Math. Nachr. 173 (1995) 169-175

On Linear Functionals of Rational Type over H ( D )

By RICHARD FOURNIER') of Montreal

(Received May 3, 1993) (Revised Version April 14, 1994)

Abstract. Let H(ID) denote the set of analytic functions in the unit disk ID of the complex plane, endowed with the topology of uniform convergence on compact subsets of ID. Let A denote the space of continuous linear functionals over H(ID). In this paper we obtain a characterization of those i E A such that

A(zkB(z)) = 0 for k = 0, 1, 2, ... ,

where B is a fixed function in H ( Q . We also include a discussion of the sharpness of this characterization and an application to some extremal problem concerning A .

1. Introduction

Let H(ID) denote the set of analytic functions in the unit disk ID of the complex plane (c. Endowed with the topology of uniform convergence on compact subsets of ID, H(ID) is a locally convex topological vector space. For functions

a m

f (z) = 1 f n z n , g(z) = C gnzn in H(D) n = O i l = O

the convolution (or Hadamard product) f * g, defined by m

f * g ( 4 = c fngnz" > n = O

also belongs to H(lD). It is well known [7] that if at least one functionfor g lies in the set H J D ) of functions analytic in the closed unit disc, thenf * g also lies in H,(ID). The dual space A of all continuous linear functionals on H(ID) may be identified with Hc(ID), due to the following basic theorem of TOEPLITZ [8].

' ) This work was carried out while the author visited Universitat Wiirzburg (Germany). The author would like to thank ST. RUSCHEWEYH for interesting discussions. Support from an ACC (Quebec) grant is acknowledged.

170 Math. Nachr. 173 (1995)

Theorem 1’. i E A if and only if there exists a function g E H,(ID) such that

i(f) = g * f(1) .for all f~ H(lD) .

The object of this note is to study a condition on a sequence { f n } c H(D) forcing a functional i E A to be of a certain type if I. annihilates { f n } . Except for the trivial condition i (z”) = 0 for all n large enough, very few such conditions are explicitly given in the literature. We mention the following result due to HALLENBECK and MACGREGOR ([6], Theorem 7.29) whose proof is based on Runge’s approximation Theorem.

Theorem 1“. Let I E A such that I.(f”) = 0 (n = 2, 3,4, . . .) where J’ E H(ID) is univalent andf’(0) = 0. Then

d(h) = nh(0) + bh’(0) ,for all h E H(D) ,

,for some a , h E CC.

DUREN ([4], page 187) alors considered a condition of this type. We shall prove.

Theorem 1. Let I E A and B E H(ID) such that A(zkB(z)) = 0 ( k = 0, 1,2, ...,). Then

where PI, p2, ..., PN are zeros ?f B in I D of multiplicity at leasf resprctively M(1), M(2), ..., M(N) and the K , , , are arbitrary complex numbers.

Remark. BRICKMAN [2] studied functionals of rational type and gave a list of equivalent definitions for these functionals. What we are really doing in this paper is adding a new equivalent condition (namely the hypothesis of Theorem 1) to this list. The author thanks T. H. MACGREGOR for bringing the work of BRICKMAN to his attention.

Remark. The result of Theorem 1 is reminiscent of invariant subspace theory according to ARNE BEURLINC [I]. BEURLING’S results are formulated in the Hilbert space H 2 of square summable power series. A closed subspace of H , is invariant under the difference-quotient

transformation, which takes f ( z ) into f(z) - j(’), if and only if its orthogonal complement

is invariant under the shift, which takes f ( z ) into zJ’(z). BEURLINC [l], almongst other important results, determines the invariant subspaces of the shift using the Nevanlinna factorization theory for functions which are analytic and of bounded type in the unit disk. The duality of H , with itself can be replaced by a duality between a larger space and a smaller space, namely H(ID) and A, by using Toeplitz Theorem 1’. Remark that H ( Q , endowed with the topology of uniform convergence on compact subsets of ID, is not a normable space. However the closed subspaces of H(lD) which are invariant under the shift are in one-to-one correspondance with the closed subspaces of A which are invariant under the difference-quotient transformation. What is shown in Theorem 1 is that every proper closed subspace of A which is invariant under the difference-quotient transformation has

Z

Fournier, On Linear Functionals of Rational Type over H(lD) 171

with 1

finite dimension; such a subspace is the vector span of functions of the form (1 - Bz)'

f i E ID and t a positive integer. As a consequence, every nontrivial closed subspace of H(D) which is invariant under the shift has finite codimension; such a subspace is determined by the common zeros of its elements and their multiplicities. The author would like to thank an anonymous referee for his useful remarks concerning the relation between invariant subspaces and Theorem 1.

2. Proof of Theorem 1

a,

Let B(z) = 1 Bnzn. Due to Theorem l', the hypothesis l ( zkB(z) ) ) = 0 amounts to n = O

f A n + k B , = O , k = 0 , 1 , 2 , . . . , n = O

(1) m

where the power series A(z) = Anzn has radius of convergence e > 1. It is seen from n = O

(1) that A(z) B(l/z) is an anaiytic function of l/z in the annulus 1 < IzI < e. Equivalently there exists an identity

( f A$) ( f = n = f 1 c t n Z - n , 1 < (ZI < e, n=O n = O

a

where the crbs can be computed in terms of the Ahs and Ks. The power series 1 and

converges for all u in the annulus l/e < IuI < 1 and is therefore also convergent for ( u I < 1. We define

( 2 )

Clearly A(z) = A(z) if 1 < IzI < e and A" is a meromorphic extension of A outside the disk IzI < e whose poles correspond to zeros of B in the disk IzI 5 l/e < 1. It follows that A is meromorphic over (c with only a finite number of poles in (c. Moreover, we see clearly from (2) that lim A"(z) exists and is either finite of infinite. The function A is therefore rational

and by (2) , if [ is a pole (order rn) of A in the extended complex plane then - is a zero

(order 2 rn) of B in IzI I l/e. In other words

n = 1

a A(z) = 1 ct,Z-" / 2 B,z-", 1 < I Z I .

n = 1 n = O

:-.a 1 . i

with K r , j E C, each P j is a zero (+ 0) of B of multiplicity at least M ( j ) in (zI 5 lie and p is a polynomial of degree less or equal to the multiplicity of 0 (considered as a zero of B). The conclusion of our Theorem follows from Theorem 1' upon noticing that

172 Math. Nachr. 173 (1995)

Remark. If 1 < Q < n3 is the radius of convergence of A,z“, the function A has at

least one pole on the circle IzI = Q ; this means that condition (1) implies that B has a zero on IzI = lie. If B(0) + 0, p(z) is a polynomial ofdegree 0; it is then easily seen that p(z) = 0.

n = O

Remark. Once Theorem 1 is established it is not very difficult to see that the following

Let 2” E A , f ’ ~ H(ID) u n i o a h t withf’(0) = 0 and B E H ( I D ) such that extension of both Theorem 1 and Theorem 1” holds (we omit the proof):

I . ( ~ ( Z ) ~ B ( z ) ) = 0 , k = K , K + 1 ..., ~ h u c . K is cm integrr 2 0 .

T l m , for all h E H(ID). N M ( j ) - 1 K t M - 1

i ( h ) = c c Kj,rh‘”(pj) + c KO,rh(t’(o) j = l t = O t = O

here pl, f l 2 , ..., P N N Y P zeros of B in ID \ (0 ) of nzultiplicity at least respwtively M(1), M (2), . . . , M ( N ) , B has ci z ~ r o of multiplicity nt l a s t M at the origin and K i , , E C.

3. Sharpness of condition (1)

In this section we verify by an example that the conclusion (3) may not follow from (1) for a power series A ( z ) with radius of convergence Q = 1. Let [r,);Ll be an increasing sequence of real numbers with 0 < r, < 1 and

(4) *, 1 (1 - r , ) < so.

n = l

For each N > 1 we define N 1 (1 - r,)2

AN(Z) := c - n ! 1 - r,z

Clearly A ( z ) := lim A N ( z ) exists uniformly on every compact set K of the plane such that N-CX

1 1 - 4 K , ti = 1,2, . . . . In particular, A E H(ID) and has a pole of order 1 at each ~ and an r n r. essential singularity at z = 1. The function A is clearly not meromorphic. Because

and because the convergence is uniform vn compact subsets of ID we have r*m ( I - r,)’ rj, ~~~. A(z) := f A j z j , JzI < 1 , with Aj =

j = O n ! Now let

Fournier, On Linear Functionals of Rational Type over H(D) 173

By (4), B is a convergent Blashke product in IzJ < 1. For each k 2 0 and N 2 1, AN(z) * zkB(z)

is analytic in IzI < - and therefore 1

On the other hand, if z = 1,

From ( 5 ) and (6) follows, for each N 2 1,

B j = O , k = 0 , 1 , 2 ,..., I and in order to obtain (1) we need to establish that

5

= 1 Aj+,Bj . j = O

A standard argument of “dominated convergence” type would show that (7) holds if there

exists a convergent series a

cj such that j = 1

This is indeed the case because the left-hand side of (8) is bounded above by

4(e - 1) (2 + j + k)2 4(e - 1)

2 --.

I

j 2

To summarize, we have obtained a non-rational function A analytic in ID (but in no disk containing ID) and an infinite Blashke product B satisfying (1). Remark that the cor- respondance between the poles of A in C and the zeros of B, as described in the proof of

Math. Nachr. 173 (1995) 174

Theorem 1, is still valid. However z = 1 is an essential singularity of A and not a zero of B (even in a limiting sense): in fact the cluster set of B at z = 1 is the closed unit disk! (see [3], Theorem 5.41.

Remark. Assume that the bounded function '4

I ( z ) := c cnzn E H(D) " = O

is such that

I ( t ) := lim I ( r 4 ) r - l . r < 1

exists and equals one in modulus for almost all 5 E aD. Such a function is called an inner function (see [3], although the word inner is not used there). Any Blashke product is an inner function; the function E ( z ) = l/e'+'/' -' is also an inner function. If k 2 Ois an integer,

= 271i C C,,+~+~C,, n = O

In other words by choosing

I(Z) - I(0) A(z ) = and B(z) = Z(z)

Z

where I is inner but does not belong to Hc(D), we also obtain a class of examples showing that (3) does not in general follow from (1) for functions analytic in ID.

4. An application of Theorem 1

We wish to end this paper by a simple application of Theorem 1. Let

9 = { F E H(D) I F ( 0 ) = 1 and (F'(z)l I Re (F(z ) ) , IzJ < 1) . The class 9' is related to a recent convolution conjecture for univalent functions [4]. It would in particular be interesting to identify extreme points and support points of 9' (we refer to [6] for appropriate definitions). 9' is a compact, convex subset of H(D) and it is known [5] that extreme points F E H,(D) of ID' are characterized by the equation

(9) IF'(z)( = Re ( F ( z ) ) , IzI = 1 . Let us assume that a function F E H,(ID) is a support point of .9', i.e., there exists A E A ,

with Re 1. non constant over 9', such that

(10) Re i ( F ) = max Re I.(G). GE*'

Fournier, On Linear Functionals of Rational Type over H(D) 175

If F is not an extreme point of 9, it is implicitely contained in the proof of the main Theorem in ([4], page 39) that

(1 1)

where dj > 0, c is an arbitrary complex number with IcI = 1, and w F E H(D) does not vanish in ID except for a simple zero at the origin. It follows from (10) and (11) that

F(z) + r:cSjzjwp(z) E 9’ , j = 0, 1, 2, ...,

Re L(&zJwF(z)) I 0 ,

and because arg (c) is unrestricted we obtain

j = 0, 1, 2, ...

j = 0, 1, 2, ... ,

L(zJwF(z)) = 0 ,

However by our Theorem 1,

L(h) = K h ( 0 ) , for any h E H(ID) ,

where K E C. Due to the normalization imposed on 9’, this means that i is constant over 9‘. We therefore have proved that any support point F E H,(ID) of 9‘ is also an extreme

point of g’, and in particular satisfies equation (9). We mention finally that we d o not yet know explicitly all functions F E 9’ n H,(ID) that verify (9).

References

[I] A. BEURLJNG, On Two Problems Concerning Linear Transformations in Hilbert Spaces, Acta Math.

[2] L. BRICKMAN, Functionals of Rational Type over the Class S, Proc. Amer Math. Soc. 92 (1984),

[3] E. F. COLLINGWOOD and A. J. LOHWATER. The Theory of Cluster Sets, Cambridge University

[4] P. L. DUREN, Extremal Problems for Univalent Functions, in Aspect of Contemporary Complex

[5] R. FOURNIER and ST. RUSCHEWEYH, Remarks on a Multiplier Conjecture for Univalent Functions,

[6] D. J. HALLENBECK and T. H. MACGREGOR, Linear Problem and Convexity Techniques in Geometric

[7] ST. RUSCHEWEYH, Convolutions in Geometric Function Theory, Les Presses de I’Universite de

[8] 0. TOEPLITZ, Die linearen vollkommenen Riume der Funktionentheorie, Comment. Math. Helv.

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DPpprirtc.ment (kc Matlii~mutique et (kc Statistiyue UrzivcwitP ( k ~ MontrPal MontrPal ( Q C ) H3C 3J7 Canada