SOME PROPERTIES OF THE CANONICAL DIVISOR IN THE BERGMAN SPACE

Transcription

1 SOME PROPERTIES OF THE CANONICAL DIVISOR IN THE BERGMAN SPACE Cyrus Luciano 1, Lothar Narins 2, Alexander Schuster 3 1 Department of Mathematics, SFSU, San Francisco, CA 94132,USA lucianca@sfsu.edu 2 Department of Mathematics, SFSU, San Francisco, CA 94132,USA lnarins@sfsu.edu 3 Department of Mathematics, SFSU, San Francisco, CA 94132,USA schuster@sfsu.edu Abstract In this paper we develop a formula for the canonical divisor of a finite A 2 zero set that includes points of order greater than one. Using this formula, we examine properties of the canonical divisor. In particular, we show that a sufficient condition of Aleman and Richter [1] for the boundedness of the canonical divisor is not necessary. AMS Subject Classification: 30H05, 46E15 Key Words: Bergman space, extremal function, canonical divisor 1. Introduction For 0 < p < +, the Bergman space A p is the set of functions analytic in the unit disk D with f p p = 1 fz p daz <, π D where da denotes Lebesgue area measure. The space obtained for p = 2, A 2, is a Hilbert space with inner product f, g = 1 fzgzdaz π and reproducing kernel D k ζ z = 1 ζ z 2, corresponding to the point evaluation functional for ζ D. That is, we have fa = f, k a,

2 or, in other words, fa = 1 π for all f A 2 and all a D. Taking the derivative of 1, we get for all f A 2 and natural numbers j, where D fz1 za 2 daz 1 f j a = f, k j a, 2 k j az = j + 1!z j 1 az 2 j. The norm of an A 2 function can be given in terms of its Taylor coefficients. Namely, if fz = c n z n, f 2 = c n 2 n We have suppressed the subscript when writing the norm because we will be dealing almost exclusively with the case p = 2. We say that a sequence Γ of points in D is an A p zero sequence if there is a nontrivial function in A p that vanishes precisely on Γ. If Γ contains points of order m, then we require that the function along with its m 1 first derivatives vanish there. Associated to each A p zero sequence is a canonical divisor. If Γ = {z j } j is an A p zero sequence which avoids the origin and where m j is the order of z j, we consider the associated subspace IΓ = {f A p : fz j = f z j = f 2 z j = = f m j 1 z j = 0, j = 1,... } and the extremal problem sup{ g0 : g IΓ, g p 1}. 4 If 0 occurs in Γ with multiplicity n, we consider instead the problem of maximizing g n 0, subject to the same constraints as above. This problem has a solution G Γ for 0 < p < + which is unique if we stipulate that G Γ 0 > 0 and as was found by Hedenmalm [6] for p = 2 and later generalized by Duren, Khavinson, Shapiro and Sundberg [3] to all other values of p plays a role in A p similar to that of the Blaschke product in the Hardy space H p. In particular, if Γ is an A p zero sequence, then G Γ vanishes precisely on Γ and is a contractive divisor in A p, that is, f/g Γ p f p

3 for all f A p that vanish along Γ. It is in general difficult to determine a formula for G Γ except in some very special cases. For example, if Γ consists of the single point α occurring with multiplicity m, where ϕ α z = α z 1 αz and Gz = C [ϕ α z] m{ 1 + mp 2 C = α m { α 1 + mp 1 2 α 2 } 1/p. 1 α ϕα z } 2/p, 5 See also Hansbo [5], who found a formula for the canonical divisor of a set consisting of two distinct points. 2. The extremal function for finite zero sets Let now Γ = {a 1,..., a n }, where a i occurs with multiplicity m i. We then have the following theorem, which is known, but we have not found a proof of it, so we include it here for the sake of completeness. The argument given below is a modification of a discussion on p of [4]. Theorem 1. When p = 2, the extremal function G Γ can be expressed as a linear combination of the functions { ka j i },...,n,j=0,...,m i plus a constant. 1 Proof. First, we note that g is a solution to the extremal problem 4 if and only if g/g 0 is a solution to the following extremal problem: min f. f IΓ f0=1 Then the problem is to find the function f F of smallest norm, where F = {f IΓ : f 0 = 1}. Since the family F is a closed convex subset of the uniformly convex space A 2, it has a unique element F of smallest norm. See [4] for definitions and a proof of this fact. Note that F = because it contains suitable polynomials. A variational argument now shows that F must be orthogonal to every function in the family F 0 = {f IΓ : f 0 = 0}. To see this, let g F 0 be arbitrary and note that F + λg F for each λ C. Thus, since G Γ is the solution to 4, it is clear that F 2 F + λg 2 = F 2 + 2Re {λ g, F } + λ 2 g 2,

4 so that 2Re {λ g, F }+ λ 2 g 2 0. Now write λ = re iθ to arrive at 2Re { e iθ g, F } + r g 2 0. Letting r 0, we conclude that Re { e iθ g, F } = 0 for every e iθ. Thus g, F = 0 for each g F 0, as asserted. Observe now that by virtue of the reproducing property 2, each of the functions k j a i, where i = 1,..., n, j = 0,..., m i 1 as well as the function 1 is orthogonal to F 0. We claim that F is actually a linear combination of these functions. To see this, let M be the set of all such linear combinations h = λ 0 + n m i 1 j=0 λ ij k j a i, λ ij C. Since M is a closed subspace of A 2, our extremal function has a unique decomposition as F = h + g, where h M and g M. Here M denotes the orthogonal complement of M, the set of all functions in A 2 that are orthogonal to every function in M. But it is easy to see that M = F 0. Since the extremal function F is orthogonal to F 0, it follows that F = h M, as asserted. Indeed, g 2 = g, g + h = g, F = Calculating the canonical divisor As in the previous section, we assume that Γ = {a 1,..., a n }, where a i occurs with multiplicity m i. By Theorem 1, we can write G Γ = λ 0 + n m i 1 j=0 λ ij k j a i for some coefficients λ 0, λ ij C. By adapting a technique from Cima and Derrick [2], who considered zero-sets whose members all have order 1, we can find these coefficients. For convenience, let us reindex the second sum above to start at j = 1, and reindex the coefficients the same way: G Γ z = λ 0 + n m i j!λ ij z j 1 1 a i z j+1.

5 Expanding each z j 1 1 a i z j 1 as a power series yields G Γ z = λ 0 + n m i λ ij k=j 1 j 1 k + 1 l a k+1 j i z k. Since the product on the inside gives zero for k < j 1, we may start the power series from 0 for all i and j without changing the result. Then, moving the constant terms outside the series, we can rewrite this as n n m i j 1 G Γ z = λ 0 + λ i1 + k + 1 λ ij k + 1 l a k+1 j i z k = G Γ 0 + k=1 l=0 n m i j 1 k + 1 λ ij k + 1 l a k+1 j i z k z k. k=1 By 3, and since the extremal function has unit norm, we get G Γ 2 = 1 = G Γ 0 2 n m i j k + 1 λ ij k + 1 l a k+1 j i z k. 6 k=1 Now for each pair p, q such that 1 p n and 1 q m p, we take the q 1th derivative of G Γ at a p, yielding G q 1 Γ a p = 0 = G Γ 0 δ q1 + k + 1 k=1 q 1 k + 1 l n m i j 1 λ ij k + 1 l a i k+1 j a k+1 q p, where δ ij is the Kronecker delta function. Multiplying each of the above equations by λ pq and summing over all pairs, we find z k

6 m n p λ pq G q 1 Γ a p = 0 p=1 q=1 n n m i j 1 = G Γ 0 λ p1 + k + 1 λ ij k + 1 l p=1 k=1 = G Γ 0 G Γ 0 λ 0 n m i j 1 2 k+1 j + k + 1 λ ij k + 1 l a i. k=1 Substituting 6 into this equation yields G Γ 0 G Γ 0 λ GΓ 0 2 = 0. a i k+1 j From this, the following generalization of the theorem of Cima and Derrick becomes apparent: Theorem 2. λ 0 G Γ 0 = 1. Using this result, we can find all the other coefficients as follows. For each i and j, define j!z j 1 F ij z = 1 a i z j+1. Then for each p and q, let S pq be the m p m q matrix given by S pq ij = F i 1 qj a p. Now we define the 1 + n k=1 m k 1 + n k=1 m k block matrix M by 1 r 1 r n c 1 S 11 S 1n M =......, c n S n1 S nn where each r k is a 1 m k matrix with the first entry 1 and all other entries 0, and each 2

7 c k is a m k 1 matrix with the first entry 1 and all other entries 0. In this vein, define λ 0 λ 11. λ λ = 1m1. λ n1. and b = λ nmn G Γ 0 0. Then, the system of equations above can be represented by the matrix equation Mλ = b, which we can rewrite as λ = M 1 b. Let M 1 = m ij. Then in particular we have λ 0 = G Γ 0m 11, which by Theorem 2 implies that λ 0 = m 11 and that m We therefore have where t = 1 + n k=1 m k. λ = 1 m11 0. m 11 m 21. m t1, 4. Properties of the canonical divisor Using the results outlined in the previous section, we wrote a Mathematica R 6 program that calculates the canonical divisor of a finite A 2 zero set. This program can be found at and was employed to examine and discover properties of the canonical divisor. 4.1 The boundedness of G Γ. It is an open problem to find a necessary and sufficient condition on Γ for the canonical divisor G Γ to be bounded. In this subsection we show that the sufficient condition given in [1] is not necessary.

8 It follows from a theorem of Horowitz [7] see p. 146 of [4], for example that a necessary condition on Γ is that it be a finite union of uniformly separated sequences. We say that Γ = {z j } is uniformly separated if there is a constant δ > 0 such that z i z j 1 z i z j δ for j = 1, 2,.... On the other hand, Aleman and Richter [1] showed that if i j 1 z j 2 sup z D 1 z j j z <, 7 then G Γ is bounded. The boundedness of the canonical divisor of an infinite zero set Γ may be deduced from the behaviour of the divisors of finite subsets of Γ. If Γ = {z j } is ordered so that z 1 z 2..., we define Γ n = {z 1,..., z n }. Then G Γn converges locally uniformly on D to G Γ cf. p. 145 of [4]. Consider now the sequence Γ = {z j }, where z j = 2 j + ij2 j. For a sequence Γ, we define MΓ = sup z D G Γ z. The table below shows MΓ n for selected values of n. Table 1: A sequence with bounded divisor that doesn t satisfy 7 n MΓ n It appears that G Γn z 6 for all z D and all n, which implies that G Γ is bounded. On the other hand, it is not difficult to show that the condition 7 does not hold for Γ. To see this note that it follows from Fatou s lemma that 7 holds if and only if But 1 z j 2 sup z D 1 z j j z j 1 z j 2 sup z D 1 z j j z <, 1 z j 2 1 z j = j 2 j+1 1 j 2 log j 1 + j 2 log 2 2. This last sum is comparable to 1 and so diverges. This indicates that the condition j 7 is not necessary for G Γ to be bounded.

9 4.2 The location of the maximum of G Γ. One can see by Theorem 1 that for a finite set Γ, the canonical divisor can be extended analytically to the closed unit disk D, and so, by the Maximum Modulus Principle, the maximum of G Γ will occur on the boundary circle, that is MΓ = sup z D G Γ z. We define mγ to be the point or points z on the boundary of D such that G Γ z = MΓ. It follows from the representation 5 that when Γ consists of the single point α with arbitrary multiplicity, then mγ lies on the ray from the origin through α. Our simulations indicate that this holds for an arbitrary sequence on a ray. In addition, if Γ lies on a diameter through the origin, then mγ will occur on that diameter. It is not however true that for arbitrary Γ, the maximum mγ will lie on a ray through one of the members of Γ. For example, m{1/2e iπ 4, 1/2e iπ 4 } occurs at approximately e 0.73i and e 0.73i. If Γ lies on a diameter through the origin, how does one determine which of the two endpoints constitutes mγ? In general this is most likely a difficult question. Using our program, we are able to make some conjectures about special cases. Assume that Γ = {a, b}, where a < 0 < b. Suppose that m is the order of a, and n is the order of b. A role will be played by p m,n, which is defined as p m,n = mn + 1 nm + 1. We determine mγ for various values of a, b, m, n and give our results in the table below. Table 2: The location of the maximum of the divisor of a sequence of two points The order of a and b The magnitude of a and b mγ m = n a < b 1 m = n a > b -1 m < n a < b 1 m < n b < a, p m,n? m < n b > p m,n 1 The question mark in the fourth row indicates that mγ could be 1 or 1, namely, if b < p m,n, then mγ = 1 if a is sufficiently close to 1, and otherwise mγ = 1. We point out to the reader the surprising fact that if b > p m,n, then mγ is independent of a.

10 4.3 The maximum value of G Γ In this subsection we examine how the maximum value of G Γ can depend on Γ. The formula 5 indicates that for a single point α of multiplicity m, MΓ increases as α increases to 1. A natural question is whether this is true for arbitrary zero sets. Does increasing the modulus of one or more members of Γ increase the maximum value of G Γ? Suprisingly, the answer is no, even for a set consisting of two points. For example, our calculations reveal that M{0.8, 0.95} > M{0.95, 0.95}. On the other hand, it does appear that lim a 1 M{a, c} > M{b, c} for all b, c 0, 1. This would generalize to the following conjecture: If Γ is a zero-set contained in the positive real axis, and a 0, 1 is not in Γ, then lim a 1 M Γ {a} > M Γ {b} for any b 0, 1. Acknowledgement. All three authors were supported by NSF grant DMS References [1] Aleman, A. and Richter, S. Single point extremal functions in Bergman-type spaces. Indiana Univ. Math. J , no. 3, [2] Cima, J. A.; Derrick, W. R. Extremals for subspaces of the Bergman space. Houston J. Math , no. 4, [3] Duren, P.; Khavinson, D.; Shapiro, H. S.; Sundberg, C. Contractive zero-divisors in Bergman spaces. Pacific J. Math , no. 1, [4] Duren, Peter and Schuster, Alexander, Bergman spaces. American Mathematical Society, Providence, RI, [5] Hansbo, J. Reproducing kernels and contractive divisors in Bergman spaces. English, Russian summary Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. POMI , Issled. po Linein. Oper. i Teor. Funktsii. 24, , 217; translation in J. Math. Sci. New York , no. 1, [6] Hedenmalm, H. A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math , [7] Horowitz, C. Factorization theorems for functions in the Bergman spaces. Duke Math. J ,