composition operators between weighted banach spaces ... - CiteSeerX

J. Austral. Math. Soc. (Series A) 64 (1998), 101–118

COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS ´ ¨ and J. TASKINEN J. BONET, P. DOMANSKI, M. LINDSTROM

(Received 19 February 1997)

Communicated by P. G. Dodds

Abstract We characterize those analytic self-maps ' of the unit disc which generate bounded or compact composition operators C ' between given weighted Banach spaces Hv1 or Hv0 of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 47B38; secondary 30D55, 46E15.

0. Introduction The aim of this paper is to study boundedness and compactness of composition operators C' , C' . f / D f Ž ' on weighted Banach spaces of analytic functions, where ' : D ! D is an analytic map on the unit disc D. We are interested in complex spaces of the form (1)

Hv1 :D Hv1 .D/ :D f f 2 H .D/ : k f kv :D sup v.z/j f .z/j < 1g;

(2)

Hv0 :D Hv0 .D/ :D f f 2 H .D/ : lim v.z/j f .z/j D 0g;

z2D

jzj!1

endowed with the norm k Ð kv , where H .D/ denotes the space of analytic functions on D and v : D ! RC is an arbitrary weight, that is, bounded continuous positive (which means strictly positive throughout the paper) function. The research of the first named author was partially supported by DGICYT, grant. no. PB94 - 0541. The research of the second named author was mainly done while he visited Universidad Politécnica de Valencia, Spain, supported by DGICYT (Spain), grant no. SAB 95-0092 in the academic year 1995/96. c 1998 Australian Mathematical Society 0263-6115/98 $A2.00 + 0.00

101

102

J. Bonet , P. Domański , M. Lindström and J. Taskinen

[2]

Composition operators were extensively studied on various ‘integral type’ spaces of analytic functions on the disc like Hardy spaces, Bergman spaces, Dirichlet spaces or Bloch spaces (see, for example, [CM, Sh] and[J]) and weighted spaces of continuous functions (see [SS]). The spaces Hv1 or Hv0 are connected with the study of growth conditions of analytic functions and were also studied in detail (see [SW1, SW2, RS, BS, BBT, L1, L2]). Our purpose is to connect both topics. Especially interesting are radial weights v, that is, v.z/ D v.jzj/. In that case, if v 1, then we get Hv1 D H 1 the space of all bounded analytic functions and Hv0 D f0g. If lim supjzj!1 v.z/ > 0, then obviously Hv1 D H 1 with an equivalent norm and also Hv0 D f0g. Moreover, C' : H 1 ! H 1 is always bounded and it is compact if and only if '.D/ D (see [CM, Ex. 3.3.2]). Now, if limjzj!1 v.z/ D 0, the so-called non-increasing majorant of v, that is, the radial function u : D ! RC , u.r / D supfv.R/ : r R < 1g, is also continuous and tends to zero at the boundary. By the maximum principle Hu1 D Hv1 , Hu0 D Hv0 , and the corresponding spaces are isometric. On account of what was just said, we call any radial, positive continuous function v : D ! RC , which is non-increasing with respect to jzj and is such that limjzj!1 v.z/ D 0, a typical weight. We will be mostly interested in the radial weights but as we have seen we may assume that the weight is typical. Nevertheless, we formulate our results in a more general (even non-radial) setting whenever it is possible. Let us explain the organization of the paper. In Section 1 we summarize preliminaries on spaces Hv1 and Hv0 . In Section 2 we collect results on boundedness of C' and in Section 3 analogous results on compactness. The last Section 4 contains some results on p-integral composition operators. We finish the introduction with some notation. If 1 p < 1 and v is a weight, we define the Banach space Z p p p Av :D Av .D/ :D f f 2 H .D/ : k f k :D (3) j f .z/j p v.z/ p d A.z/ < 1g; z2D

where d A is the 2–dimensional Lebesgue measure. If f : D ! C is an analytic function, then M. f; r / :D supjzjDr j f .z/j is a log-convex non-decreasing function (see the Hadamard Three Circle Theorem [C, V.3.13]). We reserve the letters v, w for weights. We denote the natural numbers by N D f1; 2; 3; : : : g. By C, C 0 , c etcetera we denote positive constants which may vary from place to place but do not depend on indices or variables in given formulas or inequalities. 1. The spaces Hv1 and Hv0 Notice that the norm topology on Hv1 is stronger than the topology −co of uniform convergence on the compact sets of D. Assume for a while that limjzj!1 v.z/ D 0.

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Weighted Banach spaces of analytic functions

103

Since f r .z/ :D f .r z/ tends in −co to f , the closed unit ball Bv0 of Hv0 is −co -dense in the unit ball Bv of Hv1 . This implies (see [BS, Th. 1.1], compare [RS, Th. 1]) that .Hv0 /00 D Hv1 isometrically and the embedding of Hv0 into Hv1 is the canonical embedding into its bidual. Moreover, since point evaluations are continuous functionals on Hv0 , the pointwise convergence topology (denoted by −p ) on Hv0 is weaker than the weak topology. Looking at the representation of .Hv0 /0 (see [BS, Th. 1.1]), we realize immediately that if Žz is a point evaluation at z on Hv0 , then for f 2 Hv1 we have h f; Žz i D f .z/. Thus the pointwise convergence on Hv1 is weaker than its weak-star topology. Since Hv0 is −p -dense in Hv1 and C' is always −p -continuous, C' : Hv1 ! Hw1 is equal to the bi-adjoint map of C' : Hv0 ! Hw0 whenever both operators are well defined and limjzj!1 v.z/ D limjzj!1 w.z/ D 0. In fact, from the papers of Lusky [L1, L2], we know that for radial weights and under quite general assumptions Hv0 ' c0 and Hv1 ' l1 . To each weight v corresponds the so-called growth condition u : D ! RC , u D 1=v and Bv D f f 2 H .D/ : j f j ug. In [BBT] the new function uQ : D ! RC is defined by u.z/ Q :D sup f 2Bv j f .z/j and the weight associated with v is defined by vQ :D 1=u. Q It is shown there that uQ and vQ have the following useful properties: .i/ 0 < uQ u and 0 < v v, Q vQ is bounded; .ii/ uQ and vQ are continuous and, respectively, radial, non-decreasing and nonincreasing whenever u and v are so; .iii/ k f kv 1 if and only if k f kvQ 1 for f 2 H .D/; .iv/ for every z 2 D there is f z 2 Bv with u.z/ Q D j fz .z/j; .v/ if limjzj!1 v.z/ D 0, then u.z/ Q D supf 2Bv0 j f .z/j. As in [T] a weight v is called essential if there exists a C > 0 such that v.z/ v.z/ Q Cv.z/ for all z 2 D. There are many known criteria for v to be essential (see [BBT], especially Proposition 3.4 there). In particular, if v.z/ D 1=M. f; jzj/ for some analytic function f : D ! C, then v D v. Q It turns out that tending to zero at the boundary is preserved by the tilde operation. PROPOSITION 1.1. Let v be a weight on D. Then limjzj!1 v.z/ Q D 0, whenever limjzj!1 v.z/ D 0. In particular, if v is a typical weight, then vQ is typical as well. COROLLARY 1.2. If ½ : D ! RC is a radial weight then H½1 is strictly bigger than H 1 if and only if limjzj!1 ½.z/ D 0. PROOF OF PROPOSITION 1.1. Let us take u D 1=v as usual; then the growth condition u tends to C1 at the boundary. Since there is a radial non-decreasing function u tending to C1 at the boundary, we may (and we will) assume that u is radial. Let rn 2 .0; 1/, u.rn / D n. We take f 0 0 and we define inductively a sequence of functions . f n / analytic on a neighbourhood of D such that:

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.a/ M. f j ; ri / < i for i j ; .b/ M. f j ; 1/ D j and f j .1/ D j ; .c/ M. f j f j 1; r j / 2 j . Assume that f 1, : : : , f n 1 are defined satisfying (a) – (c) for j n 1. Then we define bn , 0 < bn < 2 n , such that bn < i M. f n 1; ri / for i D 1; : : : ; n. We define a function g, analytic on a neighbourhood of D such that g.z/ :D an =.Rn z/n , where 1 rn n 1 rn an D ; Rn D 1 C 1=n : 1=n bn 1 bn 1 It is easily seen that: M.g; rn / bn ;

g.1/ D 1;

M.g; 1/ D 1:

By the choice of bn we obtain that f n D f n 1 Cg satisfies conditions (a) – (c) for j D n. Let us define f :D limn!1 f n . Since M. f; rn / n D u.rn / for n 2 N, we have j f .z/j u.z/ C 1 for any z 2 D and there is a constant C such that j f .z/j Cu.z/ for any z 2 D. Clearly, u.z/ Q ½ j f .z/j=C and limjzj!1 u.z/ Q D C1. This completes the proof. In any case we can substitute v by vQ but unfortunately we have no easy way of calculating vQ from v. PROPOSITION 1.3. For every weight v we have isometrically Hv1 D HvQ1 and, if limjzj!1 v.z/ D 0, then Hv0 D HvQ0 . PROOF. By the property (iii), Hv1 D HvQ1 isometrically and HvQ0 is an isometric (closed) subspace of Hv0 . As we mentioned before, if T D Cid : HvQ0 ! Hv0 , then T 00 : HvQ1 ! Hv1 is also equal to Cid : HvQ1 ! Hv1 . Clearly, if T were not onto, then T 00 would have not been onto as well.

2. Boundedness of C' We give firstly necessary and sufficient conditions for the operator C' to be bounded on Hv1 . PROPOSITION 2.1. Let v and w be weights. The following statements are equivalent: .i/ the operator C' : Hv1 ! Hw1 is bounded; .ii/ supz2D w.z/=v.'.z// Q D M < 1,

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105

.iii/ supz2D w.z/= Q v.'.z// Q D M < 1. If v and w satisfy limjzj!1 v.z/ D limjzj!1 w.z/ D 0, then the above conditions are equivalent to (iv) the operator C' : Hv0 ! Hw0 is bounded. REMARK. Contrary to many cases of classical function spaces, an analytic self-map ' : D ! D does not necessarily induce a bounded composition operator for general 1 weights. For example, consider v.z/ D w.z/ D e .1 jzj/ and '.z/ D .z C 1/=2. Then v D vQ and, for z D r 2 R, v.r /=v.'.r // D e1=.1 r/ for 0 < r < 1. Thus v.r /=v.'.r // ! 1, when r ! 1, so C' is not bounded on Hv1 . Q Cv.z/, and the necessity If v is essential, then for every z 2 D we have that v.z/ of (4) in the next corollary follows from this. COROLLARY 2.2. Assume that v is an essential weight. The operator C' : Hv1 ! Hw1 is bounded if and only if (4)

supz2D w.z/=v.'.z// < 1:

If v and w satisfy limjzj!1 v.z/ D limjzj!1 w.z/ D 0, then C' : Hv0 ! Hw0 is bounded if and only if (4) holds. REMARKS. (1) If Hv1 D H 1 , then C' : Hv1 ! Hw1 is continuous for all weights w and, if w tends to zero at the boundary C' .Hv1 / Hw0 . On the other hand, by Proposition 1.1 and by properties of associated weights, one can prove that if.zn / D, limn!1 jzn j D 1 and v is a weight on D tending to zero at the boundary, then there is a sequence of functions f n 2 Bv such that j f n .zn /j ! 1 as n ! 1. Thus, if Hw1 D H 1 and @'.D/ \ @ D 6D ; then C' f n is unbounded in Hw1 . Finally, in that case C' : Hv1 ! Hw1 is bounded if and only if '.D/ D. (2) The condition in the corollary is no more necessary for boundedness of C' whenever v is not essential. Let us take an arbitrary non-essential weight v and w D v. Q Then, clearly lim supjzj!1 w.z/=v.z/ D C1 but Cid : Hv1 ! Hw1 is an isometry. PROOF OF PROPOSITION 2.1. (iii) implies (ii) is trivial as w w. Q (ii) implies (i): By assumption, we have w.z/ M v.'.z// Q for all z 2 D. Thus w.z/j f .'.z//j D

w.z/ v.'.z//j Q f .'.z//j Mk f kvQ v.'.z// Q

(i) implies (iii) and (iv) implies (iii): If not, then there is a sequence .zn / ² D, with w.z Q n / > n v.'.z Q n // for all n. For all n, there exists f n 2 Bv (which can be chosen in

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Bv0 , whenever limjzj!1 v.z/ D 0, by −p -density of Bv0 in Bv ) such that j f n .'.zn //j > 1 1 u.'.z Q n //=2. By (i) or (iv), . fn Ž '/ is bounded in Hw .D/ D HwQ .D/, so there is C > 0 such that j fn .'.z//jw.z/ Q C for all z 2 D and all n 2 N. On the other hand, j f n .'.zn //jw.z Q n / D j f n .'.zn //jv.'.z Q Q n /=v.'.z Q n //w.z n // > n=2 for all n, so we have a contradiction. (iii) implies (iv): By (iii) implies (i) and Proposition 1.3, it suffices to show that C' . f / 2 Hw0Q for each f 2 HvQ0 . Take f 2 HvQ0 . Given " > 0; there is r1 2]0; 1[, such that v.z/j Q f .z/j < "=M for jzj > r1 . For jzj > r1 we consider two cases: If j'.z/j > r1 , then w.z/j Q f .'.z//j D v.'.z//j Q f .'.z//j

w.z/ Q < ": v.'.z// Q

For j'.z/j r1 , we have that there is r2 ½ r1 , 0 < r2 < 1, such that w.z/j Q f .'.z//j w.z/sup Q jzjr1 j f .z/j < "

for jzj > r2 :

Thus w.z/j Q f .'.z//j < " for jzj > r2 . THEOREM 2.3. Let v be a typical weight. The following assertions are equivalent: .i/ all operators C' : Hv1 ! Hv1 are bounded; .ii/ all operators C' : Hv0 ! Hv0 are bounded; .iii/ the following inequality holds: (5)

v.1 Q 2 n 1/ > 0: n2N v.1 Q 2 n/ inf

REMARK. For example, an essential weight v.z/ D .1 jzj/Þ , Þ > 0, satisfies the conditions of Theorem 2.3 The condition (5) was used by Lusky [L2, p. 310], when he studied the isomorphism Hv0 ' c0 . Also see [SW2]. PROOF. By Proposition 2.1, (i) holds if and only if (ii) does. By the Schwarz lemma, if '.0/ D 0, then j'.z/j jzj for every z 2 D and therefore C' : Hv1 ! Hv1 is bounded. For p 2 D, let Þp .z/ D . p

z/=.1

pz/; N

that takes D onto itself. If each CÞp is bounded, then all C' are bounded. Indeed, each ' D Þp Ž , where D Þp Ž ', p D '.0/ and .0/ D 0. We have to show that for any p 2 D, CÞp is bounded on Hv1 .D/ if and only if vQ satisfies (5).

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(i) implies (iii): Let us assume that all CÞp are bounded. Then, by Proposition 2.1, for every p 2 D there exist Mp > 0 such that v.z/ Q < Mp v.Þ Q p .z// for all z 2 D. Since it is easily seen that supjzjDr jÞp .z/j D .j pj C r /=.1 C j pjr /, we get that v.z/ Q < M p v..j Q pj C r /=.1 C j pjr // for all jzj D r . Let us define l.r / D v.1 Q r /, s D 1 r . Since 1 .j pj C 1 s/=.1 C j pj.1 s// D s.1 j pj/=.1 C j pj j pjs/; we obtain for s < 1=2 that 1 j pj j pj C 1 s .1 j pj/ l s (6) l 1 l s : 1 C j pj 1 C j pj.1 s/ 1 C j pj=2 Finally, for p D 2=5, we use the second inequality in (6) to get M > 0 and s0 > 0 such that l.s/ Ml. 2s / for all s 2]0; s0[. This immediately implies (5). (iii) implies (i): If (5) is satisfied, then l defined as above has the property that there are M > 0 and t0 2]0; 1[ with l.t/ Ml.t=2/ for all t > t0 . Then for any c < 1 we find n 2 N such that c < 2n and hence l.t/ M n l. tc /. We take c D .1Cj pj/=.1 j pj/. Then by the first inequality in (6), for all p 2 D there is Mp > 0 with j pj C 1 t ; l.t/ Mp l 1 1 C j pj.1 t/ for all t > t0 . Clearly, this implies that for all p 2 D there exists Mp > 0 such that for every jzj D r we have that v.z/ Q < Mp v.Þ Q p .z// by the argument above. THEOREM 2.4. Let ' : D ! D be an analytic map. The following statements are equivalent: .i/ for any typical weight v the map C' is bounded on Hv1 ; .ii/ for every 2] ³; ³], Þ Ž ' fixes a point in D, where Þ .z/ D ei z; .iii/ either ' is a rotation or for any 2] ³; ³], D Þ Ž ' fixes an attracting point p 2 D, that is, . p / D p and n .z/ ! p uniformly on compact sets, where Ž Ð Ð Ð Ž , .n times/; n :D .iv/ either ' is a rotation or lim inf jzj!1 .1 j'.z/j/=.1 jzj/ > 1; .v/ there is an r0 2]0; 1[ such that j'.z/j jzj for every z 2 D with jzj ½ r0 . For the proof of Theorem 2.4 we need the following result. LEMMA 2.5. For any two increasing sequences rn ! 1 and Rn ! 1 such that r0 < R0 < r1 < R1 < r3 < R3 < Ð Ð Ð there is an analytic map f : D ! C with lim

n!1

where M. f; r / D supjzjDr j f .z/j.

M. f; Rn / D 1; M. f; rn /

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PROOF. We define f 0 1. Assume that we have already found polynomials f 1 ; f 2 ; : : : ; f n 1 such that .a/

fk jrk D < 1=2k ;

Then put M :D M chosen positive and .c/

.b/

P n

1 i D1

.1

M. f k ; Rk / >k Pk 1 M. i D1 f i ; rk /

f i ; rn and fQn .z/ D A=..1

A "/Rn

rn

D

1 ; 22n

.d/

for k D 1; : : : ; n "/Rn

1:

z/, where A, " are

A D 2Mn: " Rn

In fact, it suffices to take AD

2Mn.Rn rn / .22nC1 Mn C 1/

and

"D

.Rn rn / : Rn .22nC1 Mn C 1/

Clearly fQn jrn D < 1=22n because of (c). Moreover, by (d), fQn .Rn / D 2n M. Now, by Runge’s Theorem, we can approximate fQn on Rn D by a polynomial f n satisfying P (a) and (b). We define f D i1D1 f i . The series converges almost uniformly. The condition limn M. f; Rn /=M. f; rn / D 1 follows from (b). PROOF OF THEOREM 2.4. (i) implies (v): Assume that there is a sequence .zn / in

D with jzn j ! 1 such that j'.zn /j > jzn j for all n. Define rn D jzn j, Rn D j'.zn /j and

without loss of generality we may assume that r0 < R0 < r1 < R1 < r2 < R2 < Ð Ð Ð . By Lemma 2.5, there exists f 2 H .D/ such that limn!1 M. f; Rn /=M. f; rn / D 1. Take v.z/ D 1=M. f; jzj/ and assume that C' : Hv1 ! Hv1 is bounded. Then there is c > 0 such that kC' k c. Moreover, with Þ .z/ D ei z, kCÞ k D 1 and k f kv D 1, so kCÞ Ž' . f /kv c for all 2]

³; ³]:

Now, sup j f Ž Þ Ž '.z/j z2D

1 1 ½ j f Ž Þ Ž '.zn /j M. f; jzj/ M. f; jzn j/

for all n and all . Choosing n in a suitable way, we get j f Ž Þn Ž '.zn /j D M. f; j'.zn /j/ D M. f; Rn /. Thus, kCÞn Ž' . f /kv ½ M. f; Rn /=M. f; rn / for all n, and we have a contradiction. (v) implies (i) is obvious.

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(i) implies (ii): Let us assume firstly that ' is an automorphism. Then either ' is a rotation and (ii) is satisfied or '.z/ D Þ .. p z/=.1 pz//, N where p 6D 0. Since supjzjDr j'.z/j D .j pj C r /=.1 C j pjr / > r for r < 1, it follows that (v) is not satisfied. We have proved that if ' is an automorphism, then C' satisfies (i) only for rotations, and then (ii) is satisfied as well. Now let ' be a non-automorphism. Then every Þ Ž ' is also a non-automorphism. Hence, by [Sh, Section 5.4], either Þ Ž ' has a fixed point or n :D Ž Ð Ð Ð Ž , (n times), where D Þ Ž ', satisfies j n .z/j ! 1 uniformly on compact sets. Suppose that some Þ Ž ' has no fixed point. Let K :D fz : jzj r 0 g, where r0 is given in .v/. By the Maximum Modulus Theorem, supz2K j'.z/j D supz2@ K j'.z/j and therefore it follows from .v/ that supz2@ K j n .z/j r0 . Since j n .z/j ! 1 uniformly on compact sets, we obtain a contradiction. (ii) implies (iii): Assume that ' is an automorphism. Hence '.z/ D ei

p z 1 pz N

for some

p 2 D; 2]

³; ³]:

If p 6D 0, then for a suitable chosen 0 , j pj D cos.0 =2/ and by [Sh, Section 0.5.4], ' is a parabolic automorphism, meaning (see [Sh, p. 5]) that ' has a single fixed point lying on the boundary of D. If p D 0, then ' is a rotation. Now, let ' be a non-automorphism. Then every Þ Ž ' is also a non-automorphism. Hence the assumption and [Sh, Section 5.2.1], give that every Þ Ž ' fixes an attracting point in D. (iii) implies (iv): Let us assume that each Þ Ž ' has an attracting fixed point p in D. If lim inf jzj!1 .1 j'.z/j/=.1 jzj/ 1, then there is a sequence .zn / ² D, jzn j ! 1 and limn .1 j'.zn /j/=.1 jzn j/ D Ž 1. Without loss of generality, we may assume that zn ! w 2 @ D. Clearly, j'.zn /j ! 1, and we may assume that '.zn / ! 2 @ D. Choosing 2] ³; ³] suitable, we get Þ Ž '.zn / ! w. Now, by Julia’s theorem [Sh, p. 63],

Þ Ž '

½ w DC 1C½ 1C½

²

½ w DC 1C½ 1C½

for all ½ > 0:

Since p is an attracting point of Þ Ž ', it is in any disc .½=.1 C ½//D C w=.1 C ½/, which gives a contradiction by taking ½ > 0 small enough. (iv) implies (v): The condition is obviously satisfied for rotations. Now, if lim infjzj!1 .1 j'.z/j/=.1 jzj/ > 1, then there is r0 2]0; 1[ such that .1 j'.z/j/= .1 jzj/ > 1 for jzj > r0 . Clearly, then j'.z/j < jzj, and we are done.

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3. Compactness of C' To deal with the compactness we need the following form of the Weak Compactness Theorem. The proof is similar to the case of H 2 . The reader is asked to refer to [Sh, Section 2.4]. LEMMA 3.1. Let X and Y be Hv1 and Hw1 or Hv0 and Hw0 respectively. A bounded operator C' : X ! Y is compact, if and only if, given any bounded sequence . f n / ² X which converges to 0 uniformly on the compact subsets of D, also the sequence .C' . f n // converges to 0 in Y . COROLLARY 3.2. Let v and w be weights. If there exists an r , 0 < r < 1, such that j'.z/j r for all z 2 D, then C' : Hv1 ! Hw1 is compact. THEOREM 3.3. Let v and w be weights. The following assertions are equivalent: .i/ the operator C' : Hv1 ! Hw1 is compact; .ii/ limr!1 supj'.z/j>r w.z/=v.'.z// Q D 0 or '.D/ D. If v and w satisfy limjzj!1 v.z/ D limjzj!1 w.z/ D 0, then the above conditions are also equivalent to the following ones: (iii) the operator C' : Hv0 ! Hw0 is compact; (iv) limjzj!1 w.z/=v.'.z// Q D 0, (v) limjzj!1 w.z/= Q v.'.z// Q D 0. REMARKS. (1) It is not difficult to show that if Hv1 D H 1 , then C' : Hv1 ! Hw1 is compact for any weight w tending to zero at the boundary. On the other hand, if Hw1 D H 1 , then C' : Hv1 ! Hw1 is compact if and only if '.D/ D (compare the remark after Proposition 2.1). (2) If for a fixed typical weight v, C' : Hv1 ! Hv1 is compact, then there is r < 1 such that j'.z/j < jzj for jzj > r . Thus C' : Hw1 ! Hw1 is bounded for any typical weight w. (3) If for some weights v, w tending to zero at the boundary, C' : Hv1 ! Hw1 is compact, then it is weakly compact and C' : Hv1 ! Hw0 . PROOF. We assume first that v and w satisfy limjzj!1 v.z/ D limjzj!1 w.z/ D 0. (v) implies (iv) is obvious. (iv) implies (i): By Proposition 2.1, C' is bounded. We shall use Lemma 3.1. Take a bounded sequence . f n / ² Bv and assume that f n ! 0 in the −co -topology. For given " > 0 there is r0 2]0; 1[ with w.z/ < "v.'.z//=2 Q for all jzj > r0 . Put

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111

C 0 :D sup z2D w.z/ < 1. For n big enough, we have that supjzjr0 j f n .'.z//jC 0 < "=2 and thus kC' . f n /kw supjzjr0 j f n .'.z//jw.z/ C supjzj>r0 j f n .'.z//jw.z/ 12 " C 12 "k f n kvQ "; so (i) follows. (i) implies (v): If not, then there are c > 0 and a sequence .zn / ² D, jzn j ! 1, with w.z Q n / > cv.'.z Q n // for all n. For all n, there exists f n 2 Bv such that j f n .'.zn //j D u.'.z Q //. By going to a subsequence we can assume that '.zn / ! z0 for some z0 2 D, n when n ! 1. If jz0j 6D 1, then by assumption, Q n / ½ c lim v.'.z Q Q 0 / > 0; 0 D lim w.z n // D c v.z n

n

which is a contradiction. Thus jz0j D 1. Now, since j'.zn /j ! 1, there exist natural numbers Þ.n/ with limn Þ.n/ D 1 and such that j'.zn /jÞ.n/ ½ 1=2 for all n. We define the analytic functions gn .z/ :D zÞ.n/ f n .z/ for all n. Clearly .gn / is a bounded sequence in Hv1 . It converges pointwise to 0 because of the factor z Þ.n/ . Hence, supposing that C' is compact, Lemma 3.1 implies that kC' .gn /kw ! 0 as n ! 1. On the other hand, we get for all n, kC' .gn /kwQ ½ jgn .'.zn //jw.z Q n / D j'.zn /jÞ.n/ j f n .'.zn //jw.z Q n/ 1 ½ 12 j f n .'.zn //jcv.'.z Q n // ½ 2 c;

which is a contradiction. (i) if and only if (iii): By Proposition 2.1, if one of the considered operators is compact, then both are bounded and C' : Hv1 ! Hw1 is the bi-adjoint of C' : Hv0 ! Hw0 (see Section 1). Apply the Schauder Theorem. Now, in the general case, (i) implies (ii) is similar to (i) ) (v) and (ii) implies (i) is similar to (iv) implies (i). COROLLARY 3.4. Let v and w be essential weights and assume that limjzj!1 w.z/ D 0. Then C' : Hv1 ! Hw1 is compact .or, equivalently, C' : Hv0 ! Hw0 is compact/, if and only if (7)

lim

jzj!1

w.z/ D 0: v.'.z//

We see below that for general weights our simple characterization of compactness in 3.4 fails. Of course, in that case v is no more essential and this fact gives another motivation for the concept of an essential weight.

112


[12]

EXAMPLE 3.5. There exist a typical weight w and a compact composition operator C' : Hw1 ! Hw1 which does not satisfy (7). CONSTRUCTION. Let us define increasing sequences .rn /, . pn / tending to 1 as follows: r0 D

1 ; 2

rnC1 D

2rn ; 1 C rn

p0 D 0;

pn < rn < pnC1

for n 2 N:

Choose an increasing sequence .an / of natural numbers such that (8)

an .log rnC1

log rn / ½ n:

We define three non-decreasing, unbounded and continuous functions , , Q : R ! RC as follows: .i/ 0 on . 1; log r0 ] and for each n 2 N the function is affine on [log rn ; log rnC1 ] with the derivative an ; .ii/ for n even Q on [log pn ; log pnC1 ]; .iii/ for n odd Q is affine on [log pn ; log pnC1 ]; .iv/ for n odd is affine on [log pn ; log rn ] and constant on [log rn ; log pnC1 ]. We could choose . pn / in such a way that: (v) j.log pn / .log rn /j 1 and jQ .s/

.s/j 1 for all s 2 R .

Now, we can define our typical weights: v.z/ :D e

.log jzj/

;

w.z/ :D e

.log jzj/

:

Clearly, our assumptions imply that .s/ ½ Q .s/ ½ .s/ for s 2 R . By the Hadamard Three Circle Theorem and (v), w.z/ Q ½e

Q .log jzj/

½ e 1v.z/:

On the other hand, on the annulus fz : rn jzj rnC1 g, we have v.z/ D bn jzj an with an natural. Moreover, 1=v.z/ is a supremum of a sequence M.bn 1 z an ; jzj/, which means that v D v. Q Taking the above into account, we obtain that wQ is equivalent to v D v. Q We will show that C' : Hv1 D Hw1 ! Hw1 D Hv1 is compact for '.z/ :D z=.2 z/ but still limjzj!1 w.z/=w.'.z// 6D 0. Observe, that j'.z/j '.jzj/ D jzj=.2 jzj/, thus if jzj 2 .rn ; rnC1 /, then j'.z/j '.jzj/ 2 .rn 1 ; rn /. Hence v.z/ .2 jzj/an v.z/ DC v.'.z// v.'.jzj// jzjanC1 an

1

[13]


113

and the function on the right hand side is decreasing with respect to jzj. Summarizing, by (8) for jzj 2 .rn ; rnC1 /, v.rn / v.rn / v.z/ D e v.'.z// v.'.rn // v.rn 1 /

.n 1/

:

By Corollary 3.4, C' is compact. On the other hand, for n even w.rn / v.rn / w.rn / D D ½ e 1: w.'.rn // w.rn 1 / v. pn / This completes the proof. Theorem 3.3 yields a direct method to deduce the compactness of C' in some weighted spaces once we know the compactness with respect to other weights. Precisely, assume that v is essential and that C' : Hv1 ! Hw1 is compact. If the weights ¹ and ! on D satisfy (9)

w ! C ¹Ž' vŽ'

for some constant C > 0, then C' : H¹1 ! H!1 is compact. Namely, the assumptions imply that (ii) of Theorem 3.3 is satisfied for v and w, hence, by (9), it is satisfied by ¹ and ! as well. We especially have the following corollary. COROLLARY 3.6. Assume that v and w are typical weights, v is essential, v=w is increasing as jzj ! 1, '.0/ D 0 and that C' : Hv1 ! Hv1 is compact. Then also C' : Hw1 ! Hw1 is compact. PROOF. In this case (9) is equivalent to v w C ; wŽ' vŽ'

or to

v v Ž' C : w w

But this holds with C D 1 by the Schwarz lemma, since v.z/=w.z/ is radial and increasing as jzj ! 1. THEOREM 3.7. Let ' : D ! D be an analytic map such that @'.D/ \ @ D 6D ;. Then there exists an essential typical weight v such that C' : Hv1 ! Hv1 is bounded but not compact. COROLLARY 3.8. Let ' : D ! D be an analytic map. The following assertions are equivalent: .i/ C' : Hv1 ! Hv1 is compact for any .typical/ weight v;

114


[14]

.ii/ C' : Hv0 ! Hv0 is compact for any weight v tending to zero at the boundary; .iii/ C' : H 1 ! H 1 is compact; .iv/ '.D/ r D for some r < 1. PROOF OF THEOREM 3.7. The weight w.z/ D .1 jzj/ is typical and essential and C' : Hw1 ! Hw1 is always bounded by Theorem 2.3. Let us assume that C' is compact in that case. By Theorem 3.3, there is r0 < 1 such that for any r > r0 : jzj < 12 .1

1

j'.z/j/

for jzj D r:

Let us define M.r / :D supjzjDr j'.z/j, then we have 1

(10)

r < 12 .1

M.r // for r > r0 :

The function M.r / is non-decreasing, tends to 1 as r ! 1 and, by the Hadamard Three Circle Theorem, it is logarithmically convex. In particular, log M.r /=log r is non-decreasing as r ! 1 . By (10), there is Ž > 1 such that for r ½ r0 we have log M.r /=log r > Ž. We define inductively a sequence .rn /n2N such that rn 1 D M.rn /. Obviously, rn

(11)

1

< rnŽ

for n 2 N

and limn!1 rn D 1. We define an increasing function u : [0; 1/ ! RC which is equal to 1 on [0; r0 ], u.rn / D 2n and it is affine on each interval [rn 1 ; rn ]. The weight we are looking for is defined as v.z/ D 1=u.jzj/. Firstly, we show that v is essential. By [BBT, Proposition 3.4], it suffices to prove the so-called condition (U) for v, that is, to find Þ > 0 and C > 0 such that y/Þ Cu.x/.1

x/Þ

for all 0 x < y < 1: ÐÞ We take C D 4 and arbitrary Þ such that .1 r0 /=.1 r01=Ž / > 2. Indeed, for any x, y either u.y/ 2u.x/ or we find n k such that (12)

u.y/.1

rn y; u.y/ 2u.rn / and

x rk ; u.rk / 2u.x/:

Thus (13) Since g.t/ :D .1

u.y/ u.rn / 1 r0 4 r0 we find n 2 N such that rn jzj < rnC1 , then u.M.jzj// u.M.rnC1 // v.z/ D 1; v.'.z// u.jzj/ u.rn / by the definition of .rn /. Thus C' : Hv1 ! Hv1 is bounded by Proposition 2.1. On the other hand, for any n 2 N there is zn , jzn j D rn such that j'.zn /j D rn 1 . Clearly, u.rn 1 / 1 v.zn / D D v.'.zn // u.rn / 2 and (since jzn j ! 1 as n ! 1) C' : Hv1 ! Hv1 is not compact by Corollary 3.4.

4. Integral operators C' Unfortunately we are not able to give a characterization of nuclear, integral or absolutely summing composition operators for general weights. It is, however, not too complicated to find sufficient conditions for example as follows. PROPOSITION 4.1. Let 1 p < 1, 1= p C 1=q D 1, let v and w be radial weights and let ' : D ! D be analytic. The operator C' : Hv1 ! Hw1 is p-integral, if Z w.z/q sup (14) d A.¾ / < 1: v Ž '.¾ /q j1 z ¾N j2q j' 0 .¾ /j2q= p z2D ¾ 2D

In the case p D 1 the integral is replaced by a supremum in the usual way. PROOF. Since the identity operator Hv1 ! Avp ² L p .v.z/d A.z// is order bounded, it is enough to show that C' : Hv1 ! Hw1 factorizes through the above defined map; see [DJT, Propositions 5.18 and 5.5]. To this end it suffices to prove that

116


[16]

C' : Avp ! Hw1 is bounded. Indeed, using the Bergman reproducing kernel ([Z, Section 4.1]) and the dominated convergence theorem, þ þ þZ þ þ þ f Ž '.¾ / þ sup j f Ž '.z/jw.z/ D sup þ d A.¾ /þþ w.z/ 2 N .1 z ¾ / z2D z2D þ þ D 0 11=q Z q w.z/ j J' .¾ /j q= p d A.¾ /A sup @ v Ž '.¾ /q j1 z ¾N j2q z2D ¾ 2D

0 11= p Z ð @ j f Ž '.¾ /j p v Ž '.¾ / p j J' .¾ /jd A.¾ /A ; D

where j J' j is the 2-dimensional Jacobian determinant of '. The identity j J' j D j' 0 j2 and (14) permit us to conclude. EXAMPLE 4.2. An example of integral composition operators. CONSTRUCTION. Let v.z/ :D .1 jzj/Þ and w.z/ :D .1 jzj/þ , where Þ; þ > 0. Let ² D be an open subset such that @ \ @ D D f1g. We also assume that @ is Dini-smooth ([Po, Theorem 3.5]) except at the point 1 where it has a Dini-smooth corner of opening ³, where < 1, in the sense of [Po, p. 51]; we make the technical assumption that for some c > 0 the inequality (15)

j Im.z/j c.1

Re.z//

is satisfied for every z 2 . Let ' be a Riemann conformal mapping D ! such that limz!1 '.z/ D 1, '.0/ D 0 ([Po, Theorem 2.6]); for example '.z/ :D 1 a.1 z/ for a suitable a 2 C. We show that (14) is satisfied for p D 1 and the operator C' is thus 1-integral, if .a/ þ ½ 2 ½ .Þ C 2/ , or .b/ þ ½ .Þ C 2/ ½ 2. By [Po, Theorem 3.9], there exists a neighbourhood U of 1 such that for some constants c, C > 0, (16) (17)

þ þ þ 1 '.z/ þ þ þ 0 such that j1 ¾ j C 0 .1 j¾ j/ for ¾ 2 ; this and (16) imply that for c0 > 0, 1 j'.z/j ½ c0 j1 zj for z 2 D. Hence, for c 00 > 0 j'.z/j/Þ ½ c00j1

v Ž '.z/ D .1

(18)

zj Þ :

Hence, (19)

w.z/ C sup z¾N j2j' 0 .¾ /j z;¾ 2D v Ž '.¾ /j1 z;¾ 2D j1 sup

.1 jzj/þ ¾ j.ÞC2/ 2j1

N2 z ¾j

:

We now distinguish between the two cases (a) and (b) mentioned above. If (a) holds, (19) is not larger than .1 z;¾ 2D j1

C sup

jzj/þ .1 C sup 2 z ¾N j z2D .1

jzj/þ < 1: jzj/2

In the case (b) (19) can be estimated by C sup z;¾ 2D

.1 jzj/þ j¾ j/.ÞC2/ 2j1

.1

N2 z ¾j

:

Clearly we can here take the sup only over the values 0 < z; ¾ < 1. The partial derivative of (20)

.1 .1

z/þ

¾ /.ÞC2/ 2 .1

z¾ /2

;

0 < z; ¾ < 1;

with respect to ¾ vanishes at the point ¾ D ...Þ C 2/ and at this point (20) becomes equal to a constant times .1 expression is also finite, if the condition (b) is satisfied.

2 C 2z/=..Þ C 2/ z//þ , z/þ .ÞC2/ z .ÞC2/ 2 . This

References [BBT] [BS] [C] [CM]

K. D. Bierstedt, J. Bonet and J. Taskinen, ‘Spaces of holomorphic functions with growth conditions and associated weights’, Studia Math. (to appear). K. D. Bierstedt and W. H. Summers, ‘Biduals of weighted Banach spaces of analytic functions’, J. Austral. Math. Soc. (Series A) 54 (1993), 70–79. J. Conway, Functions of one complex variable (Springer, Berlin, 1978). C. Cowen, B. MacCluer, Composition operators on spaces of analytic functions (CRC Press, Boca Raton, 1995).

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J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics 43 (Cambrige University Press, Cambridge, 1995). H. Jarchow, ‘Some functional analytic properties of composition operators’, Quaestiones Math. 18 (1995), 229–256. W. Lusky, ‘On the structure of H v0 .D/ and hv0 .D/’, Math. Nachr. 159 (1992), 279–289. , ‘On weighted spaces of harmonic and holomorphic functions’, J. London Math. Soc. 51 (1995), 309–320. Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss. 299 (Springer, Berlin, 1992). L. A. Rubel, A. L. Shields, ‘The second duals of certain spaces of analytic functions’, J. Austral. Math. Soc. 11 (1970), 276–280. J. H. Shapiro, Composition operators and classical function theory (Springer, Berlin, 1993). A. L. Shields, D. L. Williams, ‘Bounded projections, duality, and multipliers in spaces of harmonic functions’, J. Reine Angew. Math. 299/300 (1978), 256–279. , ‘Bounded projections and the growth of harmonic conjugates in the disk’, Michigan Math. J. 29 (1982), 3–25. R. K. Singh and W. H. Summers, ‘Composition operators on weighted spaces of continuous functions’, J. Austral. Math. Soc. (Series A) 45 (1988), 303–319. J. Taskinen, Compact composition operators on general weighted spaces, Department of Mathematics, Univ. Helsinki Preprint 121, 1996. K. Zhu, Operator theory in function spaces (Dekker, New York, 1995).

Dept. Matemática Aplicada Univ. Politécnica de Valencia E-46071 Valencia Spain e-mail: [email protected]

Department of Mathematics Ž Abo Akademi University, FIN-20500 Ž Abo Finland e-mail: [email protected]

Faculty of Matematics and Comp. Sci. A. Mickiewicz University, ul. Matejki 48/49 60-769 Poznań Poland e-mail: [email protected]

Department of Mathematics P. O. Box 4, FIN-00014 University of Helsinki Finland e-mail: [email protected]