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Volume 2007, Article ID 61313, 8 pagesdoi:10.1155/2007/61313 Research Article Spectrum of Compact Weighted Composition Operators on the Weighted Hardy Space in the Unit Ball Ze-Hua Zhou

Volume 2007, Article ID 61313, 8 pages

doi:10.1155/2007/61313

Research Article

Spectrum of Compact Weighted Composition Operators on

the Weighted Hardy Space in the Unit Ball

Ze-Hua Zhou and Cheng Yuan

Received 28 February 2007; Accepted 19 October 2007

Recommended by Andr´as Ront ´o

LetB Nbe the unit ball in theN-dimensional complex space, for ψ, a holomorphic

func-tion inB N, andϕ, a holomorphic map from B Ninto itself, the weighted composition op-erator on the weighted Hardy spaceH2(β, B N) is given by (C ψ,ϕ)f = ψ(z) f (ϕ(z)), where

f ∈ H2(β, B N) This paper discusses the spectrum ofC ψ,ϕwhen it is compact on a certain class of weighted Hardy spaces and when the composition mapϕ has only one fixed point

inside the unit ball

Copyright © 2007 Z.-H Zhou and C Yuan This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

It is well known that the general principle that the spectrum structure of the composition operatorC ϕis closely related to the fixed point behavior of the mapϕ is well illustrated

by compact composition operators Determining the spectrum of a compact operator is equivalent to finding the eigenvalues of the operator About the spectrum of a compact operator in a weighted Hardy space defined in the disk orB N, we refer the reader to see [1], where Cowen and MacCluer proved a theorem of considerable generality, which will show that, essentially, all of the spaces of interest to us these eigenvalues are determined

by the derivative ofϕ at the Denjoy-Wol ff fixed point of ϕ Weighting a composition

oper-ator as a generalization of a multiplication operoper-ator and a composition operoper-ator, recently, Gunatillake in [2] obtained some results for the spectrum of weighted composition op-erators on the weighted Hardy spaces of the unit disk It is, therefore, natural to wonder what results can be obtained for the spectrum of weighted composition operators on the weighted Hardy spaces onB N In our paper, we almost completely answer the above ques-tion, the fundamental ideas of the proof are those used by Gunatillake in [2] and Cowen and MacCluer in [1], but there are technical difficulties in several variables that we need

to consider before we will be ready to give the proof This statement will also need some clarification in the case of spaces defined onB N(N > 1) as the Denjoy-Wolff point may not be well defined In the proof ofLemma 2.1, a technique is inspired by the proof of [3, Theorem 7]

2 The main results

For multiindexesm =[m1,m2, , m N] andl =[l1,2, , l N], we say thatl < m for all | l | <

| m |or forl j < m jif| l | = | m |, andl n = m nfor alln < j.

Lemma 2.1 Suppose C ψ,ϕ is a compact operator on the Hardy space H2(β, B N ) If the com-position map ϕ has only one fixed point a in the unit ball, then σ(C ψ,ϕ)⊂ {0,ψ(a), ψ(a)μ } , where μ denotes all possible products of the eigenvalues of ϕ (a).

Proof Without loss of generality, we suppose a =0 In fact, if a =0, letϕ a denote the automorphism commuting 0 anda, then ϕ a ◦ ϕ a(z) = z for every z in B N, it is obvious thatC ϕ a ◦ C ϕ a = I, C ϕ ais invertible, andσ(C ψ,ϕ)= σ(C ϕ a ◦ C ψ,ϕ ◦ C ϕ a)

Let ψ0= ψ ◦ ϕ a, ϕ0= ϕ a ◦ ϕ ◦ ϕ a, then ψ0(0)= ψ a, ϕ0(0)=0, and ϕ 0(0)= ϕ  a(a) ·

ϕ (a) · ϕ  a(0) By ϕ a ◦ ϕ a(z) = z, it follows that ϕ  a(a) · ϕ  a(0)= I and ϕ 0(0) has the same eigenvalue withϕ (a) So

C ψ0,ϕ0(f ) = ψ

ϕ a(z)

f

ϕ a ◦ ϕ ◦ ϕ a(z)

= C ϕ a ◦ C ψ,ϕ ◦ C ϕ a(f ), (2.1)

C ψ,ϕandC ψ0,ϕ0are similar and have the same spectrum

Suppose C ψ,ϕ is compact For anyλ ∈ σ(C ψ,ϕ), then λ is an eigenvalue, and for the

eigenvectorg of λ, C ψ,ϕ g(z) = λg(z), that is,

ψ(z)g

ϕ(z)

Ifg(0) =0, thenψ(0)g(0) = λg(0), λ = ψ(0) If g(0) =0 andψ(0) =0, then



s ≥1

Ψs(z)



t ≥1

G t



ϕ(z)

= λ



t ≥1

G t(z)



whereΨsandG t are the homogeneous expansion ofψ(z) and g(z), and by the

assump-tiona =0 and Schwarz lemma, it follows that lim| z |→0(| ϕ(z) | / | z |)< + ∞(in fact,≤1) Comparing the lowest power terms of two sides, it is easy to know thatλ =0

Ifg(0) =0 andψ(0) =0, differentiating (2.2) with respect toz jthen leads to

g

ϕ(z)∂ψ

∂z j +ψ(z)ϕ

s j

∂g

∂ϕ s = λ ∂g

here,ψ(z)ϕ s

j(∂g/∂ϕ s) stands forψ(z)N

s =1((∂ϕ s /∂z j)(∂g/∂z s)) by Einstein’s convention For the higher-order differentiation, we get



t<m

α t(z) + ψ(z)ϕ s1s2··· s N

j1j2··· s N

∂ m g

∂ϕ s1

1 ··· ∂ϕ s n

N = λ ∂

m g

∂z j1

1 ··· ∂z j N

N

t<m α t(z) denotes the sum of all the terms which have the differential order less thanm.

Now, letm be the multiindex that ∂ m g/∂z m =0 and, for anyl < m, ∂ l g/∂z l =0

Byg =0 andg(0) =0,m > 0, it follows that

ψ(0) · ϕ (0)⊗ ϕ (0)⊗ ··· ⊗ ϕ (0)

| m |copies

∂ m g

ϕ(0)

Notice that 0 is the fixed point ofϕ and ∂ m g/∂z m = ∂ m g/∂ϕ m =0, it follows thatλ must

If [l1,2, , l N] is anN-tuple of the integers 1, 2, , N, let κ[l1 ,l2 , ,l N]

for evaluation of the corresponding partial derivative ata, that is,

f , κ[l1 ,l2 , ,l N]

l | f

∂z l1

1∂z l2

2··· ∂z l N

N

for all f in H2(β, B N) For any positive integerm, let᏷mbe the subspace spanned byK a

and the derivative evaluation kernel ata for total order up to and including m, that is,

᏷m:=span



K a,κ[1]a , , κ[a N],κ[a N,N], , κ[a N,N], , κ

[1, 1, , 1

N copies

]

a , , κ

[N, N, , N

N copies

]

a



For the details of the space᏷m, we also refer the reader to see [1, page 272], in fact, we have the following lemma

Lemma 2.2 ᏷m is an invariant subspace of C ψ,ϕ ∗

Proof First, we show that᏷0is invariant as follows:

so᏷0is invariant underC ψ,ϕ ∗

For᏷1, let f be any function on H2(β, B N), then



f , C ψ,ϕ ∗ κ[a j]



=ψ · f ◦ ϕ, κ[a j]



= f ◦ ϕ(a) ∂ψ

∂z j(a) + ψ(a)

N



k =1



D k f

ϕ(a)

D j ϕ k

(a)

= f (a) ∂ψ

∂z j(a) + ψ(a)

N



k =1



D k f

(a)

D j ϕ k

(a)

=



f , ∂ψ

∂z j(a)K a+ψ(a)

N



k =1



D j ϕ k

(a)κ[a k]



.

(2.10)

That is,

C ∗ ψ,ϕ κ[a j] = ∂ψ

∂z j(a) K a+ψ(a)

N



k =1

(D j ϕ k)(a)κ[k]

or we can denote this by Einstein’s convention

C ∗ ψ,ϕ κ[a j] = ∂ψ

∂z j(a) K a+ψ(a)ϕ

k

So᏷1is invariant underC ∗ ψ,ϕ

We can induct this to the higher order and get

C ∗ ψ,ϕ κ[j1 ,j2 ]

a = α1(a) + ψ(a)ϕ k1 ,k2

j1 ,j2(a)κ[k1 ,k2 ]

whereα1(a) denotes the lower-order terms which belongs to᏷1, as well as

C ψ,ϕ ∗ κ[j1 ,j2 , , j m]

a = α m −1(a) + ψ(a)ϕ k1 ,k2 , ,k m

j1 ,j2 , , j m(a)κ[k1 ,k2 , ,k m]

whereα m −1(a) belongs to᏷m −1

Thus we have proved that, for any finite positive integerm,᏷mis an invariant subspace

Lemma 2.3 Suppose C ψ,ϕ is a bounded operator on H2(β, B N ) with only one fixed point of

ϕ in the unit ball Then { ψ(a), ψ(a)μ } ⊂ σ(C ψ,ϕ ), where μ denotes the possible product of eigenvalues of ϕ (a).

Proof First, we use (2.9), (2.12), (2.13), and (2.14) to compute the matrix representation

ofC ∗ ψ,ϕrestricted to the subspace᏷m That is,

⎛

⎜

⎜

⎜

⎜

⎜

⎜

0 0 0 ··· ψ(a) · ϕ (a) ⊗ ϕ (a) ⊗ ··· ⊗ ϕ (a)

m copies

⎞

⎟

⎟

⎟

⎟

⎟

⎟

.

(2.15) Let us call this matrixA m ThenA m is an (1 +m + m2+···+m N)×(1 +m + m2+

···+m N) upper-triangular matrix The∗  s denote α j(a)  s.

The subspace ᏷m is finite dimensional and, therefore, is closed The Hardy space

H2(β, B N) can be decomposed asH2(β, B N)=᏷m ⊕᏷⊥

m The block matrix ofC ∗ ψ,ϕwith respect to this decomposition is





The fact that᏷m is invariant underC ψ,ϕ ∗ makes the lower-left corner of this decom-position 0 Since there is a 0 at the lower left and the subspace is finite dimensional, the spectrum ofC ψ,ϕ ∗ is the union of the spectrum ofA m and the spectrum ofC m[1, page 270] SinceA mis a finite dimensional upper-triangular matrix, its spectrum is the eigen-value ofA m By the arguments in [1, pages 274-275], we can conclude that the spectrum

ofC ∗ ψ,ϕcontains the set



ψ(a), ψ(a)μ

whereμ denotes the product of m eigenvalues of ϕ (a) So { ψ(a), ψ(a)μ }is contained in

Remark 2.4 The set



K a,κ[1]

a , , κ[N]

a ,κ[1,1]

a , , κ[N,1]

a , , κ[N,N]

a , , κ

[1, 1, ., 1

Ncopies

]

a , , κ

[N, N, , N

Ncopies

]

a



(2.18)

is only the generated element set instead of the basis So the matrix representation of

C ∗ ψ,ϕ |᏷m is not unique This matrix is called the redundant matrix It can also be used to

proveLemma 2.1

By Lemmas2.1and2.3, we can easily get the following theorem, which is the main theorem of this paper

Theorem 2.5 Let C ψ,ϕ be a compact operator on the weighted Hardy space H2(β, B N ) If ϕ has only one fixed point in the unit ball, then the spectrum of C ψ,ϕ is the set



0,ψ(a), ψ(a)μ

where μ is all possible products of ϕ (a) and a is the only fixed point of ϕ.

As we will see in the next theorem, compactness ofC ψ,ϕon someH2(β, B N) for some weight functionsψ implies that ϕ has only one fixed point in the unit ball.

Theorem 2.6 Let C ψ,ϕ be a compact operator on H2(β, B N ), where

∞



s =0

(N −1 +s)!

(N −1)!s!

1

If lim inf r →1− | ψ(rζ) | > 0 for ζ is the fixed point of ϕ, then ϕ has only one fixed point in the unit ball.

Proof By contradiction, first, suppose ϕ has no fixed point, so ϕ must have its

Denjoy-Wollf point denoted byξ on ∂B N, and letr belong to the interval (0, 1).

Now, we apply the adjoint of C ψ,ϕ to the normalized kernel function K rξ /  K rξ  as follows:

C ∗ ψ,ϕ

K rξ

K rξ =ψ(rξ)K ϕ(rξ)

Sinceξ is the Denjoy-Wollf point on the boundary, there exits a sequence { ξ n }tending

toξ such that | ϕ(rξ n)| ≥ | rξ n | ButKw  =∞

s =0(((N −1 +s)!/(N −!)!s!)( | w |2s /β(s)2))

is an increasing function of| w |, K ϕ(rξ n) ≥  K rξ n , it follows that

C ψ,ϕ ∗

K rξ n

K rξ

n ≥ψ

By [4, Lemma 3.11], it follows thatK rξ n /  K rξ n converges weakly to zero asr tends to 1

andn tends to ∞

SinceC ψ,ϕ ∗ is compact, the left-hand side of (2.21) tends to 0, but the right-hand side

of (2.21) is larger thanδ ξ > 0 That is a contradiction, so ϕ must have its fixed point in

B N

Now, we show the singleness of the fixed point ofϕ By contradiction, suppose ϕ has

more than one fixed point, then the fixed point set is an affine set, we denote it by E,

which must be uncountable if not single Then C ψ,ϕ ∗ K a = ψ(a)K a for alla ∈ E E is an

affine set, it is connected, so ψ(E) is an single point set or an uncountable set

(i) Ifψ(E) is a constant, then ψ(a) is the eigenvalue of C ψ,ϕ ∗ , which is infinite multi-plicity This contradicts to the compactness ofC ∗ ψ,ϕ

(ii) If (ψ(E) is not a constant, then it has uncountable elements That is to say, C ψ,ϕ ∗

has uncountable eigenvalues That is impossible

Hence, it must be the case thatϕ has only one fixed point in the unit ball and the proof

Theorem 1 in [5] gives a method to find ψ so that C ψ,ϕ is compact on the Hardy spaceH2(B N) whenϕ has fixed points on the boundary, we discuss the spectrum for such

operators First, we quote the theorem as a lemma

Lemma 2.7 Suppose ϕ is a linear-fractional map of B N with ϕ(e1)= e1and for ζ ∈ ∂B N ,

| ϕ(ζ) | = 1 if and only if ζ = e1 If b(z) is continuous on B N with b(e1)= 0, then the operator

T b C ϕ is compact on H2(B N ).

Ifϕ has a fixed point inside the ball,Theorem 2.5gives the spectrum Therefore, we compute the spectrum whenϕ has no fixed point inside the unit ball We will denote the

composition ofϕ with itself n times by ϕ n, that is,ϕ n = ϕ ◦ ϕ ◦ ··· ◦ ϕ (n times) Now,

we give the last theorem of this paper

Theorem 2.8 Suppose ψ and ϕ satisfy the hypothesis in Lemma 2.7, and ϕ is one-to-one which has no fixed point inside the unit ball Then σ(C ψ,ϕ)= {0}

Proof We will show that the spectral radius of this operator is 0 Since ϕ is a

nonauto-morphism linear fractional map with a fixed point ate1, it takes the unit sphere to an ellipsoid sphere by [6, Theorem 6] which is tangent to the unit sphere ate1.e1is the only fixed point ofϕ, so it is the Denjoy-Wollf point.

Let > 0, there exists δ > 0 such that | ψ(z) | < whenever| z − e1| < δ and z is in the

closed unit ball LetW = { z : | z − e1| < δ, | z | ≤1}, clearly, W is open in B N LetU =

ϕ(B N), thenU is tangent to the unit sphere at e1 LetV = U − W, then V is a compact

subset of the unit ball Therefore, the sequence{ ϕ n }converges uniformly toe1 onV

Considering a pointξ on the unit sphere, then ϕ(ξ) is either in W or V If ϕ(ξ) is in V ,

then there is anN0that does not depend onξ such that ϕ j(ξ) is in W for all j > N0 Ifϕ(ξ)

is not inV , consider the sequence { ϕ j(ξ) } ∞ j =1, eitherϕ j(ξ) is in W for all j, or ϕ j(ξ) will

be inV for some j If ϕ j(ξ) is in V for some j, take j to be the smallest integer such that

ϕ j(ξ) is in V Then ϕ(ξ) is in W for all j > j +N0 Therefore, for anyξ on the unit sphere,

at mostN0terms of the sequence{ ϕ j(ξ) } ∞ j =1will be outsideW Hence, at most N0terms

of the sequence{| ψ(ϕ j(ξ)) |} ∞ j =1will be larger thanfor anyξ Also ψ is bounded on B N, therefore,| ψ(ϕ j(ξ)) | < M for some M > 0 Now, if f is in H2(B N) andn > N0, then

C n

ψ,ϕ(f ) 2

=sup

0<r<1

S

ψ(ζ) 2 ψ

ϕ(ζ) 2

···ψ

ϕ n −1(ζ) 2 f

ϕ n(ζ) 2

d(ζ)

≤ 2(n − N0−1)M2(N0 +1)sup

0<r<1

S

f

ϕ n(ζ) 2

d(ζ)

= 2(n − N0−1)M2(N0 +1) C ϕ

n(f ) 2

≤ 2(n − N0−1)M2(N0 +1) C ϕ

n 2

 f 2 ,

(2.23)

butC ϕ n = C n, therefore

C n ψ,ϕ  ≤ (n − N0−1)M(N0 +1) C ϕ

Hence, for alln large enough,

C n ψ,ϕ 1/n

≤ ·2C n 1/n

By [6, Theorem 14],C ϕis bounded

So we can get that the spectral radius of the operator on H2(B N) is 0, therefore,

Acknowledgment

This work is supported in part by the National Natural Science Foundation of China (Grants no.10671141, 10371091)

References

[1] C C Cowen and B D MacCluer, Composition Operators on Spaces of Analytic Functions, Studies

in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.

[2] G Gunatillake, “Spectrum of a compact weighted composition operator,” Proceedings of the

American Mathematical Society, vol 135, no 2, pp 461–467, 2007.

[3] R Aron and M Lindstr¨om, “Spectra of weighted composition operators on weighted banach

spaces of analytic functions,” Israel Journal of Mathematics, vol 141, pp 263–276, 2004.

[4] D D Clahane, “Spectra of compact composition operators over bounded symmetric domains,”

Integral Equations and Operator Theory, vol 51, no 1, pp 41–56, 2005.

[5] B D MacCluer and R J Weir, “Linear-fractional composition operators in several variables,”

Integral Equations and Operator Theory, vol 53, no 3, pp 373–402, 2005.

[6] C C Cowen and B D MacCluer, “Linear fractional maps of the ball and their composition

operators,” Acta Universitatis Szegediensis Acta Scientiarum Mathematicarum, vol 66, no 1-2,

pp 351–376, 2000.

Ze-Hua Zhou: Department of Mathematics, Tianjin University, Tianjin 300072, China

Email address:zehuazhou2003@yahoo.com.cn

Cheng Yuan: Department of Mathematics, Tianjin University, Tianjin 300072, China

Email address:yuancheng1984@163.com

Considering a pointξ on the unit sphere, then ϕ(ξ) is either in W or V If ϕ(ξ) is in V ,

then there... be a compact operator on the weighted Hardy space H2(β, B N ) If ϕ has only one fixed point in the unit ball, then the spectrum of C ψ,ϕ is the. .. must be the case thatϕ has only one fixed point in the unit ball and the proof

Theorem in [5] gives a method to find ψ so that C ψ,ϕ is compact on the Hardy space< i>H2(B