Spectra of weighted composition operators on abstract Hardy spaces

In the paper, the spectra of weighted composition operators on Hardy-type spaces of holomorphic functions on the unit disc of the complex plane are studied. The spectra of invertible operators induced by elliptic and parabolic automorphisms are described, for weighted composition operators acting on abstract Hardy spaces generated by Banach lattices. We also study spectra of weighted composition operators (not necessarily invertible) on Hardy–Lorentz spaces.


Introduction
If u ≡ 1, then the corresponding map C ϕ is called a composition operator.
The study of composition operators on various spaces of holomorphic functions was initiated at the beginning of the twentieth century by the works of Hardy, Littlewood, and Riesz (see [6]). Over the years the questions related to the operator-theoretic properties of uC ϕ have been a great motivation for the development of complex and functional analysis. Very recently the spectra of weighted composition operators on different spaces of holomorphic functions have been extensively studied. Most research was done with ϕ an automorphism of D, and the main goal was to give a description of the spectrum of uC ϕ in terms of the functions ϕ and u. In this respect the main achievement was obtained for the Hardy, Bergman, and growth spaces (see [11]). The same problem has recently been considered for other types of spaces of holomorphic functions, namely the Dirichlet (see [5,9]) and the Bloch spaces (see [9]).
Moreover, weighted composition operators (and their operator-theoretic properties) have recently been treated in the context of more abstract Hardy classes, like Hardy-Orlicz spaces (see for example [15]), and general Hardy spaces generated by rearrangement-invariant Banach spaces (see [19]).
The aim of this paper is to study the spectra of weighted composition operators on abstract variants of Hardy spaces. The paper is organized as follows. In the preliminary section we gather the essential notions and discuss in detail the tools we will use underneath. The main results of this paper are contained in Sect. 3, where the study of the spectra of weighted composition operators uC ϕ on abstract Hardy spaces is presented, when the generating symbol ϕ is an automorphism of the disc. A complete description of the spectra for Hardy-Lorentz spaces is given, and then a characterization of spectra of weighted composition operators uC ϕ on abstract Hardy spaces is provided with ϕ parabolic or elliptic. In the final section, it is shown that the spectra of weighted composition operators on Hardy-Lorentz and Hardy spaces coincide.

Preliminaries
Let ( , , μ) be a complete σ -finite measure space and let L 0 ( ) := L 0 ( , , μ) denotes the space of real valued measurable functions on with the topology of convergence in measure on μ-finite sets. The order | f | |g| means that | f (ω)| |g(ω)| for μ-almost all ω ∈ . If a real Banach space X ⊂ L 0 ( ) is such that there exists f ∈ X with f > 0 μ-a.e. on and | f | |g| with f ∈ L 0 ( ) and g ∈ X implies f ∈ X with f X g X , then X is said to be a Banach lattice (on or on ( , μ)). An important class of Banach lattices are rearrangement-invariant spaces. Given f ∈ L 0 ( ), its distribution function is given by μ f (λ) = μ {t ∈ : | f (t)| > λ} , λ 0. A Banach lattice X is called to be rearrangement invariant (r.i. space for short) if for any f ∈ X and g ∈ L 0 ( ) such that μ f = μ g we have g ∈ X and f X = g X . It is well known that if X is an r.i. space for some finite measure space , then (see [14]) Throughout the paper we will consider complex r.i. spaces. The term complex r.i. space refers to the complexification of a real r.i. space, i.e., if X denotes the (real) r.i. space, the complexification X (C) of X is the Banach space of all complex valued measurable functions f on such that the element | f | defined by | f |(ω) = | f (ω)| for ω ∈ is in X and f = | f | X . For simplicity of presentation, we will often write r.i. space instead of complex r.i. space and avoid using the symbol X (C). An r.i. space X is said to be maximal (or to have the Fatou property), if for any sequence ( f n ) of non-negative elements from X such that f n ↑ f for f ∈ L 0 ( ) and sup f n X : n ∈ N < ∞, one has f ∈ X and f n X → f X . If X is an r.i. space then for any measurable set B, the expression χ B depends only of μ(B). Thus for every t ∈ {μ(B) : B ∈ } we define a function φ X by the formula φ X (t) = χ [0,t] . This function is called the fundamental function of X .
We collect here the basic properties of φ X . If X is an r.i. function space on a non-atomic measure space ( , , μ), then φ X is quasi-concave on [0, τ ) with τ = μ( ), i.e., φ X (0) = 0, φ X is positive, non-decreasing, and t → φ X (t)/t is non-increasing on (0, τ ) (see [14]). Notice that for a quasi-concave function ψ there exists a concave function ψ given by Thus we may assume that φ X is a concave function. Note also that φ X is continuous at 0 if and only if X = L ∞ .

Interpolation of operators
In the Sect. 4 we will use interpolation theory to study the spectra of weighted composition operators. Here we present some basic concepts; for a more detailed study we refer the reader to the monograph [3].
Let E 0 and E 1 be Banach spaces.
be Banach couples. If for any intermediate spaces E and F with respect to E and F, and every linear map T : E 0 + E 1 → E 0 + E 1 such that T | E j is a bounded operator from E j into F j , j = 0, 1, the operator T : E → F is bounded, then E and F are called interpolation spaces (with respect to E and F).

Hardy spaces on a disc
Let X be an r.i. space on T := [0, 2π) with normalized Lebesgue measure. An abstract Hardy space HX = HX (D) consists of holomorphic functions on the unit disc D such that for t ∈ T and r ∈ [0, 1). With the norm · HX given by f HX := sup f r X : r ∈ [0, 1) , HX is a Banach space. For a study of general variants of Hardy spaces, see [18,19]. For particular spaces X , the above method produces variants of Hardy spaces widely studied in the literature. For example, if X = L p , p ∈ [1, ∞], then HX is the standard Hardy space H p (see [6]).
We shall give two important examples of r.i. spaces, namely Lorentz and Orlicz spaces. Let ψ be a positive, increasing and concave function on [0, τ ). Let ( , , μ) be a non-atomic measure space. The Lorentz space (ψ) consists of all f ∈ L 0 (μ) such that It is clear that the fundamental function of the Lorentz space (ψ) equals ψ.
The systematic study of composition operators on spaces more general than the classical Hardy spaces H p dates back to the papers on Hardy-Orlicz spaces by Lefèvre, Li, Queffélec, and Rodríguez-Piazza from the beginning of the twenty-first century (see the book [15] and the list of references therein). Recently, Carleson embeddings of abstract Hardy spaces into Banach lattices with an application to the study of analytic properties of composition operators on very general Hardy classes have been treated by Mastyło and Rodríguez-Piazza in [19]. In the latter paper it was proved, among other things, that if X is a maximal r.i. space on T then every ϕ ∈ H (D), ϕ(D) ⊂ D, induces a composition operator C ϕ : HX → HX (see [19,Proposition 1.4]) and In what follows, we will need a particular estimate of the norm of the evaluation functional δ z on HX . Notice that if X is an r.i. space and φ X is a fundamental function, then the following inequalities, proved in [19, Lemma 1.2], hold: Note that from these inequalities follow the well-known estimates of the evaluation functionals since the fundamental functions can be written explicitly.

Motivation
In [11,12], the spectra of composition operators were studied within the framework of a very general axiomatic approach. One of the key assumptions was that the space E ⊂ H (D) satisfy the following condition: It appears that there are important spaces for which the above condition does not hold. For example, this is the case for a Hardy-Orlicz space. Indeed, it follows from (2) that for the The main goal of this paper is to study the spectra of weighted composition operators on Hardy spaces generated by r.i. spaces on T. In other words, (cf. (2)) we will study spaces of analytic functions satisfying the property: Our proofs rely on the approach taken in [9] (cf. also [11]). Let us note, however, that the much more complicated structure of the norm of the spaces under consideration forced us to take a slightly different approach. Results from Sect. 4 are obtained utilizing the outcome of [1,2].

Multiplication operator on HX
In what follows we will need a description of a bounded multiplication operator. The following result is well known for the classical Hardy spaces H p and, for example, for Hardy-Orlicz spaces (see [17]). Below we enclose the proof of the general variant.
Theorem 1 Let X be a maximal r.i. space on T and suppose that u ∈ H (D). Then the following statements are equivalent: (i) u HX ⊂ HX (ii) u ∈ HX and the operator M u is bounded Proof (i) ⇒ (ii). From the estimates (2) it follows that the convergence in HX norm is stronger than the uniform convergence on compact sets in D. Hence M u has a closed graph and it follows from the Closed Graph Theorem that M u : HX → HX is bounded.
(ii) ⇒ (iii). Choose z ∈ D and let us consider the functional δ z given for f ∈ HX by

Spectra of invertible weighted composition operators on abstract Hardy spaces
In this section we study the spectra of invertible weighted composition operators induced by parabolic or elliptic symbols. The next theorem, which follows directly from Bourdon The spectral analysis of invertible weighted composition operators uC ϕ depends significantly on the dynamics of the symbol ϕ (see for example [9,11]). Recall that the automorphism of D is called (see [6]) where ϕ is a parabolic or hyperbolic automorphism of the unit disk. For any n 1 we have From this it follows that (uC ϕ ) n = u (n) C ϕ n , where and u (0) = 1. Throughout the paper the symbol A(D) denotes the disc algebra.

The parabolic case
In this section we will give a description of the spectrum of weighted composition operator uC ϕ : HX → HX , where X is an r.i. space and φ : D → D is a parabolic automorphism. The special case when X = L p , p ∈ [1, ∞) was treated in [11,Section 4.1].
In the proof of the main theorem of this section we will use the following lemma.
Lemma 1 Let ψ : [0, τ ) → [0, τ ) with τ > 0 be a quasi-concave function. Let {a n }, {b n } ⊂ (0, τ ) be sequences such that a n → 0, b n → 0, as n → ∞. If lim n→∞ a n b n = 1, Proof Since a n /b n → 1 and b n /a n → 1 as n → ∞, it follows that for every ε ∈ (0, 1) there exists n 0 ∈ N such that for each n > n 0 we have a n (1 + ε)b n and b n (1 + ε)a n . For a given n > n 0 we have either a n b n or a n < b n . In the first case we obtain by quasi-concavity of ψ and so In the second case (that is, when a n < b n ) we obtain ψ(b n ) ψ(a n ) 1 + ε.
Recall that an interpolating sequence for H ∞ (D) is a sequence {z j } ⊂ D such that for any bounded sequence of complex numbers {c j }, there is a function f ∈ H ∞ (D) such that f (z j ) = c j . It was proved in [7,Proposition 4.9] that if z 0 ∈ D, then the sequence of iterates {ϕ n (z 0 )}, n ∈ N, is an interpolating sequence for H ∞ (D).
Proof To prove that r (uC ϕ ) |u(a)| we note that by (1) it follows that Since ϕ is a parabolic automorphism by (3) we conclude that = 1 as n → ∞.
We will prove the reverse inequality. Let a ∈ ∂D be the fixed point of ϕ and take ε ∈ (0, 1). Notice that (see [11, p. 1759 From the formula (2) it follows that for every ε ∈ (0, 1) we have Since φ X is a quasi-concave function on [0, 1), the inequality r (uC ϕ ) |u(a)| now follows from Lemma 1 (with τ = 1) and the fact that To give a description of the spectrum of uC ϕ : HX → HX , observe that it is enough to justify the inclusion λ : |λ| ∈ r (uC ϕ ) ⊂ σ HX (uC ϕ ).
We will prove the formula (8). Take a sequence {z n } defined by z n = {ϕ n (0)}, n ∈ N 0 (cf. [11,Section 4.1]). Since {z n } is an interpolating sequence for H ∞ , by the Open Mapping Theorem there exist a constant c > 0 and a sequence { f n } ⊂ H ∞ such that f n ∞ c for all n ∈ N 0 and Since L ∞ → X → L 1 , it follows that the sequence { f n } is bounded in HX and hence also in H 1 . Take λ ∈ C with |λ| = |u(a)|. From the above we get Now using the binomial expansion and formula (9) we obtain Combining the above equality with (10) and using the substitution w(z) = u(z) ϕ (z) (note that w (n) (z) = u (n) (z) ϕ n (z) −1 by the proof of [11,Theorem 4.3]) we get In the proof of [11,Theorem 4.3] it was shown that lim n→∞ w (n) (z n ) This proves that r HX (λ − uC ϕ ) 2r HX (uC ϕ ), and hence by the Spectral Mapping Theorem −λ ∈ σ HX (uC ϕ ) showing that λ ∈ C : |λ| = |u(a)| ⊂ σ HX (uC ϕ ).

The elliptic case
The approach with invertible weighted composition operators induced by an elliptic automorphism is standard. We include the result and a sketch of the proof for the sake of completeness. Proof The proof of (i) is an easy modification of [11,Theorem 4.11].
To show that {λ ∈ C : |λ| = |u(a)|} ⊂ σ HX (uC ϕ ) notice that e n (z) = z n ∈ HX for every n ∈ N 0 . Using a similar argument as in the proof of [

Spectra of weighted composition operators on Hardy-Lorentz spaces
In this paragraph we consider a very general approach based on interpolation theory restricted to the special case of standard Hardy-Lorentz spaces. In particular we show that the spectrum of an operator on the Hardy-Lorentz space H p,q does not depend on the parameter q and hence the spectra of weighted composition operators on Hardy-Lorentz H p,q and Hardy H p spaces are the same. This holds also for the general case when a weighted composition operator is not necessarily invertible.
Theorem 5 Let ϕ be a holomorphic self-map of D, u ∈ A(D), and p, q ∈ (1, ∞). Then Proof Notice that by [19,Proposition 1.4] and Theorem 1 the operator uC ϕ is bounded on H p,q .
The reverse inclusion is obtained in a similar way and so σ H p (uC ϕ ) = σ H p,q (uC ϕ ).
From [11, Corollary 5.1] we obtain a description of the spectra of invertible weighted composition operators on Hardy-Lorentz spaces. = λ ∈ C : min |u(a)| ϕ (a) 1/ p , Let us also note that Theorem 5 can be applied to not necessarily invertible weighted composition operators. The corollary below follows from [12, Theorem 6] (cf. also [8]).