On the Generalized Hardy Spaces

Abstract

We introduce new spaces that are extensions of the Hardy spaces and we investigate the continuity of the point evaluations as well as the boundedness and the compactness of the composition operators on these spaces.

1. Introduction

Let 𝑈 be the open unit disk in the complex plane ℂ, 𝜕𝑈 its boundary, and 𝐻(𝑈) the space of all analytic functions on the unit disk.

For an analytic function 𝑓 on the unit disk and 0<𝑟<1, we define the delay function 𝑓_𝑟 by 𝑓_𝑟(𝑒^𝑖𝜃)=𝑓(𝑟𝑒^𝑖𝜃). It is easy to see that the functions 𝑓_𝑟 are continuous for each 𝑟, hence they are in 𝐿^𝑝(𝜕𝑈,𝑑𝜃/2𝜋), where 𝑑𝜃/2𝜋 is the normalized arc length on the unit circle.

For 0<𝑝<∞, the Hardy space 𝐻^𝑝(𝑈)=𝐻^𝑝 is the set of all 𝑓∈𝐻(𝑈) such that

‖𝑓‖^𝑝_𝑝=sup_0<𝑟<10^2𝜋|𝑓_𝑟(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋<∞.(1.1) Also we recall that 𝐻^∞(𝑈)=𝐻^∞ is the space of all bounded analytic functions defined on 𝑈, with supremum norm ‖𝑓‖_∞=sup_𝑧∈𝑈|𝑓(𝑧)|. We know that for 𝑝≥1, 𝐻^𝑝 is a Banach space (see, e.g., [1, page 37]).

By the Littlewood Subordination Theorem (see [2, Corollary 2.23]), we see that the supremum in the above definition of 𝐻^𝑝 is actually a limit, that is,

‖𝑓‖^𝑝_𝑝=lim_𝑟→10^2𝜋|𝑓_𝑟(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋<∞.(1.2) Another important result is Fatou's Radial Limit Theorem (see [1, Theorems 2.2 and 2.6]), which says, if 𝑓 is in 𝐻^𝑝 for some 𝑝>0, then the radial limit

𝑓^∗(𝑒^𝑖𝜃)=lim_𝑟→1𝑓(𝑟𝑒^𝑖𝜃)(1.3) exists for almost all 𝜃 and the mapping 𝑓→𝑓^∗ is an isometry of 𝐻^𝑝 to a closed subspace of 𝐿^𝑝(𝜕𝑈,𝑑𝜃/2𝜋). Therefore,

‖𝑓‖^𝑝_𝑝=₀^2𝜋|𝑓^∗(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋<∞.(1.4) We will also write 𝑓(𝑒^𝑖𝜃) for 𝑓^∗(𝑒^𝑖𝜃). If 𝑝=2 and 𝑓(𝑛) are the 𝑛th coefficients of 𝑓 in its Maclaurin series, then we have another representation for the norm of 𝑓 on 𝐻² as follows:

‖𝑓‖²₂=^∞_𝑛=0|𝑓(𝑛)|²<∞.(1.5) The formula above defines a norm that turns 𝐻² into a Hilbert space whose inner product is given by

⟨𝑓,𝑔⟩_𝐻2=^∞_𝑛=0𝑓(𝑛)𝑔(𝑛)=₀^2𝜋𝑓(𝑒^𝑖𝜃)𝑔(𝑒^𝑖𝜃)𝑑𝜃2𝜋(1.6) for each 𝑓,𝑔∈𝐻² (see, e.g., [2]).

Let 𝑒_𝑤 be the point evaluation at 𝑤, that is, 𝑒_𝑤(𝑓)=𝑓(𝑤). It is well known that point evaluations at the points of 𝑈 are all continuous on 𝐻^𝑝 (see, e.g., [1, page 36]).

Let 𝑤∈𝑈 and 𝐻 be a Hilbert space of analytic functions on 𝑈. If 𝑒_𝑤 is a bounded linear functional on 𝐻, then the Riesz Representation Theorem implies that there is a function (which is usually called 𝐾_𝑤) in 𝐻 that induces this linear functional, that is, 𝑒_𝑤(𝑓)=⟨𝑓,𝐾_𝑤⟩.

Let 𝜑 be an analytic self-map of the unit disk. The linear composition operator 𝐶_𝜑 is defined by 𝐶_𝜑(𝑓)=𝑓∘𝜑 for 𝑓∈𝐻(𝑈). It is well known (see, e.g., [1, page 29] or [3, Theorem 1]) that the composition operators are bounded on each of the Hardy spaces 𝐻^𝑝 (0<𝑝<∞). One of the first papers in this research area is [3], while Schwartz in [4] begun the research on compact composition operators on 𝐻^𝑝. Shapiro and Taylor in [5] have studied the role of angular derivative for compactness of 𝐶_𝜑 in 𝐻^𝑝. For some other classical results, see [2, 6].

The boundedness and compactness of composition operators, as well as weighted composition operators and other natural extensions of them, on various spaces of analytic functions have been investigated by many authors; see, books [2, 6], and, for example, the following recent papers [7–28] and the references therein.

Throughout this paper, 𝑃 denotes the set of all analytic polynomials and for a function 𝐹, 𝑅_𝐹 denotes the range of 𝐹.

2. Generalized Hardy Spaces

In this section, we define new spaces and investigate some basic properties of these spaces.

Definition 2.1. Let 𝐹∶𝐻(𝑈)→𝐻(𝑈) be a linear operator such that 𝐹(𝑓)=0 if and only if 𝑓=0, that is, 𝐹 is 1-1. For 1≤𝑝<∞, the generalized Hardy space 𝐻_𝐹,𝑝(𝑈)=𝐻_𝐹,𝑝 is defined to be the collection of all analytic functions 𝑓 on 𝑈 for which sup_0<𝑟<10^2𝜋|(𝐹(𝑓))_𝑟(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋<∞.(2.1) Denote the 𝑝th root of this supremum by ‖𝑓‖_{𝐻𝐹,𝑝}. Since |𝐹(𝑓)|^𝑝 is a subharmonic function, so by [2, Corollary 2.23], we have ‖𝑓‖^𝑝_{𝐻𝐹,𝑝}=lim_𝑟→10^2𝜋|(𝐹(𝑓))_𝑟(𝑒^𝑖𝜃)|^𝑝𝑑𝜃.2𝜋(2.2) Therefore, 𝑓∈𝐻_𝐹,𝑝 if and only if 𝐹(𝑓)∈𝐻^𝑝 and ‖𝑓‖^𝑝_{𝐻𝐹,𝑝}=‖𝐹(𝑓)‖^𝑝_𝑝=₀^2𝜋|𝐹(𝑓)(𝑒^𝑖𝜃)|^𝑝𝑑𝜃.2𝜋(2.3) It is easy to see that 𝐻_𝐹,𝑝 is a normed space with the norm ‖⋅‖_{𝐻𝐹,𝑝}.

From now on, unless otherwise stated, we assume that 𝐹 satisfies the conditions of Definition 2.1.

In this section, we first set some conditions on 𝐹 such that 𝐻_𝐹,𝑝 becomes a Banach space. In the following theorem, we obtain a necessary and sufficient condition for 𝐻_𝐹,𝑝 to be a Banach space.

Theorem 2.2. Let 1≤𝑝<∞ and 𝑃⊆𝑅_𝐹. Then 𝐻^𝑝 is a subspace of 𝑅_𝐹 if and only if 𝐻_𝐹,𝑝 is a Banach space.

Proof. Suppose that 𝐻^𝑝⊆𝑅_𝐹. Since 𝐻_𝐹,𝑝 is a normed space, it suffices to show that it is complete. Let {𝑓_𝑛} be a Cauchy sequence in 𝐻_𝐹,𝑝 and set 𝐹(𝑓_𝑛)=𝑔_𝑛. Then {𝑔_𝑛} is a Cauchy sequence in 𝐻^𝑝. Since 𝐻^𝑝 is complete, there is a 𝑔∈𝐻^𝑝 such that ‖𝑔_𝑛−𝑔‖_𝑝→0,as𝑛→∞.(2.4) Since 𝐻^𝑝⊆𝑅_𝐹, there is an 𝑓∈𝐻(𝑈) such that 𝐹(𝑓)=𝑔. Now we show that this 𝑓 is the 𝐻_𝐹,𝑝-limit of {𝑓_𝑛}. We have ‖𝑓_𝑛−𝑓‖_{𝐻𝐹,𝑝}=‖𝑔_𝑛−𝑔‖_𝑝→0,as𝑛→∞.(2.5) Hence 𝑓_𝑛−𝑓∈𝐻_𝐹,𝑝 for sufficiently large positive integer 𝑛, which implies that 𝑓∈𝐻_𝐹,𝑝. So 𝑓_𝑛→𝑓 in 𝐻_𝐹,𝑝 as 𝑛→∞.
Conversely, suppose that 𝐻_𝐹,𝑝 is a Banach space. If 𝐻^𝑝⊆𝑅̸_𝐹, then there is a 𝑔∈𝐻^𝑝 such that 𝑔 is not in 𝑅_𝐹. Since the polynomials are dense in 𝐻^𝑝, there is a sequence {𝑝_𝑛} in 𝑃 such that ‖𝑝_𝑛−𝑔‖_𝑝→0 as 𝑛→∞. Let 𝑞_𝑛=𝐹⁻¹(𝑝_𝑛). Then {𝑞_𝑛} is a Cauchy sequence in 𝐻_𝐹,𝑝 and so there is a 𝑞∈𝐻_𝐹,𝑝 such that ‖𝑞_𝑛−𝑞‖_{𝐻𝐹,𝑝}→0 as 𝑛→∞. Hence ‖𝐹(𝑞_𝑛)−𝐹(𝑞)‖_𝑝→0 as 𝑛→∞. On the other hand, ‖𝐹(𝑞_𝑛)−𝑔‖_𝑝→0 as 𝑛→∞. This shows that 𝑔=𝐹(𝑞) which is a contradiction.

Example 2.3. Let 𝐹(∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=∑^∞_𝑗=0(𝑎_𝑗/2^𝑗)𝑧^𝑗, where ∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻(𝑈). It is not hard to see that 𝑔(𝑧)=∑^∞_𝑛=02^−𝑛/2𝑧^𝑛∈𝐻² and 𝑔 is not in 𝑅_𝐹. So by Theorem 2.2, 𝐻_𝐹,2 is not a Banach space.

Proposition 2.4. If 𝐻²⊆𝑅_𝐹, then 𝐻_𝐹,2 is a Hilbert space.

Proof. We define a scalar product on 𝐻_𝐹,2 by ⟨𝑓,𝑔⟩_𝐻𝐹,2=⟨𝐹(𝑓),𝐹(𝑔)⟩_𝐻2.(2.6) It is easy to show that this scalar product defines an inner product on 𝐻_𝐹,2.

There is a Banach space 𝐻_𝐹,𝑝 such that it does not satisfy the condition of Theorem 2.2. For example, let 1≤𝑝<∞, 𝐹(𝑓)=𝑧𝑓 for each 𝑓∈𝐻(𝑈). Then 1∉𝑅_𝐹. By the following proposition, we see that although 𝐻^𝑝⊆𝑅̸_𝐹, 𝐻_𝐹,𝑝 is a Banach space.

Proposition 2.5. Suppose 1≤𝑝<∞, ℎ∈𝐻(𝑈), ℎ≢0, and 𝐹(𝑓)=𝑓ℎ for every 𝑓∈𝐻(𝑈). Then 𝐻_𝐹,𝑝 is a Banach space.

Proof. If 𝐻^𝑝⊆𝑅_𝐹, then by Theorem 2.2, the proposition holds. Otherwise, let {𝑓_𝑛} be a Cauchy sequence in 𝐻_𝐹,𝑝. Seting 𝐹(𝑓_𝑛)=𝑔_𝑛, so {𝑔_𝑛} is a Cauchy sequence in 𝐻^𝑝. Therefore, there is a 𝑔∈𝐻^𝑝 such that ‖𝑔_𝑛−𝑔‖_𝑝→0 as 𝑛→∞. If 𝑔∈𝑅_𝐹, then the proof is similar to the proof of Theorem 2.2.
Now suppose that 𝑔 is not in 𝑅_𝐹. Then there are 𝑧₀∈𝑈, 𝑚₁≥0, and 𝑚₂>𝑚₁ such that 𝑔(𝑧)=(𝑧−𝑧₀)^𝑚1𝑔₀(𝑧),ℎ(𝑧)=(𝑧−𝑧₀)^𝑚2ℎ₀(𝑧),(2.7) where ℎ₀,𝑔₀∈𝐻(𝑈), 𝑔₀(𝑧₀)≠0, and ℎ₀(𝑧₀)≠0. Therefore, we have ‖𝑔_𝑛−𝑔‖_𝑝=‖ℎ𝑓_𝑛−𝑔‖_𝑝=₀^2𝜋|(𝑒^𝑖𝜃−𝑧₀)^𝑚2ℎ₀(𝑒^𝑖𝜃)𝑓_𝑛(𝑒^𝑖𝜃)−(𝑒^𝑖𝜃−𝑧₀)^𝑚1𝑔₀(𝑒^𝑖𝜃)|^𝑝𝑑𝜃=2𝜋_02𝜋|(𝑒^𝑖𝜃−𝑧₀)|^𝑚1|(𝑒^𝑖𝜃−𝑧₀)^{𝑚2−𝑚1}ℎ₀(𝑒^𝑖𝜃)𝑓_𝑛(𝑒^𝑖𝜃)−𝑔₀(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋≥(1−|𝑧₀|)^𝑚1₀^2𝜋|(𝑒^𝑖𝜃−𝑧₀)^{𝑚2−𝑚1}ℎ₀(𝑒^𝑖𝜃)𝑓_𝑛(𝑒^𝑖𝜃)−𝑔₀(𝑒^𝑖𝜃)|^𝑝𝑑𝜃.2𝜋(2.8) Since ‖𝑔_𝑛−𝑔‖_𝑝→0 as 𝑛→∞, we have lim_𝑛→∞0^2𝜋|(𝑒^𝑖𝜃−𝑧₀)^{𝑚2−𝑚1}ℎ₀(𝑒^𝑖𝜃)𝑓_𝑛(𝑒^𝑖𝜃)−𝑔₀(𝑒^𝑖𝜃)|^𝑝𝑑𝜃2𝜋=0.(2.9) Hence ‖(𝑧−𝑧₀)^{𝑚2−𝑚1}ℎ₀𝑓_𝑛−𝑔₀‖_𝑝→0 as 𝑛→∞. Since the point evaluation at 𝑧₀ is a bounded linear functional on 𝐻^𝑝, we have (𝑧₀−𝑧₀)^{𝑚2−𝑚1}ℎ₀(𝑧₀)𝑓_𝑛(𝑧₀)−𝑔₀(𝑧₀)→0,as𝑛→∞.(2.10) So 𝑔₀(𝑧₀)=0, which is a contradiction.

The set of all analytic polynomials is dense in 𝐻^𝑝, but this is not the case for each space 𝐻_𝐹,𝑝. Also it is possible that 𝑃⊆𝐻̸_𝐹,𝑝. For example, let 𝑝=2, 𝑔(𝑧)=1/(1−𝑧), and 𝐹(𝑓)=𝑓𝑔. Then 1 is not in 𝐻_𝐹,2, for if 1∈𝐻_𝐹,2, then 𝐹(1)=𝑔∈𝐻², which is a contradiction.

In the following proposition, we will find a dense subset in 𝐻_𝐹,𝑝, whenever 𝑃⊆𝑅_𝐹.

Proposition 2.6. Suppose 1≤𝑝<∞ and 𝑃⊆𝑅_𝐹. Then {𝐹⁻¹(𝑝)∶𝑝∈𝑃}=𝐻_𝐹,𝑝.

Proof. It is clear that {𝐹⁻¹(𝑝)∶𝑝∈𝑃}⊆𝐻_𝐹,𝑝. Suppose that 𝑓∈𝐻_𝐹,𝑝. Then there is a sequence {ℎ_𝑛} in 𝑃 such that ‖ℎ_𝑛−𝐹(𝑓)‖_𝑝→0 as 𝑛→∞. Setting 𝑓_𝑛=𝐹⁻¹(ℎ_𝑛), we have ‖𝑓_𝑛−𝑓‖_{𝐻𝐹,𝑝}=‖ℎ_𝑛−𝐹(𝑓)‖_𝑝,(2.11) so the result follows.

Corollary 2.7. Suppose 1≤𝑝<∞, 𝑃⊆𝑅_𝐹, and 𝐹⁻¹(𝑝)∈𝑃 for each 𝑝∈𝑃. Then 𝑃∩𝐻_𝐹,𝑝=𝐻_𝐹,𝑝.

3. Point Evaluations

In this section, we investigate the continuity of the point evaluations on 𝐻_𝐹,𝑝. The idea behind Theorem 3.1 is similar to the one found in [29, page 51].

Theorem 3.1. Suppose that 𝑤∈𝑈 and 𝐻²⊆𝑅_𝐹. Then we have the following.(a)If 𝑝≥2 and ∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗∈𝐻², then 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝. (b)Let 1≤𝑝<2 and ∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗∈𝐻^∞. If for each 0<𝑟<1 and 𝑓∈𝐻_𝐹,1, (𝐹(𝑓))_𝑟=𝐹(𝑓_𝑟), then 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝.

Proof. (a) By Proposition 2.4, 𝐻_𝐹,2 is a Hilbert space. Since ⟨𝐹⁻¹(𝑧^𝑗),𝐹⁻¹(𝑧^𝑘)⟩_𝐻𝐹,2=⟨𝑧^𝑗,𝑧^𝑘⟩_𝐻²=0,‖𝐹⁻¹(𝑧^𝑗)‖_𝐻𝐹,2=‖𝑧^𝑗‖_𝐻²=1(3.1) for each 𝑗,𝑘∈ℕ∪{0}, where 𝑗≠𝑘, {𝐹⁻¹(𝑧^𝑗)∶𝑗∈ℕ∪{0}} is an orthonormal set in 𝐻_𝐹,2. Also if 𝑔∈𝐻_𝐹,2 and for each 𝑗∈ℕ∪{0}, ⟨𝑔,𝐹⁻¹(𝑧^𝑗)⟩_𝐻𝐹,2=0, then for each 𝑗∈ℕ∪{0}, ⟨𝐹(𝑔),𝑧^𝑗⟩_𝐻²=0. Since {𝑧^𝑗∶𝑗∈ℕ∪{0}} is a basis for 𝐻², 𝑔≡0. Therefore, {𝐹⁻¹(𝑧^𝑗)∶𝑗∈ℕ∪{0}} is a basis for 𝐻_𝐹,2. Since ∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗∈𝐻², there is an ℎ∈𝐻_𝐹,2 such that 𝐹(ℎ)=∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗. For each 𝑗∈ℕ∪{0}, we have ⟨𝐹⁻¹(𝑧^𝑗),ℎ⟩_𝐻𝐹,2=⟨𝑧^𝑗,𝐹(ℎ)⟩_𝐻²=⟨𝑧^𝑗,^∞_𝑘=0𝐹⁻¹(𝑧^𝑘)(𝑤)𝑧^𝑘⟩_𝐻2=𝐹⁻¹(𝑧^𝑗)(𝑤).(3.2) Hence ℎ=𝐾_𝑤∈𝐻_𝐹,2 and 𝑒_𝑤 is continuous on 𝐻_𝐹,2.
Let 𝑝≥2. If 𝑓∈𝐻_𝐹,𝑝, then |𝑓(𝑤)|≤‖𝐾_𝑤‖_𝐻𝐹,2‖𝑓‖_𝐻𝐹,2≤‖𝐾_𝑤‖_𝐻𝐹,2‖𝑓‖_{𝐻𝐹,𝑝},(3.3) so 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝.
(b) Let 𝑓∈𝐻_𝐹,1. Then for each 0<𝑟<1, 𝑓_𝑟∈𝐻_𝐹,2 and so 𝑓_𝑟(𝑤)=⟨𝑓_𝑟,𝐾_𝑤⟩_𝐻𝐹,2=⟨𝐹(𝑓_𝑟),𝐹(𝐾_𝑤)⟩_𝐻2=₀^2𝜋𝐹(𝑓_𝑟)(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃.2𝜋(3.4) Also by [1, Theorem 2.6], ‖(𝐹(𝑓))_𝑟−𝐹(𝑓)‖₁→0 as 𝑟→1. Therefore, by Holder's inequality and the fact that 𝐹(𝐾_𝑤)=∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗, we have |₀^2𝜋(𝐹(𝑓))_𝑟(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃2𝜋−₀^2𝜋𝐹(𝑓)(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃|2𝜋≤‖𝐹(𝐾_𝑤)‖_∞0^2𝜋|(𝐹(𝑓_𝑟)−𝐹(𝑓))(𝑒^𝑖𝜃)|𝑑𝜃2𝜋≤‖𝐹(𝐾_𝑤)‖_∞‖(𝐹(𝑓))_𝑟−𝐹(𝑓)‖₁→0,(3.5) as 𝑟→1. So we obtain 𝑓(𝑤)=lim_𝑟→1𝑓_𝑟(𝑤)=lim_𝑟→10^2𝜋𝐹(𝑓_𝑟)(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃2𝜋=₀^2𝜋𝐹(𝑓)(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃.2𝜋(3.6) Hence |𝑓(𝑤)|=|₀^2𝜋𝐹(𝑓)(𝑒^𝑖𝜃)𝐹(𝐾_𝑤)(𝑒^𝑖𝜃)𝑑𝜃|2𝜋≤‖𝐹(𝐾_𝑤)‖_∞0^2𝜋|𝐹(𝑓)(𝑒^𝑖𝜃)|𝑑𝜃2𝜋=‖𝐹(𝐾_𝑤)‖_∞‖𝑓‖_𝐻𝐹,1(3.7) for each 𝑓∈𝐻_𝐹,1. Now let 1≤𝑝<2. If 𝑓∈𝐻_𝐹,𝑝, then |𝑓(𝑤)|≤‖𝐹(𝐾_𝑤)‖_∞‖𝑓‖_𝐻𝐹,1≤‖𝐹(𝐾_𝑤)‖_∞‖𝑓‖_{𝐻𝐹,𝑝},(3.8) so the result follows.

Suppose that 𝑤 and 𝐹 satisfy the hypotheses in the first line of Theorem 3.1. It is easy to see that if 𝑒_𝑤 is continuous on 𝐻_𝐹,2, then according to the proof of the previous theorem, 𝐹(𝐾_𝑤)=∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗.

Now we give two examples for the preceding theorem.

Example 3.2. (a) Let 𝑤∈𝑈 and 𝐹(∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=𝑎₀+∑^∞_𝑗=1𝑗𝑎_𝑗𝑧^𝑗, where ∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻(𝑈). It is easy to see that ∑^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗=1+∑^∞_𝑗=1((𝑤)^𝑗/𝑗)𝑧^𝑗∈𝐻^∞ and for each 𝑓∈𝐻_𝐹,1, 𝐹(𝑓_𝑟)=(𝐹(𝑓))_𝑟. Therefore, by Theorem 3.1, 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝 for each 1≤𝑝<∞.
(b) Let 𝑝≥2 and 𝐹(∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=∑^∞_𝑗=0𝑏_𝑗𝑧^𝑗 such that (𝑏₀,𝑏₁,…,𝑏_𝑛) is a permutation of (𝑎₀,𝑎₁,…,𝑎_𝑛) for some 𝑛∈ℕ and 𝑏_𝑘=𝑎_𝑘 for 𝑘>𝑛. Then by Theorem 3.1, for each point 𝑤 in the open unit disk, 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝. Also if 𝑝=2 and 𝑆 is the above permutation, then 𝐹(𝐾_𝑤)=^∞_𝑗=0𝐹⁻¹(𝑧^𝑗)(𝑤)𝑧^𝑗=^∞_𝑗=0(𝑤)^{𝑆−1(𝑗)}𝑧^𝑗.(3.9) Therefore, 𝐾_𝑤(𝑧)=∑^∞_𝑗=0(𝑤)^{𝑆−1(𝑗)}𝑧^{𝑆−1(𝑗)}.

There is a Banach space 𝐻_𝐹,𝑝 such that it does not satisfy the conditions of Theorem 3.1, but for each 𝑤∈𝑈, 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝. For example, let 1≤𝑝<∞, 𝑔(𝑧)=1/2−𝑧, and 𝐹(𝑓)=𝑓𝑔 for each 𝑓∈𝐻(𝑈). Then 1 is not in 𝑅_𝐹 and 𝐹((1)_𝑟)≢(𝐹(1))_𝑟. By the following theorem, we see that although the hypotheses of Theorem 3.1 do not hold, 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝 for each 𝑤∈𝑈.

Theorem 3.3. Let 1≤𝑝<∞, 𝑤∈𝑈, ℎ∈𝐻(𝑈), ℎ≢0, and for each 𝑓∈𝐻(𝑈), 𝐹(𝑓)=𝑓ℎ. Then 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝.

Proof. We break the proof into two parts.
(1) Let ℎ(𝑤)≠0. If |𝑤|<𝑟<1 and Γ_𝑟 is the circle of radius 𝑟 with center at the origin, then the Cauchy formula shows that for any 𝑓 in 𝐻_𝐹,𝑝, 1𝑓(𝑤)ℎ(𝑤)=2𝜋𝑖_Γ𝑟𝑓(𝜁)ℎ(𝜁)=1𝜁−𝑤𝑑𝜁2𝜋𝑖₀^2𝜋𝑓(𝑟𝑒^𝑖𝜃)ℎ(𝑟𝑒^𝑖𝜃)𝑟𝑒^𝑖𝜃−𝑤𝑟𝑖𝑒^𝑖𝜃=𝑑𝜃_02𝜋𝑓(𝑟𝑒^𝑖𝜃)ℎ(𝑟𝑒^𝑖𝜃)𝑟𝑟−𝑤𝑒^−𝑖𝜃𝑑𝜃.2𝜋(3.10) It follows that |𝑓(𝑤)||ℎ(𝑤)|≤‖(𝑓ℎ)_𝑟‖_𝑝‖𝑟𝑟−𝑤𝑒^−𝑖𝜃‖_𝑞,(3.11) where 1/𝑝+1/𝑞=1. Now if 𝑟 tends to 1, |𝑟/(𝑟−𝑤𝑒^−𝑖𝜃)| converges uniformly to the bounded function |1−𝑤𝑒^𝑖𝜃|⁻¹ and ‖(𝑓ℎ)_𝑟‖_𝑝≤‖𝑓ℎ‖_𝑝. Hence there is an 𝑀<∞ such that 𝑀|𝑓(𝑤)|≤|ℎ(𝑤)|‖𝑓‖_{𝐻𝐹,𝑝},(3.12) and the result follows.
(2) Let ℎ(𝑤)=0. Then ℎ(𝑧)=(𝑧−𝑤)^𝑚ℎ₀(𝑧), where 𝑚∈ℕ, ℎ₀∈𝐻(𝑈), and ℎ₀(𝑤)≠0. Let 𝐹₁(𝑓)=𝑓ℎ₀ for each 𝑓∈𝐻(𝑈), it is easy to see that 𝐻_𝐹,𝑝⊆𝐻_𝐹1,𝑝. Then by the preceding part, there is a constant 0<𝐶<∞ such that |𝑓(𝑤)|^𝑝≤𝐶‖𝑓ℎ₀‖^𝑝_𝑝=𝐶₀^2𝜋|𝑓(𝑒^𝑖𝜃)|^𝑝|ℎ₀(𝑒^𝑖𝜃)|^𝑝|𝑒^𝑖𝜃−𝑤|^𝑚𝑝|𝑒^𝑖𝜃−𝑤|^𝑚𝑝𝑑𝜃≤𝐶2𝜋(1−|𝑤|)^𝑚𝑝₀^2𝜋|𝑓(𝑒^𝑖𝜃)ℎ(𝑒^𝑖𝜃)|^𝑝𝑑𝜃=𝐶2𝜋(1−|𝑤|)^𝑚𝑝‖𝑓‖^𝑝_{𝐻𝐹,𝑝}(3.13) for each 𝑓∈𝐻_𝐹,𝑝. So 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝.

There is an example of 𝐻_𝐹,𝑝 such that 𝐻_𝐹,2 is not a Hilbert space and 𝑒_𝑤 is continuous for some 𝑤∈𝑈.

Example 3.4. Let 𝑝=2 and 𝐹(∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=∑^∞_𝑗=0(𝑎_𝑗/2^𝑗)𝑧^𝑗, where ∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻(𝑈). We can show that for each 𝑤∈(1/2)𝑈, 𝑒_𝑤 is continuous on 𝐻_𝐹,2. Suppose 𝑓(𝑧)=∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻_𝐹,2, we have |𝑓(𝑤)|=|^∞_𝑗=0𝑎_𝑗𝑤^𝑗|=|^∞_𝑗=0𝑎_𝑗2^𝑗(2𝑤)^𝑗|≤(^∞_𝑗=0|𝑎_𝑗2^𝑗|²)^1/2(^∞_𝑗=0|(2𝑤)^𝑗|²)^1/2=‖𝑓‖_𝐻𝐹,2(^∞_𝑗=0|2𝑤|^2𝑗)^1/2,(3.14) as desired. Also it is easy to see that 𝑒_𝑤 is continuous on 𝐻_𝐹,𝑝 for each 𝑝>2 and 𝑤∈(1/2)𝑈.

4. Continuity of the Composition Operators on 𝐻_𝐹,𝑝

In the most important classical spaces, all analytic maps of the unit disk into itself induce bounded composition operators, but there are analytic self-maps of the unit disk which do not give bounded composition operators on some generalized Hardy spaces.

Example 4.1. (a) Suppose that 𝜑(𝑧)=𝑧² and for each 𝑓∈𝐻(𝑈), 𝐹(𝑓)=𝑓𝑔, where 𝑔(𝑧)=1−𝑧. Let 𝑓_𝑗(𝑧)=1+𝑧+𝑧²+⋯+𝑧^𝑗 for each 𝑗∈ℕ∪{0}. Then 𝑓_𝑗∈𝐻_𝐹,2. We see that ‖𝐶_𝑧2(𝑓_𝑗)‖²_𝐻𝐹,2=‖𝑓_𝑗(𝑧²)𝑔(𝑧)‖²₂=‖(1+𝑧²+⋯+𝑧^2𝑗)(1−𝑧)‖²₂=2𝑗+2,‖𝑓_𝑗‖²_𝐻𝐹,2=‖𝑓_𝑗𝑔‖²₂=‖1−𝑧^𝑗+1‖²₂=2.(4.1) So ‖𝐶_𝑧2‖²≥𝑗+1 and 𝐶_𝑧2 is not bounded on 𝐻_𝐹,2.
(b) Let 𝐹(∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=∑^∞_𝑗=0𝑎_𝑗𝛽(𝑗)𝑧^𝑗, where ∑^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻(𝑈) and {𝛽(𝑗)} is given in [2, Example 3.4]. If 𝜑(𝑧)=𝑧², then in [2, Example 3.4] shows that 𝐶_𝜑 is not bounded on 𝐻_𝐹,2.

In this section, we investigate the continuity of the composition operator on some spaces 𝐻_𝐹,𝑝 in terms of a Carleson measure criterion. This criterion has been used to characterize the boundedness and the compactness of the composition operators in different papers (see, e.g., [16, 30]).

Definition 4.2. A positive measure 𝜇 on 𝑈 is called a Carleson measure (in 𝑈) if there is a constant 𝐾<∞ such that 𝜇(𝑆(𝑏,ℎ))<𝐾ℎ for all 𝑏∈𝜕𝑈 and 0<ℎ<1, where 𝑆(𝑏,ℎ)={𝑧∈𝑈∶|𝑧−𝑏|<ℎ}.

Proposition 4.3. Let 1≤𝑝<∞ and 𝑃⊆𝑅_𝐹. If 𝜇 is a finite, positive Borel measure on 𝑈, then the following conditions are equivalent.(a)𝜇 is a Carleson measure in 𝑈.(b)There is a constant 𝐶<∞ such that _𝑈|𝐹(𝑓)|^𝑝𝑑𝜇≤𝐶‖𝑓‖^𝑝_{𝐻𝐹,𝑝}(4.2) for all 𝑓 in 𝐻_𝐹,𝑝.

Proof. By [2, Theorem 2.33], (a)⇒(b) is clear. Now we prove that (b) implies (a). For each 𝑝₀∈𝑃, there is a 𝑞_𝑝0∈𝐻_𝐹,𝑝 such that 𝐹(𝑞_𝑝0)=𝑝₀ and _𝑈|𝑝₀|^𝑝𝑑𝜇≤𝐶‖𝑝₀‖^𝑝_𝑝.(4.3) If 𝑔∈𝐻^𝑝, then there is a sequence {𝑝_𝑛} in 𝑃 such that ‖𝑝_𝑛−𝑔‖_𝑝→0 as 𝑛→∞. So ‖𝑝_𝑛‖^𝑝_𝑝→‖𝑔‖^𝑝_𝑝,as𝑛→∞.(4.4) By (4.3), we obtain _𝑈|𝑝_𝑛|^𝑝𝑑𝜇≤𝐶‖𝑝_𝑛‖^𝑝_𝑝(4.5) for each 𝑛∈ℕ. The right-hand side of (4.5) is bounded, hence there is a subsequence {𝑝_𝑛𝑘} such that ∫_𝑈|𝑝_𝑛𝑘|^𝑝𝑑𝜇 converges. Since point evaluations at the points of 𝑈 are all continuous on 𝐻^𝑝, the Principle of Uniform Boundedness implies that {𝑝_𝑛𝑘} converges to 𝑔 uniformly on compact subsets of 𝑈. In particular, for each 0<𝑟<1, the convergence is uniform on 𝑟𝑈. Therefore, for each 0<𝑟<1, ∫_𝑟𝑈|𝑝_𝑛𝑘|^𝑝𝑑𝜇→∫_𝑟𝑈|𝑔|^𝑝𝑑𝜇 as 𝑘→∞. Since {∫_𝑈|𝑝_𝑛𝑘|^𝑝𝑑𝜇} is a bounded sequence, there is an 𝑀>0 such that ∫_𝑟𝑈|𝑔|^𝑝𝑑𝜇≤𝑀 for each 0<𝑟<1. Hence ∫_𝑈|𝑔|^𝑝𝑑𝜇≤𝑀 and 𝑔∈𝐿^𝑝(𝜇). Now let {𝑟_𝑚} be an increasing sequence which converges to 1 and let 𝜖>0. By [31, page 32], there is an integer 𝑚₀ such that ∫_𝐴|𝑔|^𝑝𝑑𝜇<𝜖, where 𝐴=𝑈−𝑟_𝑚0𝑈. Let 𝐵=𝑟_𝑚0𝑈. Then by [31, page 73], we have limsup_𝑘→∞∫_𝐴|𝑝_𝑛𝑘|^𝑝𝑑𝜇≤𝜖. So there is a 𝐾₀ such that for each 𝑘>𝐾₀, ∫_𝐴|𝑝_𝑛𝑘|^𝑝𝑑𝜇≤𝜖. Again apply [31, page 32] for |𝑝_𝑛1|^𝑝,|𝑝_𝑛2|^𝑝,…,|𝑝_𝑛𝐾0|^𝑝. Therefore, there are 𝑚₁,𝑚₂,…,𝑚_𝐾0 such that _{𝑈−𝑟𝑚𝑘𝑈}|𝑝_𝑛𝑘|^𝑝𝑑𝜇≤𝜖(4.6) for each 1≤𝑘≤𝐾₀. Let 𝑚^′=max{𝑚₀,𝑚₁,…,𝑚_𝐾0}. Therefore, for each 𝑚>𝑚^′ and 𝑘∈ℕ, ∫_{𝑈−𝑟𝑚𝑈}|𝑝_𝑛𝑘|^𝑝𝑑𝜇≤𝜖. So by [32, Theorem 19.10], we have lim_{𝑘→∞𝑈}|𝑝_𝑛𝑘|^𝑝𝑑𝜇=lim_𝑘→∞lim_{𝑚→∞𝑟𝑚𝑈}|𝑝_𝑛𝑘|^𝑝𝑑𝜇=lim_𝑚→∞lim_{𝑘→∞𝑟𝑚𝑈}|𝑝_𝑛𝑘|^𝑝𝑑𝜇=lim_{𝑚→∞𝑟𝑚𝑈}|𝑔|^𝑝=𝑑𝜇_𝑈|𝑔|^𝑝𝑑𝜇.(4.7) So by the preceding relation and (4.4) and (4.5), we have _𝑈|𝑔|^𝑝𝑑𝜇≤𝐶‖𝑔‖^𝑝_𝑝(4.8) for every 𝑔∈𝐻^𝑝. Thus the result follows from [2, Theorem 2.33].

Definition 4.4. For 𝑏 on the unit circle and 0<ℎ<1, let 𝒮(𝑏,ℎ)={𝑧∈𝑈∶|𝑧−𝑏|<ℎ}.(4.9)
From now on, unless otherwise stated, we assume that 𝐻_𝐹,𝑝 is a Banach space.

Proposition 4.5. Let 1≤𝑝<∞ and 𝑃⊆𝑅_𝐹. If 𝜇 is a finite, positive Borel measure on 𝑈, then the following conditions are equivalent.(a)There is a constant 𝐾<∞ such that 𝜇(𝒮(𝑏,ℎ))<𝐾ℎ for all 𝑏∈𝜕𝑈 and 0<ℎ<1. (b)There is a constant 𝐶<∞ such that _𝑈|𝐹(𝑓)|^𝑝𝑑𝜇≤𝐶‖𝑓‖^𝑝_{𝐻𝐹,𝑝}(4.10) for all 𝑓∈𝐻_𝐹,𝑝.

Proof. The implication (a)⇒(b) follows by Proposition 4.3 and is exactly the same as the proof of [2, Theorem 2.35(1)⇒(2)]. For the other direction, by Theorem 2.2, 𝐻^𝑝⊆𝑅_𝐹. Hence for each 𝑔∈𝐻^𝑝, we have _𝑈|𝑔|^𝑝𝑑𝜇≤𝐶‖𝑔‖^𝑝_𝑝.(4.11) Then the result follows from [2, Theorem 2.35(2)⇒(1)].

In the following theorem, we use the techniques used in [2, Theorem 3.12 part (1)].

Theorem 4.6. Suppose 1≤𝑝<∞, 𝜑 and 𝜑₀ are analytic self-maps of 𝑈. Let 𝑃⊆𝑅_𝐹 and for each 𝑓∈𝐻_𝐹,𝑝, 𝐹(𝑓∘𝜑)=𝐹(𝑓)∘𝜑₀. Then 𝐶_𝜑 is a bounded operator on 𝐻_𝐹,𝑝 if and only if 𝜇(𝒮(𝜁,ℎ))=○(ℎ) for all 𝜁 in 𝜕𝑈 and 0<ℎ<1, where 𝜇(𝐸)=𝜎((𝜑^∗₀)⁻¹(𝐸)), 𝜎 is normalized Lebesgue measure on the unit circle, and 𝐸 is a subset of the closed disk 𝑈.

Proof. Let 𝑓∈𝐻_𝐹,𝑝. By [2, Theorem 2.25] and [33, page 163], we obtain _𝜕𝑈|(𝐹(𝑓∘𝜑))^∗|^𝑝𝑑𝜎=_𝜕𝑈|(𝐹(𝑓)∘𝜑₀)^∗|^𝑝=𝑑𝜎_𝜕𝑈|(𝐹(𝑓))^∗∘𝜑^∗₀|^𝑝=𝑑𝜎_𝑈|(𝐹(𝑓))^∗|^𝑝𝑑𝜇.(4.12) So the result follows from Proposition 4.5.

Remark 4.7. If 𝑝, 𝜑, 𝜑₀ and 𝐹 satisfy the hypotheses in Theorem 4.6, then by [2, Theorem 3.12, part (1)], 𝐶_𝜑 is a bounded operator on 𝐻_𝐹,𝑝 if and only if 𝐶_𝜑0 is a bounded operator on 𝐻^𝑝. Since we know that the composition operators are always bounded on each of the Hardy spaces 𝐻^𝑝, 𝐶_𝜑 is a bounded operator on 𝐻_𝐹,𝑝.

5. Compactness of the Composition Operators on 𝐻_𝐹,𝑝

In this section, we investigate the compactness of the composition operators on 𝐻_𝐹,𝑝.

The idea of the proof of the following theorem is similar to the proof of Proposition 3.11 in [2]. A detailed proof of another similar result can be found, for example, in [18].

Theorem 5.1. Suppose 1≤𝑝<∞ and for each 𝑤∈𝑈, 𝑒_𝑤 is continuous. Also if 𝑓_𝑛→𝑓 as 𝑛→∞ uniformly on compact subsets of 𝑈, then 𝐹(𝑓_𝑛)→𝐹(𝑓) in 𝐻(𝑈) as 𝑛→∞. Then the following conditions are equivalent. (a)𝐶_𝜑 is compact on 𝐻_𝐹,𝑝. (b)If {𝑓_𝑛} is a bounded sequence in 𝐻_𝐹,𝑝 and 𝑓_𝑛→0 as 𝑛→∞ uniformly on compact subsets of 𝑈, then ‖𝑓_𝑛∘𝜑‖_{𝐻𝐹,𝑝}→0 as 𝑛→∞.

Proof. The implication (a)⇒(b) follows exactly as the proof of [2, Theorem 3.11].
Now we show that (b)⇒(a). Let {𝑓_𝑛} be a bounded sequence in 𝐻_𝐹,𝑝. Since 𝑒_𝑤 is continuous for each 𝑤∈𝑈, {𝑓_𝑛} is a normal family. So there is a function 𝑓∈𝐻(𝑈) and a subsequence {𝑓_𝑛𝑘} such that 𝑓_𝑛𝑘→𝑓 as 𝑘→∞ uniformly on compact subsets of 𝑈. Hence 𝐹(𝑓_𝑛𝑘)→𝐹(𝑓) in 𝐻(𝑈) as 𝑘→∞. It is easy to see that 𝑓∈𝐻_𝐹,𝑝. Therefore, {𝑓_𝑛𝑘−𝑓} is a bounded sequence in 𝐻_𝐹,𝑝 such that it converges uniformly to 0 on compact subsets of 𝑈. By the hypotheses, we conclude that 𝑓_𝑛𝑘∘𝜑→𝑓∘𝜑 in 𝐻_𝐹,𝑝 as 𝑘→∞. Thus 𝐶_𝜑 is a compact operator.

Corollary 5.2. Let 1≤𝑝<∞, 𝑔∈𝐻^𝑝, 𝑔≢0, and 𝐹(𝑓)=𝑓𝑔 for each 𝑓∈𝐻(𝑈). Let 𝜑 be an analytic self-map of 𝑈 and 𝜑(𝑈)⊆𝑈. Then 𝐶_𝜑 is a compact operator on 𝐻_𝐹,𝑝.

In the rest of this section, we investigate the relation between compactness of 𝐶_𝜑 on 𝐻_𝐹,𝑝 and 𝐻^𝑝.

Theorem 5.3. Let 𝑝, 𝜑₀, 𝜑, 𝐹, and 𝜇 satisfy the hypotheses in Theorem 4.6. Then 𝐶_𝜑 is compact on 𝐻_𝐹,𝑝 if and only if 𝜇(𝒮(𝜁,ℎ))=∘(ℎ) as ℎ→0 uniformly in 𝜁 in 𝜕𝑈.

Proof. Let 𝜇(𝒮(𝜁,ℎ))=∘(ℎ) as ℎ→0 uniformly in 𝜁 in 𝜕𝑈 and {𝑓_𝑛} be a bounded sequence in 𝐻_𝐹,𝑝. Then by [2, Theorem 3.12, part (2)], there is a subsequence {𝐹(𝑓_𝑛𝑘)} such that {𝐶_𝜑0(𝐹(𝑓_𝑛𝑘))} converges in 𝐻^𝑝. Since 𝐹(𝑓_𝑛𝑘∘𝜑)=𝐹(𝑓_𝑛𝑘)∘𝜑₀, {𝑓_𝑛𝑘∘𝜑} converges in 𝐻_𝐹,𝑝. Therefore, 𝐶_𝜑 is a compact operator on 𝐻_𝐹,𝑝.
Conversely, let {𝑓_𝑛} be a bounded sequence in 𝐻^𝑝. By Theorem 2.2, 𝐻^𝑝⊆𝑅_𝐹, so there is a sequence {𝑔_𝑛} in 𝐻_𝐹,𝑝 such that 𝐹(𝑔_𝑛)=𝑓_𝑛. Since 𝐶_𝜑 is compact on 𝐻_𝐹,𝑝, we may extract a subsequence {𝑔_𝑛𝑘} such that {𝐶_𝜑(𝑔_𝑛𝑘)} converges in 𝐻_𝐹,𝑝. So 𝐶_𝜑0 is a compact operator on 𝐻^𝑝 and [2, Theorem 3.12, part (2)] implies the result.

Remark 5.4. If 𝑝, 𝜑, 𝜑₀, and 𝐹 satisfy the hypotheses in Theorem 5.3, then by [2, Theorem 3.12, part (2)], 𝐶_𝜑 is a compact operator on 𝐻_𝐹,𝑝 if and only if 𝐶_𝜑0 is a compact operator on 𝐻^𝑝.

Now we present an example for Remarks 4.7 and 5.4.

Example 5.5. Let 𝑝=2, 𝜑₀(𝑧)=𝜑(𝑧)=𝑧², and 𝐹(Σ^∞_𝑗=0𝑎_𝑗𝑧^𝑗)=∑^∞_𝑗=0𝑎_𝑗𝑐_𝑗𝑧^𝑗, where Σ^∞_𝑗=0𝑎_𝑗𝑧^𝑗∈𝐻(𝑈) and for every odd positive integer 𝑗 and every 𝑛∈ℕ, 𝑐₂^𝑛_𝑗=𝑐_𝑗=1/𝑗 and 𝑐₀=1. It is easy to see that by Remark 4.7, 𝐶_𝜑 is a bounded operator on 𝐻_𝐹,2 and by Remark 5.4 and [2, Corollary 3.14] 𝐶_𝜑 is not a compact operator on 𝐻_𝐹,2.

Acknowledgments

The author would like to thank Dr. Bahram Khani Robati for his suggestions and comments. Also she appreciates professor Stevo Stevic for the helpful advice and notification to this work. This work is a part of the author’s doctoral thesis, written at Shiraz University under the direction of B. Khani Robati.