Fredholm Weighted Composition Operator on Weighted Hardy Space

Abstract

This paper gives a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy spaces.

1. Introduction

Let ℋ be a Hilbert space of analytic functions on the open unit disk 𝔻. For 𝜓∈ℋ and analytic self-map 𝜑 of 𝔻, the weighted composition operator 𝐶_𝜓,𝜑 on ℋ is defined as 𝐶_𝜓,𝜑𝑓=𝜓𝜑∘𝑓,𝑓∈ℋ. (1) It is natural to ask for which 𝜓 and 𝜑 the weighted composition operator 𝐶_𝜓,𝜑 is bounded and compact on ℋ. In the past ten years, this problem on various Hilbert spaces of analytic functions has been studied extensively. It seems there is no unified ways to solve the problem since different spaces have very distinguished properties; see [1–10] for the characterization of bounded and compact weighted composition operators on different function spaces. In [11], the invertibility of weighted composition operator on the classical Hardy space is characterized, as an extension, in [12, 13]; the Fredholmness of weighted composition operator on Dirichlet space and weighted Dirichlet space is characterized, respectively. The results show that the Fredholmness and invertibility of weighted composition operator are closely related to properties of the reproducing kernel functions. Inspired by the characterization of Fredholm composition operator in [6], in this paper, we give a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy space.

Now we follow some notations as in [14].

ℋ is called a weighted Hardy space if the monomials 1,𝑧,𝑧²,… constitute a complete orthogonal set of nonzero vectors in ℋ. Denote 𝛽(𝑛)=‖𝑧^𝑛‖; then for 𝑓∈ℋ with 𝑓(𝑧)=∑^∞_𝑛=0𝑓(𝑛)𝑧^𝑛, 𝑓²=^∞_𝑛=0𝑓(𝑛)²𝛽(𝑛)². (2) So we use 𝐻²(𝛽) to denote the weighed Hardy space with weighted sequence {𝛽(𝑛)}.

𝐻²(𝛽) is called automorphism invariant if, for any automorphism map 𝜑 of 𝔻, 𝑓∘𝜑∈𝐻²𝛽,𝑓∈𝐻²𝛽. (3) That is, the composition operator 𝐶_𝜑 is bounded on 𝐻²(𝛽) if 𝜑 is an automorphism map of 𝔻. It is well known that many familiar function spaces such as Hardy space, Bergman space, and Dirichlet space are automorphism-invariant weighted Hardy space. In [14, p. 127], an example of nonautomorphism-invariant weighted Hardy space is presented.

Our main result reads as follows.

Theorem 1. Let 𝐻²(𝛽) be an automorphism-invariant weighted Hardy space such that ^∞_𝑛=0𝑛^2𝑗𝛽(𝑛)²=∞ (4) for some nonnegative integer 𝑗. If 𝐶_𝜓,𝜑 is a bounded Fredholm operator on 𝐻²(𝛽), then there exist constants 𝑐, 𝑟 with 𝑐>0, 0<𝑟<1 such that |𝜓(𝑧)|≥𝑐 for 𝑟<|𝑧|<1, and 𝜑 is an automorphism of  𝔻.

As an application, in the end of this paper, we give a complete characterization of Fredholm and invertible weighed composition operator on 𝒟_𝛼 (−∞<𝛼<∞). Recall that 𝒟_𝛼 is a class of important weighed Hardy space with 𝛽(𝑛)=(𝑛+1)^𝛼.

2. Fredholm Weighted Composition Operator on 𝐻²(𝛽)

Recall that 𝐻²(𝛽) has reproducing kernel function 𝐾_𝜆(𝑧)=^∞_𝑛=0𝜆𝑧𝛽(𝑛)²,𝜆,𝑧∈𝔻. (5) That is, for 𝑓∈𝐻²(𝛽), 𝑓(𝜆)=𝑓,𝐾_𝜆. (6)

Let 𝐾_𝜆^[𝑗]=^∞_𝑛=𝑗𝑛!𝑛−𝑗!𝜆^𝑛−𝑗𝑧^𝑛𝛽(𝑛)², (7) then 𝐾_𝜆^[𝑗] is the 𝑗th derivative evaluation kernel function at 𝜆 in 𝐻²(𝛽) [14], that is, for 𝑓∈𝐻²(𝛽), 𝑓^(𝑗)(𝜆)=𝑓,𝐾_𝜆^[𝑗]. (8)

Firstly we give some discussion about weighted composition operator 𝐶_𝜓,𝜑 acting on 𝐾_𝜆^[𝑗]. The following lemmas is well known and easy to verify.

Lemma 2. Consider 𝐶^∗_𝜓,𝜑𝐾_𝜆=𝜓(𝜆)𝐾_𝜑(𝜆).

Lemma 3. Let 𝐾_𝑧^(𝑖)(𝜆)=𝐾_𝜆^[𝑖](𝑧).

By Fań di Bruno’s formula, (𝐾_𝑧∘𝜑)^(𝑖)(𝜆) can be expressed as linear combination of 𝐾_𝑧^(𝑚)𝜑(𝜆)^𝑖_𝑘=1𝜑^(𝑘)(𝜆)^𝑚𝑘, (9) where 𝑚=∑^𝑖_𝑘=1𝑚_𝑘 and ∑^𝑖_𝑘=1𝑘𝑚_𝑘=𝑖 with nonnegative integer 𝑚_𝑘.

Since 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗](𝑧)=⟨𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗],𝐾_𝑧⟩=⟨𝐾_𝜆^[𝑗],𝐶_𝜓,𝜑𝐾_𝑧⟩=(𝜓𝐾_𝑧∘𝜑)^(𝑗)(𝜆),(𝜓𝐾_𝑧∘𝜑)^(𝑗)(𝜆)=^𝑗_𝑖=0𝑗!𝑖!𝑗−𝑖!𝜓^{(𝑗−𝑖)}(𝜆)(𝐾_𝑧∘𝜑)^(𝑖)(𝜆), (10)(11) it follows from Lemma 3 and (9) that 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗] can be expressed as 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗]=^𝑗_𝑖=0𝐹_𝑖(𝜓,𝜑)(𝜆)𝐾^[𝑖]_𝜑(𝜆), (12) where 𝐹_𝑖(𝜓,𝜑) is algebraic combination of 𝜓^(𝑚) and 𝜑^(𝑘) with 0≤𝑚,  𝑘≤𝑗.

Since 𝐾^[𝑚]_𝜑(𝜆)(𝑧)=^∞_𝑛=𝑖𝑛!(𝑛−𝑚)!𝜑(𝜆)^{(𝑛−𝑚)}𝑧^𝑛𝛽(𝑛)², (13) the degree of nonzero coefficient 𝐾^[𝑚]_𝜑(𝜆)(𝑧) is at least 𝑚 and the coefficient of 𝑧^𝑘 for 𝑘≥𝑚 is 𝑘!(𝑘−𝑚)!𝜑^𝑘−𝑚(𝜆)𝛽(𝑘)², (14) so by (12) the coefficient of 𝑧^𝑛 in 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗] for 0≤𝑛≤𝑗 is ^𝑛_𝑖=0𝐹_𝑖(𝜓,𝜑)(𝜆)𝑛!(𝑛−𝑖)!𝜑^𝑛−𝑖(𝜆)𝛽(𝑛)². (15)

On the other hand, 𝜓𝐾_𝑧∘𝜑^(𝑗)(𝜆)=𝜓(𝜆)𝐾_𝑧(𝜑(𝜆))^(𝑗)=^∞_𝑛=0𝜓(𝜆)𝜑^𝑛(𝜆)^(𝑗)𝑧^𝑛𝛽(𝑛)². (16) So by (10) we have 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗](𝑧)=^∞_𝑛=0𝜓(𝜆)𝜑^𝑛(𝜆)^(𝑗)𝑧^𝑛𝛽(𝑛)². (17)

Comparing (12) and (17) with the aid of (15), we obtain 𝐹₀𝜓,𝜑(𝜆)=𝜓^(𝑗)(𝜆),𝐹₁𝜓,𝜑(𝜆)+𝐹₀𝜓,𝜑(𝜆)𝜑(𝜆)=𝜓(𝜆)𝜑(𝜆)^(𝑗),⋮𝐹_𝑖𝜓,𝜑(𝜆)+𝐹_𝑖−1𝜓,𝜑(𝜆)𝜑(𝜆)+⋯+𝐹₁(𝜆)1𝜓,𝜑()!𝜑𝑖−1^𝑖−1(𝜆)+𝐹₀𝜆)1𝜓,𝜑(𝜑𝑖!^𝑖(𝜆)=1𝜓(𝜆)𝜑𝑖!^𝑖(𝜆)^(𝑗),⋮𝐹_𝑗𝜓,𝜑(𝜆)+𝐹_𝑗−1𝜓,𝜑(𝜆)𝜑(𝜆)+⋯+𝐹₁(𝜆)1𝜓,𝜑!𝜑𝑗−1^𝑗−1(𝜆)+𝐹₀(𝜆)1𝜓,𝜑𝜑𝑗!^𝑗(𝜆)=1𝜓(𝜆)𝜑𝑗!^𝑗(𝜆)^(𝑗). (18)

By the above reasoning, we obtain the following lemma.

Lemma 4. Consider 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗]=^𝑗_𝑖=0𝐹_𝑖(𝜓,𝜑)(𝜆)𝐾^[𝑖]_𝜑(𝜆), (19) where 𝐹_𝑖(𝜓,𝜑) is linear combination of (𝜓𝜑^𝑙)^(𝑗)𝜑^𝑚 with 0≤𝑙, 𝑚≤𝑗.

Note that in [15, Lemma 2], a different form of 𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗] is given.

Now we give the characterization of Fredholm weighted composition operator on 𝐻²(𝛽).

Proposition 5. If 𝐶_𝜓,𝜑 is a bounded Fredholm operator on 𝐻²(𝛽), then 𝜓 has only finite zeroes in 𝔻, and 𝜑 is univalent.

Proof. Let 𝜆∈𝔻 such that 𝜓(𝜆)=0. By Lemma 2, 𝐶^∗_𝜓,𝜑𝐾_𝜆=0, (20) which implies that if 𝜓(𝜆)=0; then 𝐾_𝜆∈ker𝐶^∗_𝜓,𝜑. Since 𝐶_𝜓,𝜑 is a Fredholm operator, dim(ker𝐶^∗_𝜓,𝜑) is finite; it follows that 𝜓 has only finite zeroes in 𝔻.
If 𝜑(𝑎)=𝜑(𝑏) for some 𝑎,𝑏∈𝔻 with 𝑎≠𝑏, then there exists infinitely disjoint subset {𝑎_𝑛}, {𝑏_𝑛} of  𝔻 such that 𝜑(𝑎_𝑛)=𝜑(𝑏_𝑛). Since 𝜓 has only finite zeroes in 𝔻, we can assume 𝜓(𝑎_𝑛)𝜓(𝑏_𝑛)≠0. By Lemma 2, we have 𝐶^∗_𝜓,𝜑𝐾_𝑎𝑛𝜓(𝑎_𝑛)−𝐾_𝑏𝑛𝜓(𝑏_𝑛)=0, (21) which implies that 𝐾_𝑎𝑛/𝜓(𝑎_𝑛)−𝐾_𝑏𝑛/𝜓(𝑏_𝑛)∈ker(𝐶^∗_𝜓,𝜑), a contradiction to the Fredholmness of 𝐶^∗_𝜓,𝜑. So 𝜑 must be univalent.

Theorem 6. Let 𝐻²(𝛽) be a weighted Hardy space of 𝔻 such that ^∞_𝑛=0𝑛^2𝑗𝛽(𝑛)²=∞ (22) for some nonnegative integer 𝑗. If 𝐶_𝜓,𝜑 is a bounded Fredholm operator on 𝐻²(𝛽), then 𝜑 is an automorphism of 𝔻.

Proof. Let 𝑗 be the minimal nonnegative integer such that ^∞_𝑛=0𝑛^2𝑗𝛽(𝑛)²=∞. (23) Then ‖𝐾_𝜆^[𝑗]‖→∞ as |𝜆|→1⁻. As in the proof of [14, Theorem 2.17], 𝐾_𝜆^[𝑗]𝐾_𝜆^[𝑗]weaklyconvergesto0as𝜆⟶1⁻. (24) And for 0≤𝑖<𝑗,  ‖𝐾_𝜆^[𝑖]‖ is uniformly bounded for 𝜆∈𝔻.
Since 𝜓𝜑^𝑖=𝐶_𝜓,𝜑(𝑧^𝑖), it follows that 𝜓𝜑^𝑖∈𝐻(𝛽). So 𝜓𝜑^𝑖(𝑗)(𝜆)=𝜓𝜑^𝑖,𝐾_𝜆^[𝑗],lim_{|𝜆|→1−}𝜓𝜑^𝑖(𝑗)(𝜆)𝐾_𝜆^[𝑗]=0. (25)
By induction and the fact that |𝜑(𝜆)|<1, it follows from Lemma 4 that lim_{|𝜆|→1−}𝐹_𝑖𝜓,𝜑(𝜆)𝐾_𝜆^[𝑗]=0 (26) for 0≤𝑖≤𝑗.
By (19), we have lim_{|𝜆|→1−}𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗]𝐾_𝜆^[𝑗]≤^𝑗_𝑖=0lim_{|𝜆|→1−}𝐹_𝑖(𝜓,𝜑)(𝜆)𝐾_𝜆^[𝑗]𝐾^[𝑖]_𝜑(𝜆)=lim_{|𝜆|→1−}𝐹_𝑗𝜓,𝜑(𝜆)𝐾_𝜆^[𝑗]𝐾_𝜑(𝜆)^[𝑗]. (27)
As in [6, Theorem 2], we can obtain that 𝜑 is onto, otherwise lim_{|𝜆|→1−}𝐶^∗_𝜓,𝜑𝐾_𝜆^[𝑗]𝐾_𝜆^[𝑗]=0. (28) contradicts the Fredholmness of 𝐶_𝜓,𝜑. Combined with Proposition 5, it follows that 𝜑 is an automorphism of 𝔻.

Theorem 6 gives the necessary condition of 𝜑 for the weighted composition operator 𝐶_𝜓,𝜑 to be Fredholm on 𝐻²(𝛽) with condition (22). Furthermore, if 𝐻²(𝛽) is automorphism invariant, then the multiplication operator 𝑀_𝜓 on 𝐻²(𝛽) must be bounded Fredholm since in this case 𝐶_𝜑 is invertible and 𝑀_𝜓=𝐶_𝜓,𝜑𝐶_𝜑−1. (29) So in the following, we consider the condition for a multiplication operator 𝑀_𝜓 on 𝐻²(𝛽) to be Fredholm.

Denote ℳ={𝜓isanalyticin𝔻,𝜓𝑓∈𝐻²(𝛽),𝑓∈𝐻²(𝛽)}. For 𝜓∈ℳ, the multiplication operator 𝑀_𝜓 on 𝐻²(𝛽) is bounded. ℳ is called the multiplier space of 𝐻²(𝛽).

Lemma 7. Let 𝜓∈ℳ. Then 𝑀^∗_𝜓𝐾_𝜆^[𝑗]=∑^𝑗_𝑖=0𝐶^𝑖_𝑗𝜓^{(𝑗−𝑖)}(𝜆)𝐾_𝜆^[𝑖].

Proposition 8. Let 𝐻²(𝛽) be a weighted Hardy space such that ^∞_𝑛=0𝑛^2𝑗𝛽(𝑛)²=∞ (30) for some nonnegative integer 𝑗. If 𝑀_𝜓 is a bounded Fredholm operator on 𝐻²(𝛽), then there exist 𝑐>0 and 0<𝑟<1 such that |𝜓(𝑧)|≥𝑐 for 𝑟<|𝑧|<1.

Proof. If 𝑀_𝜓 is a Fredholm operator, then there exist bounded operator 𝑇 and compact operator 𝑆 such that 𝑇𝑀^∗_𝜓=𝐼+𝐾, (31) where 𝐼 is identity. Acting on 𝐾_𝜆^[𝑗]/‖𝐾_𝜆^[𝑗]‖, we have ^𝑗_𝑖=0𝐶^𝑖_𝑗𝜓^{(𝑗−𝑖)}(𝜆𝐾)^𝜆[𝑖]𝐾_𝜆^[𝑗]‖𝑇‖≥𝑇𝑀^∗_𝜓𝐾_𝜆^[𝑗]=𝐾_𝜆^[𝑗]𝐾_𝜆^[𝑗]𝐾+𝑆^𝜆[𝑗]𝐾_𝜆^[𝑗]𝑆𝐾≥1−^𝜆[𝑗]𝐾_𝜆^[𝑗]. (32) Let |𝜆|→1⁻; then |𝜓(𝜆)|≥1/‖𝑇‖, since ‖𝑆(𝐾_𝜆^[𝑗]/‖𝐾_𝜆^[𝑗]‖)‖→0.

The following example shows that generally 𝐻²(𝛽) is not automorphism invariant even if 𝐻²(𝛽) satisfies condition (22).

Example 9. Fix 𝑎∈𝔻(𝑎≠0). Let 𝛽(2𝑘)=1 and 𝛽(2𝑘−1)=|𝑎|^{−(2𝑘−1)} for 𝑘=1,2,…. It is easy to verify that ^∞_𝑛=01𝛽(𝑛)²=∞,^∞_𝑛=0𝛽(𝑛)²𝑎^𝑛2=∞, (33) which means that ‖𝜑_𝑎‖=∞ and hence 𝜑_𝑎∉𝐻²(𝛽), where 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧) is an automorphism of 𝔻.

Now, we give the proof of the main result of Theorem 1.

Proof. By Theorem 6, 𝜑 is an automorphism of 𝔻. Since 𝐻²(𝛽) is automorphism invariant, 𝐶_𝜑 is bounded and invertible with 𝐶_𝜑⁻¹=𝐶_𝜑−1. It follows that 𝑀_𝜓=𝐶_𝜓,𝜑𝐶_𝜑−1 is a bounded Fredholm operator on 𝐻²(𝛽). By Proposition 8, there exist 𝑐>0 and 0<𝑟<1 such that |𝜓(𝑧)|≥𝑐 for 𝑟<|𝑧|<1.

3. Application

In the section, we give a complete characterization of bounded Fredholm and bounded invertible weighted composition operator on 𝒟_𝛼.

First, we cite some well-known results of 𝒟_𝛼. Denotes ℳ_𝛼 by the multiplier space of 𝒟_𝛼.(1)For 𝛼>1, 𝒟_𝛼 is an algebra. 𝛼=1 is the classical Dirichlet space. For 𝛼<1,  𝒟_𝛼 is equivalent to the Hilbert space with reproducing kernel function 𝐾_𝜆(𝑧)=1/(1−𝜆𝑧)^1−𝛼. 𝛼=0 is the classical Hardy space. For 𝛼<0, 𝒟_𝛼=𝐿²_𝑎(𝔻,(1−|𝑧|²)^−1−𝛼𝑑𝐴) with equivalent norms [16, 17]. (2)For 𝛼>1,  ℳ_𝛼=𝐷_𝛼. For 𝛼≤0, ℳ_𝛼=𝐻^∞(𝔻). (For more information about ℳ_𝛼, see [18].)(3)Let 𝜓∈ℳ_𝛼. 𝑀_𝜓 is Fredholm if and only if there exist 𝑐>0, 0<𝑟<1, |𝜓(𝑧)|≥𝑐,  and 𝑟<|𝑧|<1. 𝑀_𝜓 is invertible if and only if there exists 𝑐>0 such that |𝜓(𝑧)|≥𝑐, |𝑧|<1.(4)𝒟_𝛼 is automorphism invariant.

By Theorem 1, (29), and the results (3), (4) above, we have the following corollaries.

Corollary 10. 𝐶_𝜓,𝜑 is a bounded Fredholm operator on 𝒟_𝛼 if and only if 𝜓∈ℳ_𝛼, and there exist 𝑐>0 and 0<𝑟<1 such that |𝜓(𝑧)|≥𝑐 for 𝑟<|𝑧|<1;  𝜑 is automorphism of 𝔻.

Corollary 11. 𝐶_𝜓,𝜑 is a bounded invertible operator on 𝒟_𝛼 if and only if 𝜓∈ℳ_𝛼, and there exists 𝑐>0 such that |𝜓(𝑧)|≥𝑐 for |𝑧|<1; 𝜑 is automorphism of 𝔻.

Acknowledgments

This work is supported by NSFC (10971195, 11201274) and Young Science Funds of Shanxi (2010021002-2).