Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.


Introduction
in which he collected many results obtained a few years before about weighted Banach spaces H ∞ v of analytic functions of type H ∞ and inductive limits of them, interpolation and sampling and composition, multiplication and differentiation operators between them. Since that time much progress has been obtained by many authors, and it is my feeling that it might be time to review, at least part, of all this work. In this sense, this article could be considered as a continuation of [40]. The selection of material reflects the research interests of the author. There are certainly many important related results which are not included here, and I apologize to those authors whose work is not mentioned. Despite the necessary selection, the list of references is enormous. I am also sorry about that, but it B José Bonet jbonet@mat.upv.es seems unavoidable. The author hopes that this survey might be informative for the reader, and that it might show how many interesting results have been obtained recently and, at the same time, that some questions remain open in this area of research.
We briefly describe the content of the article. Precise statements can be seen at the right place in the paper. No proofs are given in the survey. The interested reader should look at the original articles. Precise references are given. After the notation and terminology in the next section, we recall the notion of associated weight in Sect. 3. Associated weights were introduced and studied in [36] and they have played a significant role in characterizing properties of weighted Banach spaces H ∞ v . Two important results due to Abakumov and Doubtsov [3] characterizing those radial weights which are equivalent to their associated weight are stated in Theorems 2 and 3. In Sect. 4 we report on our joint work with Vukotic [69] about conditions on a non-negative function v : G → [0, ∞), not necessarily bounded or strictly positive, defined on an open connected domain G in the complex plane, to ensure that the semi-normed space H ∞ v (G) is in fact normed and complete. Section 5 is dedicated to review some results of Lusky, Taskinen and the author about the solid hull and core of H ∞ v . A distance formula from an element f ∈ H ∞ v to the closed subspace H 0 v , which is the closure of the polynomials, was obtained by Perfekt [141,142]. A direct, elementary proof was presented in [57] and it is explained in Sect. 6.
Necessary definitions about bounded and linear operators, spectrum, hypercyclicity, power boundedness and (uniform) mean ergodicity are briefly recalled in Sect. 7. They are important in the second part of the article. The operators of differentiation and integration are treated in Sect. 8. We first state characterizations of boundedness and compactness due to Harutyunyan and Lusky [110] and to Abanin and Tien [6]. Then we include a few results about the spectrum, mean ergodicity and hypercyclicity of these two operators when they act on weighted Banach spaces of entire functions, see [30]. The proper closed invariant subspaces of the integration operator on H 0 v were determined by Abanin and Tien [7]. Their statement is presented in Theorem 39. In Sect. 9 we collect some results of Albanese, Ricker and the author [9] about the Cesàro operator on weighted Banach spaces for standard weights. A few results about the Volterra integral operator are collected in Sect. 10. Several questions about weighted composition operators on H ∞ v spaces are discussed in Sect. 11: boundedness, (weak) compactness, invertibility, the spectrum, (uniform) mean ergodicity and hypercyclicity. We briefly report on the deep work of many authors. Section 12 presents a few results by Boyd and Rueda [76,77] and by Vukotic and the author [68] concerning superposition operators on H ∞ v of analytic functions on the disc. Lusky, Taskinen and the author have investigated bounded Toeplitz operators acting on H ∞ v in the case of the disc in the papers [60,61]. Some of these results are stated in Sect. 13. The final Sect. 14 includes a few results about the Hilbert matrix operator, the Libera operator and the Hausdorff operators.
Recall that a function g vanishes at infinity on G if for every ε > 0 there is a compact subset K of G such that |g(z)| < ε if z / ∈ K . If G is an open subset of C, we denote by H (G) the Fréchet space of all analytic functions on G endowed with the topology τ co of uniform convergence on the compact subsets of G.
We are mainly interested in the case of radial weights defined on the unit disc or the whole complex plane. To be more precise, we fix some notation and terminology in this case. We set R = 1 (for the case of analytic functions on the unit disc) and R = +∞ (for the case of entire functions). A weight v is a continuous function v : [0, R[→]0, ∞[, which is non-increasing on [0, R[ and satisfies lim r →R r n v(r ) = 0 for each n ∈ N. Observe that in the case of R = ∞, the condition lim r →∞ r n v(r ) = 0, for each n ∈ N, is equivalent to the fact that H ∞ v (C) contains the polynomials, and also to the fact that H 0 v (C) contains the polynomials. We extend v to D if R = 1 and to C if R = +∞ by v(z) := v(|z|). For an analytic function f ∈ H ({z ∈ C; |z| < R}) and r < R, we denote M( f , r ) := max{| f (z)| ; |z| = r }.  (1/v(r )), r → R, see e.g. [35]. It was proved in [37] that, under the present conditions on the radial weight v on the disc or the complex plane, the space H ∞ v (G) is canonically isometric to the bidual of H 0 v (G). It will be clear from the context in the rest of the article when we refer to analytic functions on the disc or entire functions. Hence, we might write simply H ∞ v and H 0 v . Anyway, if it is necessary to distinguish at some point, we will use the notations H ∞ v (D) and H ∞ v (C), respectively.
We Banach spaces of the type mentioned above appear naturally in the study of growth conditions of analytic functions and have been considered in many papers. We refer to [35][36][37]151]. Composition operators on weighted Banach spaces of this type when G = D have been studied in [49,52,88,137,155]. Pointwise multiplication operators were considered in [50], and sampling and interpolation in these spaces in [92]. Lusky presented in [125][126][127] a complete isomorphic classification of the spaces H ∞ v and H 0 v . We reported about these and related results in our survey [40]. Much progress has been done recently about these spaces and operators between them. We present some of the new interesting developments in the next pages.
For each 0 < p < ∞, the Bloch space of order p is and the little Bloch space of order p is It is a well-known fact that B p is a Banach space when it is endowed with the norm  (0)). With this identification, several operators on Bloch type spaces can be treated as operators on H ∞ v (D) spaces; in particular, composition operators on Bloch spaces can be considered as weighted composition operators. See for example [56,88]. We will not state in this survey all the possible consequences for Bloch type spaces, and refer to the original papers. Moreover, we will not report here about important, related work on weighted Hardy and Bergman spaces.
Our notation for complex analysis, functional analysis and operator theory is standard. We refer the reader to [96,112,131,146]. If E is a Hausdorff locally convex space space, for example a Banach space, its topological dual is denoted by E . The weak topology on E is denoted by σ (E, E ) and the weak* topology on E by σ (E , E). The linear span of a subset A of E is denoted by span(A). In what follows, we set N 0 := N ∪ {0}.

The associated weight and the characterization of essential weights.
Let G be an open connected subset of C and let v : G → R be a general weight on G. We assume in what follows that the norm ||δ z || of the Dirac measure, . By our assumption above,ṽ(z) is finite for every z ∈ G. Moreover v ≤ṽ on G, 1/ṽ is continuous and subharmonic, and the Banach spaces H ∞ v (G) and H ∞ v (G) coincide isometrically. A weight v is called essential if there is C ≥ 1 such that v ≤ṽ ≤ Cv on G. Associated weights were thoroughly studied in [36,52].
Example 1 (1) If G = D or G = C and v is radial, thenṽ is also radial.
(2) If G = D or G = C, v is radial and lim r →R r n v(r ) = 0 for each n ∈ N, then lim r →R r nṽ (r ) = 0. (3) The following weights are essential: , a, b > 0, and (iii) v(z) = (log e 1−r ) −α , α > 0 for the unit disc; and (iv) v(r ) = exp(−r n ) with n ∈ N, for C. See [36,Example 1.7]. (4) ( [49]) A radial and non-increasing weight v on D is essential if and only if it is equivalent to a log-convex radial weight w on D. We recall that the radial weight w is log-convex on G = D or G = C if the function t → − log w(e t ) is convex. The associated weight to a radial weight is log-convex. (5) Every normal weight on D in the sense of Shields and Williams is essential. We refer the reader to [92,151].
A thorough investigation of essential radial weights on the disc D or the complex plane was undertaken by Abakumov and Doubtsov [1][2][3]. We recall their main results. A few definitions are necessary. We consider in the rest of this section only radial, non-increasing, continuous, strictly positive weights v : G → R for G = D or G = C.
Given two functions f , g : We say that v is approximable from below by monomials if there is C > 0 such that 1/v(r ) ≤ C P v (r ) for each 0 ≤ r < R. It was proved in [49] that every essential weight on the unit disc is approximable from below by monomials. Abakumov and Doubtsov used a result of Erdös and Kövári to show in [3, Lemma 1] that the associate weightṽ of a radial, non-increasing, continuous, strictly positive weight v on C such that lim r →∞ r nṽ (r ) = 0 for each n ∈ N satisfies P v (r ) Theorem 2 (Abakumov,Doubtsov [3]) The following conditions are equivalent for a radial, non-increasing, continuous, strictly positive weight v : D → R on the unit disc such that lim r →1− v(r ) = 0: (iii) v is approximable by a finite sum of moduli of (two) analytic functions. (iv) v is approximable by the maximum of an analytic function modulus.
(v) v is approximable by power series with positive coefficients. (vi) v is approximable from below by monomials.
It follows from [36,Example 3.3] that there exist log-convex, radial, non-increasing, continuous, strictly positive weights v : C → R on the complex plane with lim r →∞ r n v(r ) = 0 for each n ∈ N, which are not approximable by a finite sum of moduli of entire functions. Accordingly condition (ii) in Theorem 2 does not imply (iii) for weights on the complex plane. We have the following result.
Theorem 3 (Abakumov,Doubtsov [3]) The following conditions are equivalent for a radial, non-increasing, continuous, strictly positive weight v : C → R on the complex plane such that lim r →∞ r n v(r ) = 0 for each n ∈ N.

approximable by a finite sum of moduli of analytic functions. (iii) v is approximable by the maximum of an analytic function modulus.
(iv) v is approximable by power series with positive coefficients.
(v) v is approximable from below by monomials.
Each of these conditions imply that v is equivalent to a log-convex radial weight w on C.
The following example is taken from [3, Example 1].
is in fact normed and complete. The completeness of weighted Bergman spaces was studied by Arcozzi and Björn [19]. They obtained complete characterizations when the We review here some of the results in [69].
For a non-negative function v : and it is endowed with the natural seminorm f v := sup z∈G v(z)| f (z)|. We use the following notation in this section: is not a Banach space. As a consequence of Corollary 9 one easily deduces the following example: the weight v(z) = max{0, Re z} is continuous on the unit disc D, vanishes in the left-hand half of the disc, it is strictly positive in the remaining open right semi-disc, and H ∞ v (D) is not a Banach space.  In particular, if v(z) := |F(|z|)|, z ∈ G, for a non-zero function F ∈ H (G), then H ∞ v (G) is a Banach space.
Let v be the weight on D defined by v(z) := a n > 0 if |z| = 1 − (1/n), and v(z) = 0 otherwise. Then H ∞ v (G) is a Banach space by Corollary 13. Observe that the sequence (a n ) n ⊂]0, ∞[ need not be bounded. Similar examples can be obtained by replacing D by C and 1 − (1/n) by n, n ∈ N.

Proposition 14 Let F ∈ H (G) be a non-zero function on a planar domain G. Define
Example 15 (1) Let q > 0 and v(z) = |Re z| q , z ∈ D. Then the normed space H ∞ v (D) is complete.
(2) Let v be a weight on D such that there is a strictly increasing sequence (r n ) n of positive numbers tending to 1 such that for each n there is a n > 0 such that v(r n e iθ ) ≥ a n almost everywhere in [0, 2π]. Then the normed space is actually a Banach space.

Solid hull and solid core of the space H
In this section we identify an analytic function f (z) = ∞ n=0 a n z n on D or C with the sequence of its Taylor coefficients (a n ) ∞ n=0 . Let A and B be vector spaces of complex sequences containing the space of all the sequences with finitely many non-zero coordi- The solid core of A is It is easy to see that the Fréchet spaces H (D) and H (C) of analytic functions on the unit disc and on the whole complex plane are solid.
In [18], We refer the reader to [97,165] for information about Hardy spaces. The solid hull of the Hardy spaces S(H p ) = H 2 , 2 ≤ p ≤ ∞ is known. The proof for H ∞ depends on the following deep result of Kislyakov from 1981.
The solid hull S(H p ) for 1 ≤ p < 2 seems to be unknown. The solid core of the Hardy spaces s(H p ) = H 2 , 1 ≤ p ≤ 2, is known. Moreover s(H ∞ ) = 1 . In particular, the space H ∞ is not solid. The disc algebra A(D) is also not solid. It is an open problem to describe the solid core s(H p ) for 2 < p < ∞.
Bennet, Stegenga and Timoney in their paper [34] determined the solid hull and the solid core of the weighted spaces The solid hull and core of spaces of analytic functions on the disc has been investigated by many authors. In addition to those mentioned above, Anderson, Dostanić, Blasco, Buckley, Jevtić, Pavlović, Ramanujan, Shields and Vukotić, among many others. We refer the reader to the book [117].
In the case of a standard weight v α (z) . The solid hull of A −α is known: This is Theorem 8.2.1 of [117]. Moreover, the solid core s A −α can also be characterized, see Theorem 8.3.4 of [117]: In our joint article [58] with Lusky and Taskinen we extended previous work in [66,67] and determined the solid hull and solid core of weighted Banach spaces H ∞ v , both in the case of the analytic functions on the disc and on the whole complex plane, for a very general class of radial weights v. The case of Bergman spaces was treated later in [59]. We present some of these results here.
Recall that a sequence (e n ) ∞ n=1 of elements of a separable Banach space X is a Schauder basis, if every element f ∈ X can be presented as a convergent sum where the numbers f n ∈ K are unique for f ∈ X .
By a theorem of Lusky [126], is never a basis for H 0 v (D). This implies the following consequence.

Corollary 18 In the case of analytic functions on the disc D, one always has S(H
Lusky [126] proved that the monomials = {z k : In the case of weighted spaces of entire functions we have the following result to be found in [59].

Theorem 19 Let v be a weight on the complex plane satisfying condition (b) (given later). The space H ∞ v is solid if and only if is a Schauder basis of H
Now we present the solid hull and solid core of weighted Banach spaces H ∞ v for concrete weights on the disc and on the complex plane.
and the solid core is These results are a consequence of a general theorem. To state it, we need a few definitions. Let r m ∈]0, R[ be a global maximum point of the function r m v(r ) for any m > 0. The weight v is said to satisfy the condition The proof is mainly based on results and techniques of Lusky [127] and methods due to Bennet, Stegenga and Timoney [34], in particular Theorem 16 of Kislyakov. We recall the main technical result of Lusky, since it is important in other contexts, for example in the study of Toeplitz operators. Let For the numbers m n as in condition (b) set Define the operators V n = R n − R n−1 for n ∈ N.
Here is the explicit calculation of the sequence m n for concrete weights. They are needed to deduce Theorems 20 and 21 from Theorem 22.
The weight v satisfies condition (b) for the sequence m n := p(log b)n 2 with K = b 5 .
, one can take m n = n 4 − n 2 . It can be shown that one can also take m n = n 4 in this case.
The investigation of solid hull and cores of spaces of type H ∞ has been continued by Schindl in [147]. He observed that some of the weights which appear in the descriptions given above arise frequently in the theory of ultradifferentiable and ultraholomorphic function classes. This connection enabled him to see which growth behaviour must satisfy the sequences (m n ) which appear in the descriptions and also to study when these numbers can exist. New examples were obtained.

Distance formulas
Lusky, Taskinen and the author investigated in [57] The proximinality in Theorem 24, i.e. the existence of the minimizer g, also appears in Perfekt [142] as a consequence of the fact that Our approach gave an elementary, direct proof of the formula of the distance. Consequences about Bloch type spaces were obtained, but will not be stated here.
Our proof depends on a technical lemma, which could be of independent interest, in which we use the following notation. Given an analytic function f on D or C, we denote by σ n f the n-th Cesàro mean of f ; i.e. the arithmetic mean of the first n Taylor polynomials of f . In this case, one has M(σ n f , r ) ≤ M( f , r ) for each 0 < r < R.
Then, for each The following simple examples show that the distance d( f , H 0 v ) can be attained at many points of H 0 v for a given function f ∈ H ∞ v .
(1) Consider the weight v(r ) = e −r , r ∈ [0, ∞[, on the complex plane and the analytic function k! for each n ∈ N. We have, for each n, P n ∈ H 0 v and Set P n (z) = n k=0 z k for each n ∈ N. We have, for each n, P n ∈ H 0 v and

A few definitions concerning bounded linear operators
We recall a few definitions concerning operator theory, mean ergodic operators and linear dynamics which will be used in the rest of the article.
and if the space is clear from the context, we simply write T e . The operator T ∈ L(X , Y ) is said to be Fredholm if ker T and Y /ImT are finite dimensional. An operator T is Fredholm if and only if there are S ∈ L(Y , X ) and compact operators K 1 ∈ L(X ) and is not injective. If we need to stress the space X , then we also write σ (T ; X ), σ pt (T ; X ) and ρ(T ; X ). Given λ, μ ∈ ρ(T ) The essential spectrum σ e (T , X ) of an operator T ∈ L(X ) on the Banach space X is the set of all λ ∈ C such that λI − T is not Fredholm. The essential spectral radius is It can be calculated as follows converges to some operator P ∈ L(X ) in the strong operator topology τ s , i.e., lim n→∞ T [ An operator T ∈ L(X ) is called uniformly mean ergodic if there exists P ∈ L(X ) such that lim n→∞ T [n] − P = 0. It is then immediate that necessarily lim n→∞ T n n = 0. A result of Lin, [120, Theorem 2.1], states that T ∈ L(X ) satisfying lim n→∞ T n n = 0 is uniformly mean ergodic if and only if Im whenever T is power bounded. An operator T ∈ L(X ), with X a separable Banach space, is called hypercyclic if there exists x ∈ X such that the orbit {T n x : n ∈ N 0 } is dense in X . If, for some z ∈ X , the projective orbit {λT n z : λ ∈ C, n ∈ N 0 } is dense in X , then T is called supercyclic. Clearly, hypercyclicity implies supercyclicity. The operator T is called chaotic if it is hypercylic and has a dense set of periodic points.
More details for mean ergodic operators can be seen in [96,120], and for linear dynamics in [26,106].

Continuity of differentiation and integration operators
The differentiation operator D( f ) := f is continuous on the Fréchet space H (G) of all analytic functions. A detailed study of continuity of the differentiation operator D( f ) := f acting in the space H ∞ v for a radial weight function on the disc or the complex plane was conducted by Harutyunyan and Lusky in [110]. They used methods developed by Lusky in [127], which were mentioned briefly in Sect. 5. As was observed by Abanin and Tien [6], the weights were assumed implicitly to be log-convex in [110]. This is a natural assumption as was explained in Sect. 3. In this section we first state some results of Abanin and Tien [6,8]. They used a more direct approach than Harutyunyan and Lusky.
We assume that the radial weight v : [0, R[→]0, ∞[ is continuous, non-increasing on [0, R[ and satisfies lim r →R r n v(r ) = 0 for each n ∈ N.
Every increasing log-convex weight on (0, R) has a right derivative everywhere on its domain of definition. Accordingly, we state the results for differentiable weights v.
(a) The following conditions are equivalent.
The following conditions are equivalent.
The differentiation operator D is continuous and surjective on H ∞ v (C) for v(r ) = e −αr , α > 0, it is continuous but not surjective for v(r ) = exp(−(log r ) 2 ) and it is not continuous for v(r ) = exp(−e r ). See also [110,Theorems 4

.1 and 4.2].
Theorem 27 [6,8,110] The following conditions are equivalent for a radial, log-convex weight v on D.
If these equivalent conditions hold, then the continuous operator D : Theorem 27 can be considered as an extension of a classical result of Hardy and Littlewood [97,Theorem 5.5 The integration operator J f (z) = z 0 f (ζ )dζ, z ∈ G, is also well-defined and continuous on the Fréchet space H (G) for G = D and G = C. The continuity of J on spaces of type H ∞ v was also investigated in [6,110]. We recall some results.

Theorem 28
Let v be a radial, log-convex weight on C.
(a) The following conditions are equivalent.
The following conditions are equivalent.
As a consequence, the integration operator J is continuous on

Corollary 29
Let v be a radial, log-convex weight on C. The following conditions are equivalent.
The following conditions are equivalent.
(i) The integration operator J :

Corollary 31
Let v be a radial, log-convex weight on C and w(r ) : Examples are given in [6,8] to show that the requirement that the weight is log-convex in the results in this section is really essential. Related results about continuity of differentiation and integration type operators in generalized Fock spaces can be seen in [132,134].

Spectrum, mean ergodicity and linear dynamics
We survey now some results about the behaviour of the differentiation and the integration operators when they act on some weighted Banach spaces of entire functions of type H ∞ v (C) and H 0 v (C). These results are taken from [30,44]. The spectrum of the differentiation operator on weighted Banach spaces of entire functions had been studied by Atzmon and Brive [24].
In   Extensions of these results for more general weighted Banach spaces of entire functions were obtained by Beltrán [27], Bonilla and the author [46] and Mengestie, Worku and the author in [64]. The case of differentiation and integration operators acting on Hörmander algebras was studied in [31]. Tien investigated in [157] the dynamical properties of translation operators on weighted Hilbert and Banach spaces of entire functions.

Invariant subspaces of the integration operator
Abanin and Tien in [7] described the proper closed invariant subspaces of the integration operator on various scales of weighted Banach spaces of analytic functions on the unit disc and the complex plane. We include some of their results for H 0 v . Let E be a Banach space of analytic functions on the open unit disc D or the complex plane C which contains the polynomials and such that the inclusion map E ⊂ H (G) is continuous. For each N ∈ N, we set

Lemma 38 Let E be a Banach space of analytic functions on the open domain G = D or G = C, such that the inclusion map E ⊂ H (G) is continuous and the polynomials are contained and dense in E. For each N ∈ N we have
Theorem 39 (Abanin, Tien [7]) Let v be a radial, continuous, non-increasing, log-convex weight v : [0, R[→ (0, ∞[, which satisfies lim r →R r n v(r ) = 0 for each n ∈ N, with R = 1 for the disc and R = +∞ for the complex plane. Assume that the integration operator J : H 0 v → H 0 v is continuous. Moreover, in the case of the complex plane assume that

Then every proper closed invariant subspace for J on H 0 v is of the form
The weight v(z) := exp(−|z| α ), α > 1, on the complex plane satisfies the assumption of Theorem 39. Galbis and the author utilized the results of Abanin and Tien to describe in [53] the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of analytic functions on the unit disc or the complex plane, in particular for Korenblum type spaces and for Hörmander algebras of entire functions.

The Cesàro operator of growth Banach spaces for standard weights
The classical Cesàro operator C is given by for f ∈ H (D). It is a Fréchet space isomorphism of H (D) onto itself. In terms of the Taylor coefficients f (n) := f (n) (0) n! , for n ∈ N 0 , of functions f (z) = ∞ n=0 f (n)z n ∈ H (D) one has the description It is known that there are many classical Banach spaces X of analytic functions on D such that the Cesàro operator C acts continuously from X into itself; for instance, the Hardy spaces H p (D), 1 ≤ p < ∞, the Bergman and the Dirichlet spaces, etc. See for example [14,143] and the references therein. On the other hand, C fails to act in H ∞ (D) since C(1)(z) = (1/z) log(1/ (1 − z)), for z ∈ D.
We collect in this section several results about the behaviour of the Cesàro operator C when it acts on growth Banach spaces A −γ and A −γ 0 , for γ > 0. We denote the norm in these spaces by The spectrum of C on A −γ and A −γ 0 was obtained by Aleman and Persson [14,143]. We keep the notations C γ,0 : A is a closed subspace of A −γ 0 and has codimension 1. Moreover, the Cesàro operator C γ : A −γ → A −γ is continuous and it satisfies Now we present several results obtained in collaboration with Albanese and Ricker [9].

Proposition 43
The Cesàro operator C γ,0 is not supercyclic and hence, also not hypercyclic, in each space A The optimal domain of the Cesàro operator C when it acts on a Banach space of analytic functions X on D is defined by which is a Banach space for the norm If C acts on X , then X ⊆ [C, X ] and the natural inclusion map is continuous. Moreover, [C, X ] is the largest of all Banach spaces of analytic functions Y on D that C maps continuously into X .

Theorem 44 Let γ > 0 and ϕ(z)
The optimal domain [C, A −γ ] of C γ : A −γ → A −γ is isometrically isomorphic to A −γ and is given by Moreover, the norm The behaviour of the Cesàro operator on the Korenblum space A −∞ and related Fréchet and (LB)-spaces of analytic functions on the unit disc was investigated by Albanese, Ricker and the author in [10]. The spectrum is completely determined and some consequences concerning mean ergodicity are deduced.

Volterra operators
The Volterra integral operator is defined on the space of analytic functions of G = D or G = C in the following way. Given g ∈ H (G), we set The analytic function g is called the symbol of V g . Clearly V g : The Volterra operator for holomorphic functions on the unit disc was introduced by Pommerenke [144] and he proved that V g is bounded on the Hardy space H 2 , if and only if g ∈ B M O A. Aleman and Siskakis [15] extended this result for H p , 1 ≤ p < ∞, and they considered later in [16] the case of weighted Bergman spaces. Volterra operators on weighted Banach spaces H ∞ v (D) of holomorphic functions on the disc of type have been investigated in [25], thus extending results in [113,153]. Lin [122] obtained some further results in this direction.
Contreras, Peláez, Pommerenke and Rättyä [87] studied thoroughly the boundedness, compactness and weak compactness of the operators V g : X → H ∞ from a Banach space of analytic functions X into H ∞ . They obtained general results which they applied to particular choices of X. Smith, Stoljarov and Volberg [152] presented a necessary and sufficient condition for the operator of integration to be bounded on the space of bounded analytic functions on a simply connected domain, thus solving a conjecture in [17] about the boundedness of V g with values on H ∞ when the function g is univalent. A counterexample to the general case of the conjecture about the boundedness of V g is also included in [152]. Motivated by the results of this article, Abakumov and Doubtsov [5] characterized the boundedness of V g : H ∞ v → H ∞ w for general weights v, w and a univalent symbol g. In the article [100] the authors characterize the continuity and (weak) compactness of V g : H ∞ v (D) → H ∞ when v(z) = (1 − |z| 2 ) α is a standard weight and g is univalent.
Aleman, Constantin, Peláez and Persson [11,13,14] investigated the spectra of Volterra and Cesàro operators on several spaces of holomorphic functions on the disc. The spectra of Volterra operators acting on growth spaces has been investigated by Malman [128]. Constantin started in [83] the study of the Volterra operator on spaces of entire functions. She characterized the continuity of V g on the classical Fock spaces and investigated its spectrum. Constantin and Peláez [84] characterize the entire functions g ∈ H (C) such that V g is bounded or compact on a large class of Fock spaces induced by smooth radial weights. See also [4,38,39,65]. The investigation of the spectrum of the Volterra operator for weighted spaces of entire functions was continued in [43,84]. Volterra operators on Korenblum type Fréchet and (LB)-spaces and on Hörmander algebras are studied in [42,43].
The first result is a consequence of more general Theorems [25, Theorems 1 and 2].

Proposition 45
Let g ∈ H (D) be an analytic function and let α > 0 and β > 0. Theorem 46 (Abakumov and Doubtsov [5]) Let g ∈ H (D) be a univalent function and let v and w be weight functions.

is continuous and if and only if
The proof of Theorem 46 uses methods of [152] and Theorem 2.

Lemma 47 Let X ⊂ H (D) be a Banach space that contains the constants and such that the inclusion X ⊂ H (D)
is continuous. Assume that V g : X → X is continuous for some non-constant entire function g such that g(0) = 0. Then X ).

The corresponding result is valid for Banach spaces X ⊂ H (C) containing the constants and such that the inclusion X ⊂ H (C) is continuous.
Theorem 48 (Malman [128]) Let g ∈ H (D), g(0) = 0, and let α > 0.
is continuous if and only if g is a polynomial of degree less than or equal to the integer part of p.

(C) is compact if and only if g is a polynomial of degree strictly less than p.
Theorem 50 [43] Assume that v(r ) = exp(−αr p ), α > 0, p > 0. Let g be a polynomial of degree n ∈ N less than or equal to the integer part of p with g(0) = 0.
(i) If the degree n of g satisfies n < p, then σ (V g , H ∞ v (C)) = {0}. (ii) If p = n and g(z) = βz n + k(z), k a polynomial of degree strictly less than n, then

Weighted composition operators
Weighted composition operators on various spaces of analytic functions on the unit disc or the complex plane have been studied very thoroughly by a number of authors. For the unit disc, the books of Cowen, MacCluer [89] and Shapiro [150] are standard references. In this section we mainly concentrate on composition operators on the spaces H ∞ v (D) and H 0 v (D) and we mention a few results about spaces of entire functions.

Continuity, (weak) compactness, isometries
We consider a non-constant self map ϕ ∈ H (D) satisfying ϕ(D) ⊂ D and a function ψ ∈ H (D) which is not identically equal zero. They induce the weighted composition operator This operator is continuous on H (D) for the topology of uniform convergence on the compact subsets of D.
If ψ = 1, then as usual we denote the composition operator W ϕ,1 by C ϕ And if ϕ(z) = z, z ∈ D, then W ϕ,ψ is the multiplication operator We first mention some results about continuity, compactness and the essential norm from [49,52,88,116,137]. All the weights we consider are radial, non-increasing and tending to zero at the boundary.

Theorem 51
Let v and w be weights on D.
(a) The following conditions are equivalent.
The following conditions are equivalent.
In this case, W ϕ,ψ = sup z∈D

Theorem 52
Let v and w be weights on D. If the operator W ϕ,ψ :  (ϕ(z)) .

Theorem 54
Let v and w be weights on D.
(1) Assume that the operator W ϕ,ψ : is continuous. Then W ϕ,ψ is either compact or an isomorphism on a subspace isomorphic to ∞ .
(2) Assume that the operator W ϕ,ψ : H 0 v (D) → H 0 w (D) is continuous. Then W ϕ,ψ is either compact or an isomorphism on a subspace isomorphic to c 0 . In particular, W ϕ,ψ is compact if and only if it is weakly compact in both cases.

Proposition 55
Let ϕ be given. The following holds: is continuous for all radial non-increasing weights v if and only if there is 0 < s < 1 such that |ϕ(z)| ≤ |z| for all |z| ≥ s.
is compact for all radial non-increasing weights v if and only if ϕ(D) ⊂ sD for some 0 < s < 1.

Proposition 56 A radial non-increasing weight v satisfies that the operator C
is continuous for every ϕ if and only if the weightṽ satisfies the condition If ϕ(z) = (z + 1)/2, z ∈ D, and v(z) = exp(−1/(1 − |z|)), z ∈ D, the composition operator C ϕ is not continuous on H ∞ v (D). We refer to Lusky [125,127] for the relevance of condition (L1) in connection with the isomorphic classification of spaces H ∞ v . Some conditions of various types that are equivalent to (L1) were stated in [6, Lemma 2.6]. Interesting extensions of Proposition 56 were obtained by Bourdon [72].
The characterization of nuclear weighted composition operators W ϕ,ψ : under some conditions on the weights has been obtained in [55]. This result was inspired and motivated by the characterization of nuclear weighted composition operators on Bloch spaces due to Fares and Lefèvre [101].
Fredholm weighted composition operators on H ∞ v and H 0 v were investigated in [102,105,115].
(a) The following conditions are equivalent.
(b) The following conditions are equivalent. A general result about invertible weighted composition operators is due to Bourdon [73]. Extensions of these results have been obtained recently by Mas and Vukotić [130].
Martín and Vukotić [129] analyzed when composition operators on the Bloch space are (not necessarily surjective) isometries, and they show that every thin Blaschke product induces an isometric composition operator on the Bloch space. Motivated by these results, together with Lindström and Wolf, we characterize in [56] isometric weighted composition operators on H ∞ v (D). As a consequence a composition operator C ϕ of H ∞ v p (D) for a standard weight v p (z) = (1 − |z|) p , p > 0, is an isometry if and only if ϕ is a rotation. Boyd and Rueda [78] present a detailed study of the question of under which conditions a given isometry between weighted spaces of holomorphic functions is surjective.
Boyd and Rueda have obtained interesting related results on isometries on weighted Banach spaces H ∞ v (U ) of holomorphic functions defined on an open subset U of C n . We refer the reader to their papers [74,75] and the references therein. They explain how the isometries of a weighted space of holomorphic functions are determined by a subgroup of the automorphisms of a subset of the domain, called the v-boundary of U . The relation between this group and the weight is investigated for bounded and unbounded domains. Examples are presented.

The spectrum
The following result was obtained in [164]. It extends theorems due to Aron, Lindström [23,54].

A similar result holds for H 0 v (D).
Further extensions and related results can be seen in [98,99,103,104]. Let ϕ be an analytic self map on D which is not an automorphism and has a (necessarily unique) fixed point a ∈ D. By Koenigs' Theorem [150,Chapter 6] As a consequence of Schwarz Lemma, an analytic self map ϕ on D which is not an automorphism and has a fixed point a ∈ D satisfies ϕ (a) = 0 if and only if 0 < |ϕ (a)| < 1. Moreover, in this case, the Koenigs' eigenfunction of ϕ can be obtained as the limit of the sequence (σ n ) n with σ n := ϕ n /ϕ (a) n , n ∈ N, which converges to σ uniformly on the compact subsets of D. Here ϕ n = ϕ • · · · • ϕ is the n-fold composition of ϕ.
The spectrum and essential spectrum of the composition operator C ϕ : H (D) → H (D) for an analytic self map ϕ on D which is not an automorphism and has a fixed point a ∈ D with 0 ≤ |ϕ (a)| < 1 have been determined recently in [20]: σ (C ϕ , H (D)) = {0} ∪ {ϕ (a) n ; n ∈ N 0 }, and its essential spectrum reduces to {0}. The proofs are based on explicit formulas for the spectral projections associated with the point spectrum found by Koenigs. As a consequence, information on the spectrum for bounded composition operators induced by a symbol as above on Banach spaces of analytic functions continuously embedded in H (D) is obtained. The definitions of spectrum and essential spectrum for an operator on a locally convex space coincide with those given in Sect. 7. Recall that an operator between locally convex spaces is compact if it maps a neighbourhood of the domain into a relatively compact set in the image. The case of composition operators induced by rotations was analyzed in [41].
In [54,98] it was investigated how the essential spectral radius of C ϕ on both H ∞ v (D) and H 0 v (D) determines whether the Koenigs eigenfunction σ of ϕ belongs to H ∞ v (D) and H 0 v respectively. Let ϕ be an analytic self map on D which is not an automorphism such that ϕ(0) = 0 and 0 < |ϕ (0)| < 1. Bourdon [71]  . Examples given in [54] show that Bourdon's characterization does not hold for more general radial weights. On the other hand, by [98,Theorem 3.1], σ ∈ H ∞ if and only if r e,H ∞ (C ϕ ) = 0.
Proposition 59 [98] Let ϕ be an analytic self map on D which is not an automorphism such that ϕ(0) = 0 and 0 < |ϕ (0)| < 1. Let v be a radial weight on D. Then Hyvärinen, Lindström, Nieminen and Saukko [115], using ideas of Kamowitz and Gunatillake, calculated the spectrum of the invertible weighted composition operator W ϕ,ψ for an automorphic symbol ϕ on a wide class of analytic function spaces; in particular for spaces of type H ∞ v (D). The analysis of the spectral behaviour depends on the type of the symbol ϕ, that is, if it is an elliptic, parabolic or hyperbolic automorphism.
An automorphism ϕ ∈ Aut(D) is an injective analytic function on D such that ϕ(D) = D. If ϕ ∈ H (D) is a self map on D that fixes a point p ∈ D, and is not a conformal automorphism, then (ϕ n ) n converges to p uniformly on the compact subsets of D.
An elliptic automorphism has one fixed point in D. A hyperbolic automorphism has two fixed points in the boundary ∂D of D, one is attractive and the other one repulsive, and a parabolic automorphism has one fixed point in the boundary of D with multiplicity 2. See more details in [79], [89] and [150]. Recall that the disc algebra A(D) is the space of continuous functions on the closed unit disc D which are analytic on D. It is a Banach space endowed with the supremum norm on D.
Theorem 61 [115] Let v p = (1 − |z| 2 ) p , p > 0 and let W ϕ,ψ : (iii) If ϕ is an automorphism such that there is a j ∈ N such that ϕ j (z) = z for each z ∈ D, then, for the smaller such n ∈ N, we have (iv) If ϕ is an automorphism such that ϕ n does not coincide with the identity function id(z) = z for each n ∈ N, and a ∈ D is the unique fixed point of ϕ, then σ (W ϕ,ψ , H ∞ v p (D)) = {λ ∈ C ; |λ| = |ψ(a)|}.
Further results can be seen in [103,104]. Many questions remain open about the spectrum of weighted composition operators.

Hypercyclicity and mean ergodicity
Miralles and Wolf [135] investigated hypercyclic continuous composition operators C ϕ : for an analytic self map ϕ on D and a radial weight on D satisfying the usual assumptions (non-increasing approaching 0 at the boundary).
is a continuous hypercyclic operator, then ϕ has no fixed point in D and it is injective.

Theorem 63 If ϕ is an automorphism which fixes no point in D and C
We refer to the book of Shapiro [150] for linear fractional transformations of the unit disc.

Theorem 64 Let ϕ be a linear fractional transformation of D such that C
is continuous. If ϕ is a hyperbolic non-automorphism, then C ϕ is hypercyclic.

Proposition 65
Let v(z) = (1 − |z|) 1/2 for every z ∈ D. If ϕ be a linear fractional transformation of D which is a parabolic non-automorphism, then C ϕ : These results were extended to the case of weighted composition operators of the form λC ϕ , λ ∈ C, by Liang and Zhou [121]. Colonna and Martínez-Avendaño [82] extend some of these results and present an informative brief summary of the literature on hypercyclic composition operators on Banach spaces of analytic functions.
Power bounded and (uniformly) mean ergodic composition operators on H (G), G an open connected subset of C, were investigated by Domanski and the author in [47]. This research was continued later for operators on spaces of real analytic functions in [48]. These papers triggered quite an amount of research. The study of mean ergodic weighted composition operators on H (G) was done by Beltrán, Gómez-Collado, Jordá and Jornet in [33]. They continued their work in [32] about mean ergodicity of composition operators on the Banach spaces H ∞ and the disc algebra A(D).
A systematic study of powers of (weighted) composition operators on Banach spaces of analytic functions on the unit disc has been done by Arendt, Chalendar, Kumar and Srivastava [21,22], Jordá and Rodríguez [118] and Tien [158]. See the references in these papers for more related work in this direction. We state some results for composition operators C ϕ on H ∞ v (D) and H 0 v (D).

Proposition 66
Let v be a log-convex weight on D satisfying condition (L1). Let ϕ be an elliptic automorphism with fixed point z(0) ∈ D. Then (i) If ϕ is equivalent to a rational rotation, then C ϕ is uniformly mean ergodic on H ∞ v (D) and H 0 v (D). Moreover, there is k ∈ N such that ((C ϕ ) [n] ) n converges to (1/k)(C ϕ + · · · + (C ϕ ) k ).
(ii) If ϕ is equivalent to a irrational rotation, then C ϕ is not uniformly mean ergodic on

Weighted composition operators on spaces of entire functions
Many properties of composition operators on spaces of entire functions have also been investigated. For instance, in the frame of Fock spaces, in 2003, Carswell, MacCluer and Schuster [81] characterized bounded and compact composition operators on the classical Fock spaces F p , 0 < p < ∞. We refer to [166] for Fock spaces. They showed that only the class of affine mappings ψ(z) = az + b, |a| ≤ 1 and b = 0 whenever |a| = 1 induce bounded composition operators. Compactness of C ψ was described by the strict requirement |a| < 1.
In 2008, Guo and Izuchi [107] studied various aspects of the composition operators on Fock type spaces. In analogy to the notion of associated weights for weighted spaces of analytic functions with sup-norms, Mangino and the author introduced in [63] p-associated weights for spaces of entire p-integrable functions, 1 ≤ p < ∞. As an application, necessary conditions for the boundedness of composition operators acting between general Fock type spaces were proved.
Seyoum, Mengestie and the author [149] proved that every bounded composition operator C ϕ defined by an analytic symbol ϕ on the complex plane when acting on generalized Fock spaces F p ϕ , 1 ≤ p ≤ ∞, and p = 0, is power bounded. Mean ergodic and uniformly mean ergodic bounded composition operators on these spaces are characterized in terms of the symbol. The behaviour for p = 0 and p = ∞ differs. The set of periodic points of these operators is also determined. This research is continued for weighted composition operators in [148].
Ueki [161,162] investigated the boundedness, compactness and essential norm of weighted composition operators W ϕ,ψ on Hilbert Fock spaces of several variables in terms of a certain integral transform. Weighted composition operators on Fock spaces were investigated by Hai and Khoi [108], and between different Fock type spaces by Tien and Khoi [160]. The characterizations of the boundedness and compactness of these operators in the Fock space setting required that the symbol ϕ and the multiplier ψ satisfy certain uniform conditions. Carroll and Gilmore [80] present a more explicit characterization of bounded weighted composition operators, as well as compact weighted composition operators, on Fock spaces in terms of the order and type of the multiplier, and obtain a complete description of zero-free multipliers that admit bounded or compact operators. An explicit asymptotics for the iterates of the operator is also given. As an application it is shown that a weighted composition operator acting on Fock spaces cannot be supercyclic.
Tien [159] completely solves several problems, such as boundedness and compactness, topological structure, ergodic and dynamical properties, for composition operators on weighted Banach spaces of entire functions with sup-norms.
The dynamics of weighted composition operators on weighted Banach spaces of entire functions H ∞ v (C) and H 0 v (C) has been investigated by Beltrán in [28]. The continuity and compactness are characterized. Moreover, in the case of affine symbols and exponential weights, it is analyzed when the operator is power bounded, (uniformly) mean ergodic and hypercyclic. This work was continued by Beltrán and Jordá [29] to investigate power boundedness, (uniform) mean ergodicity and hypercyclicity of certain weighted composition operators on spaces of entire functions of exponential and infraexponential type.

Superposition operators
The purpose of this section is to present a few results about superposition operators f → ϕ• f defined between weighted Banach spaces H ∞ v = H ∞ v (D) of holomorphic functions on the disc by means of an entire function ϕ. If X and Y are linear spaces of holomorphic functions on the unit disc D of the complex plane and ϕ is an entire function, the superposition operator S ϕ : X → Y with symbol ϕ is defined by S ϕ ( f ) := ϕ • f . Since X and Y are assumed to be linear spaces, the operator S ϕ is linear if and only if ϕ is a linear function that fixes the origin. The central question concerning superposition operators is to characterize those symbols ϕ such that the superposition operator maps X into Y . In case X and Y are Banach spaces, it is also important to determine when S ϕ is bounded, in the sense that it maps bounded subsets of X into bounded subsets of Y , when S ϕ is continuous or when it is compact, in the sense that it maps bounded sets into relatively compact sets.
We refer the reader to the introduction of [68] and to the survey [163] for references about superposition operators on different spaces of analytic functions on the disc. Superposition operators on weighted spaces of type H ∞ v (D) have been investigated in [68,76,77,93,145].

Toeplitz operators
Lusky, Taskinen and the author studied boundedness and compactness of Toeplitz operators on H ∞ v (D) in [60] and [61] for general weights on the unit disc, in particular for exponential weights v(r ) = exp(−α/(1 − r ) β ), α, β > 0. Toeplitz operators on spaces of analytic functions have been investigated by many authors. We refer the reader to [165] and, for example, to the introduction of [60] or to the survey article [156]. We recall the necessary definitions to state some results.
Given a weight v on D, we set Let μ be the Lebesgue area measure on D endowed with v as density, i.e. dμ(re iϕ ) = v(r )rdrdϕ and denote the weighted L p -and Bergman spaces by form an orthonormal basis of A 2 v . Since the convergence in the space A 2 v implies pointwise convergence, we find the reproducing kernel, i.e. a family of functions K z ∈ A 2 v , z ∈ D, such that g(z) = g, K z = D g(w)K z (w) dμ(w) for all g ∈ A 2 v . The integral operator defined by the right-hand side can be extended to L 2 v , and it actually defines the orthogonal projection from L 2 v onto A 2 v , i.e. the Bergman projection P v ; see [85,94]. Using the orthonormal basis we can write, for all z ∈ D, Some questions seems to be open concerning the operators mentioned in this last section.