Dirichlet spaces D and D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) for all β ≤ / .It is intriguing for us to find some relationship between the density of R ( C ϕ ) on two different spaces D − β and D − β for / < β , β < .We already know that if C ϕ has dense range in D − β for some / < β < , then ϕ must be a weak-star generator of H ∞ , which implies that ϕ isunivalent on the closed unit disc ¯ D . However, this does not imply that thepolynomials are dense in D ( ϕ ( D ) , (1 −| ϕ − | ) − β dA ) for all / < β < .In fact, for any given / < β < β < we can find an analytic self-map of D such that ϕ ∈ D − β \ D − β . Then the polynomials are notdense in D ( ϕ ( D ) , (1 − | ϕ − | ) − β dA ) but they are dense in D ( ϕ ( D ) , (1 −| ϕ − | ) − β dA ) . Hence, there exists an analytic self-map ϕ of D such that C ϕ has dense range in D − β but does not have dense range in D − β . Thisalso shows that ϕ being a weak-star generator of H ∞ does not imply thatpolynomials are dense in D ( ϕ ( D ) , (1 −| ϕ − | ) − β dA ) for all / < β < .It is well-known that if ϕ is a weak-star generator of H ∞ , then the poly-nomials are dense in the Bergman space A ( ϕ ( D )) , but the converse is nottrue in general. The following theorem gives a condition for the converse tohold for certain analytic self-maps of D . Theorem 18.
Suppose / < β < and ϕ ∈ D − β is an analytic map-selfof D such that the polynomials are dense in A ( ϕ ( D )) . Then the followingstatements are equivalent to each other. (i) { C ϕ p : p is a polynomial } is dense in A − β . (ii) ϕ is a weak-star generator of H ∞ . (iii) ϕ is univalent on the open unit disc.Proof. If { C ϕ p : p is a polynomial } is dense in A − β , then ϕ is clearlyunivalent on D . See the beginning of the proof of Theorem 7. This showsthat (i) implies (iii).To prove (iii) implies (ii), assume that ϕ ∈ D − β is univalent on theopen unit disc. Then ϕ is also univalent on the closed unit disc by Corol-lary 3.5 in [2] and the continuity of ϕ on D . Thus, ϕ − is continuous on ϕ ( D ) . By Lemma 12, there is a sequence { p k } of polynomials such that p k converges uniformly to ϕ − . Then p k ◦ ϕ converges uniformly to f ( z ) = z .This implies that ϕ is a weak-star generator of H ∞ since z is the weak-stargenerator of H ∞ .Finally, let us assume that (ii) holds. Then for any f ∈ H ∞ there existsa sequence { p k } of polynomials such that p k ( ϕ ( z )) ϕ ( z ) → f ( z ) pointwise OMPOSITION OPERATORS ON HARDY-SOBOLEV SPACES 13 on D and {k p k k ∞ } is bounded. By the dominated convergence theorem,we have k C ϕ p k − f k A − β → as k → ∞ . This shows (ii) implies (i) andcompletes the proof of the theorem. (cid:3)
3. F
REDHOLM COMPOSITION OPERATORS
For β > / we have H β = H ∞ β ⊂ A ( D ) , where H ∞ β is the multiplieralgebra of H β ; see [20]. Since the reproducing kernel of H β is bounded inthis case, the proof in [8] fails. However, we still have the following result. Theorem 19.
Suppose β > / and ϕ : D → D is analytic. Then C ϕ isFredholm on H β if and only if ϕ ∈ Aut ( D ) , the automorphism group of D .Proof. Obviously, we only need to prove that the Fredholmness of C ϕ on H β implies that ϕ ∈ Aut ( D ) . If C ϕ is Fredholm, then ϕ must be univalenton D (see the proof of Theorem 4 in [5]). Note ϕ ∈ H β implies ϕ ∈ A ( D ) , the disc algebra, for β > / . Thus, ϕ is univalent except for alimited number of points on ∂ D . Otherwise, we can find two sequences { z n } and { w n } in ∂ D such that ϕ ( z n ) = ϕ ( w n ) . For β > / the kernelfunction K z ( w ) is also well-defined for z ∈ ∂ D and we can easily provethat C ∗ ϕ K z = K ϕ ( z ) . Thus C ∗ ϕ ( K z n − K w n ) = K ϕ ( z n ) − K ϕ ( w n ) = 0 . However, the functions K z n − K w n are clearly linearly independent, whichcontradicts to the Fredholmness of C ϕ . This shows that ϕ must be univalentexcept for a limited number of points on ∂ D .Now ϕ ( D ) is a simply connected domain. If ϕ is not a surjection, thenthere are two cases for us to consider.The first case is when ∂ D \ ∂ϕ ( D ) = ∅ . Then for any ζ ∈ ∂ D \ ∂ϕ ( D ) there is a neighborhood U ( ζ , δ ) = { z || z − ζ | < δ } of ζ such that U ( ζ , δ ) ∩ ϕ ( D ) = ∅ . Let f ζ ( z ) = (1 + z ¯ ζ ) / be the peak function at ζ and consider f n ( z ) = f nζ ( z ) k f nζ k H β , n = 1 , , , · · · . According to [6], the sequence { f n } converges to weakly in H β . Since C ϕ f n = f n ( ϕ ( z )) = ( ϕ ( z )¯ ζ ) n k f nζ k H β , it is not difficult to check that k C ϕ f n k → , which implies that C ϕ cannotbe Fredholm, a contradiction. The second case is when ∂ D $ ∂ϕ ( D ) . Since ϕ ( D ) is simply connected,we see that ∂ϕ ( D ) \ ∂ D contains an arc L from a boundary point of D into the inner of D . As ∂ϕ ( D ) = ϕ ( ∂ D ) is a closed curve, L must be anoverlapping arc of two arcs with opposite directions. Since ϕ is univalentexcept for a limited number of points on ∂ D , we see that the pre-image of L under ϕ has two non-intersecting arcs in ∂ D . Let l and l be two arcs suchthat ϕ ( l ) = ϕ ( l ) = L as a subset of ∂ D but ϕ ( l ) and ϕ ( l ) have oppositedirections in ∂ϕ ( D ) . Choose sequences { z n } ⊂ l and { w n } ⊂ l such that ϕ ( z n ) = ϕ ( w n ) . Then C ∗ ϕ ( K z n − K w n ) = K ϕ ( z n ) − K ϕ ( w n ) = 0 . However, the functions K z n − K w n are clearly linear independent, whichcontradicts to the Fredholmness of C ϕ again.Therefore, ϕ must be a surjection, and hence an automorphism of the unitdisk. (cid:3) R EFERENCES [1] J. Akeroyd and S. Fulmer, Closed-range composition operators on weighted Bergmanspaces, IntegralEquationsOperatorTheory, (2012), 103-114.[2] P. Bourdon, Density of the polynomials in Bergman spaces, Pacific J. Math., (1987), 215-221.[3] P. 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