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

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 limr→1−v(r) = 0: (i) v is essential. (ii) v is equivalent to a log-convex radial weight w on D. (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 ap

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 limr→∞rnv(r) = 0 for each n ∈N. 123

Proposition 4 Let α > 1 and let vα be a weight on the complex plane such that vα(r) := exp(−(logr)α), e ≤r < ∞. The weight vα is essential if α ≥2 and not essential if 1 < α < 2. 4 The seminormed space H∞ v (G) for non-negative weight v, not necessarily bounded or strictly positive Vukotic and the author investigated in [69] some necessary and some sufficient conditions on a non-negative function v : G →[0, ∞[, not necessarily bounded or strictly positive, defined on an open connected domain G in

Theorem 2.1]. We review here some of the results in [69]. For a non-negative function v : G →[0, ∞[, the space H∞ v (G) associated with v is also defined by H∞ v (G) :=  f ∈H(G) ; ∥f ∥v = sup z∈G v(z)| f (z)| < +∞  , and it is endowed with the natural seminorm ∥f ∥v := supz∈G v(z)| f (z)|. We use the following notation in this section: Ev := {z ∈G ; v(z) > 0}.

Proposition 5 Let v : G →[0, ∞[ be a weight on a planar domain G. Then the space H∞ v (G) is normed if and only if Ev is not a discrete set (that is, it has a limit point in G). Moreover, if H∞ v (G) is normed, then the inclusion map J : H∞ v (G) →(H(G), τco) has closed graph.

Proposition 6 Assume that the space H∞ v (G) is normed. The following conditions are equiv- alent: (i) The space H∞ v (G) is a Banach space. (ii) The inclusion map J : H∞ v (G) →(H(G), τco) is continuous. (Equivalently, every sequence in H∞ v (G) bounded in the norm is a normal family.) (iii) The closed unit ball B∞ v of H∞ v (G) is bounded in (H(G), τco). (iv) For each z ∈G, the point evaluation functional δz( f ) := f (z), z ∈G, satisfies δz ∈H∞ v (G)′ and, moreover, supz∈K ∥δz∥′

Theorem 7 Let v : G →[0, ∞[ be a bounded weight on a planar domain G such that the space H∞ v (G) is normed. The space H∞ v (G) is complete if and only if there is a bounded, continuous, strictly positive weight ˜v on G such that H∞ v (G) = H∞ ˜v (G). 123

Proposition 8 Let v : G →[0, ∞[ be a weight on a planar domain G. If H∞ v (G) is a Banach space containing non-zero functions, then the boundary ∂G is contained in the closure Ev of Ev in C.

Corollary 9 Let v : G →[0, ∞[ be a weight on a planar domain G (other than the plane itself) such that H∞ v (G) is normed and it contains non-zero functions. (1) If Ev is contained in a convex closed proper subset A of G, then H∞ v (G) is not a Banach space. (2) If the closure of Ev in C is a compact subset of G, then H∞ v (G) 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

Proposition 10 Let v : G →[0, ∞[ be a weight on a planar domain G. Suppose that for each z ∈G there is a bounded open set U ⊂G such that z ∈U, ∂U ⊂Ev, and v is bounded away from 0 on ∂U. Then H∞ v (G) is a Banach space

Corollary 11 Let v : G →[0, ∞[ be a continuous weight on a planar domain G such that G\Ev is discrete, i.e., the zeros of v are isolated. Then H∞ v (G) is a Banach space. As a consequence of Corollary 11, if F ∈H(G) is a non-zero analytic function on a planar domain G and v(z) := |F(z)|, z ∈G, then H∞ v (G) is a Banach space.

Corollary 12 Let v : G →[0, ∞[ be a weight on a planar domain G such that G\Ev is a compact subset of G. If infz∈K v(z) > 0 for each compact subset K ⊂G\(G\Ev), then H∞ v (G) is a Banach space.

Corollary 13 Let G = D (resp. G = C). Let v be a bounded radial weight on G. The space H∞ v (G) is a Banach space if and only if Ev is not compact in G or, equivalently, if and only if there is an increasing sequence (rk)k in ]0, 1[ tending to 1 (resp. (rk)k in ]0, ∞[ tending to ∞) such that v(rk) > 0 for each k ∈N. 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) := an > 0 if |z| = 1 −(1/n), and v

Proposition 14 Let F ∈H(G) be a non-zero function on a planar domain G. Define v(z) := 0 if F(z) = 0 and v(z) := 1/|F(z)| if F(z) ̸= 0. Then H∞ v (G) is a Banach space that coincides with the set of all f ∈H(G) such that there is C = C( f ) > 0 with | f (z)| ≤ C|F(z)| for each z ∈G. 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 (rn)n of positive numbers tending to 1 such t

Theorem 16 (Kislyakov) There is C > 0 such that for each (b j)m j=n, 0 < n < m, there is a polynomial P(z) = m j=n c jz j such that |b j| ≤|c j|, j = n, . . . , m and ∥P∥∞≤ C(m j=n |b j|2)1/2. The solid hull S(H p) for 1 ≤p < 2 seems to be unknown. The solid core of the Hardy spaces s(H p) = H2, 1 ≤p ≤2, is known. Moreover s(H∞) = ℓ1. 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,

Theorem 17 If S(H∞ v ) = H∞ v , then  is a Schauder basis of H0 v . By a theorem of Lusky [126],  is never a basis for H0 v (D). This implies the following consequence.

Corollary 18 In the case of analytic functions on the disc D, one always has S(H∞ v (D)) ̸= H∞ v (D) and s(H∞ v (D)) ̸= H∞ v (D). Lusky [126] proved that the monomials  = {zk : k = 0, 1, 2, . . .} are a basis of H0 v (C) for the weight v(r) = exp(−(log(r))2). 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 H0 v . 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.

Theorem 20 For v(r) = exp(−1/(1 −r)) the solid hull of H∞ v (D) is ⎧ ⎨ ⎩(bm)∞ m=0 : sup n exp(−2n2) (n+1)4  m=n4+1 |bm|2  1 −1 n2

Theorem 21 Let v be the weight v(r) = exp(−ar p) on C, where a > 0 and p > 0 are constants. Then, the solid hull of H∞ v (C) is ⎧ ⎨ ⎩(bm)∞ m=0 : sup n∈N  pn2+1<m≤p(n+1)2 |bm|2e−2n2n4m/p(ap)−m/p < ∞ ⎫ ⎬ ⎭. and the solid core is

Theorem 22 If the weight v satisfies condition (b), we have S(H∞ v ) = ⎧ ⎪⎨ ⎪⎩ (bm)∞ m=0 : sup n v(rmn) ⎛ ⎝  mn<m≤mn+1 |bm|2r2m

Theorem 23 (Lusky) The norms ∥h∥v and sup n sup rmn−1≤r≤rmn+1 M∞(Vnh,r)v(r) are equivalent. Moreover, the operators Vn are uniformly bounded on H∞ v and there are c1 > 0 and c2 > 0 with c1 sup n

Theorem 24 For every f ∈H∞ v there is g ∈H0 v with d( f , H0 v ) = ∥f −g∥v = lim sup r→R M( f ,r)v(r). 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 σn 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(σn f ,r) ≤M( f ,r) for each 0 < r < R.

Lemma 25 Let f ∈H∞ v and assume that there is 0 < τ < 1 with τ∥f ∥v ≤lim sup r→R M( f ,r)v(r). Then, for each ε > 0 and m ∈N there is n ∈N, n > m, such that with ρ = (1 −τ)/(1 + τ) we have  1 + τ 2(1 + ε)  ∥f −ρσn f ∥v ≤lim sup r→R M( f ,r)v(r) = lim sup r→R

Theorem 26 (Abanin and Tien [6, 8]) Let v be a radial, log-convex weight on C. (a) The following conditions are equivalent. (i) The operator D : H∞ v (C) →H∞ v (C) is continuous. (ii) −log v(r) = O(r) as r →∞. (iii) lim supr→∞v(r)(1/v)′(r) < ∞. (b) The following conditions are equivalent. (i) The operator D : H∞ v (C) →H∞ v (C) is compact. (ii) −log v(r) = o(r) as r →∞. (iii) limr→∞v(r)(1/v)′(r) = 0. 123

Theorem 27 [6, 8, 110] The following conditions are equivalent for a radial, log-convex weight v on D. (i) The operator D : H∞ v (D) →H∞ w (D), with w(r) = (1 −r)v(r) is continuous. (ii) v(r)(1 −r)−α is increasing on [r0, 1) for some α > 0 and some r0 > 0. (iii) lim supr→1  −(1−r)v′(r) v(r)  < ∞. (iv) supn v(1−2−n) v(1−2−n+1) < ∞.

Theorem 27 can be considered as an extension of a classical result of Hardy and Littlewood [97, Theorem 5.5]: f ∈H(D) satisfies supz∈D(1−|z|)γ | f (z)| < ∞if and only if supz∈D(1− |z|)γ +1| f ′(z)| < ∞. 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. (i) The operator J : H∞ v (C) →H∞ v (C) is continuous. (ii) lim supr→∞v(r)  r 0 1 v(t)dt < ∞. (iii) lim infr→∞v(r)(1/v)′(r) > 0. (b) The following conditions are equivalent. (i) The operator J : H∞ v (C) →H∞ v (C) is compact.

Corollary 29 Let v be a radial, log-convex weight on C. The following conditions are equiv- alent. (i) The operator D : H∞ v (C) →H∞ v (C) is continuous and surjective. (ii) 0 < lim infr→∞v(r)(1/v)′(r) ≤lim supr→∞v(r)(1/v)′(r) < ∞. (iii) There are A, C ≥1 such that, for all 0 ≤r < ∞, 1 Ae−Cr ≤v(r) ≤Ae−r/C.

Theorem 30 Let v be a radial, log-convex weight on D and w(r) := v(r)/(1−r), 0 ≤r < 1. The following conditions are equivalent. (i) The integration operator J : H∞ w (D) →H∞ v (D) is continuous. 123

Corollary 31 Let v be a radial, log-convex weight on C and w(r) := v(r)/(1−r), 0 ≤r < 1. The differentiation operator D : H∞ v (D) →H∞ w (D) is continuous and surjective if and only if 0 < lim infr→1  −(1 −r)v′(r) v(r)  ≤lim supr→1  −(1 −r)v′(r) v(r) 

Theorem 33 Assume that the differentiation operator D : H0 v (C) →H0 v (C) is continuous. The following conditions are equivalent: (1) D has a dense set of periodic points. (2) D has a periodic point different from 0. (3) limr→∞v(r)er = 0.

Theorem 34 Assume that the differentiation operator D : H0 v (C) →H0 v (C) is continuous. The following conditions are equivalent: (1) D is hypercyclic on H0 v (C). (2) lim infn→∞ ∥zn∥v n! = 0.

Proposition 35 The operator J is never hypercyclic on H0 v (C) and it has no periodic points different from 0 in H∞ v (C).

Corollary 36 Let v be a weight such that the differentiation operator D : H0 v (C) →H0 v (C) is continuous. Then (1) If there is A > 0 such that 1/v(r) ≤Ar−1/2er, r > 0, then D is not hypercyclic on H0 v (C). In particular D is not hypercyclic for v(r) = exp(−log2 r) or for v(r) = exp(−αr), 0 < α < 1. 123

Theorem 37 (1) The differentiation operator D satisfies (i) ∥Dn∥α = n! eα n n , n ∈N. (ii) It is power bounded if and only if α < 1. (iii) The spectrum of D is the closed disc of radius α. (iv) It is uniformly mean ergodic on H∞ α (C) and H0 α(C) if α < 1, not mean ergodic if α > 1, and it is not mean ergodic on H∞ 1 (C) and not uniformly mean ergodic on

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 AN(E) = span({z j ; j ≥N}).

Theorem 39 (Abanin, Tien [7]) Let v be a radial, continuous, non-increasing, log-convex weight v : [0, R[→(0, ∞[, which satisfies limr→R rnv(r) = 0 for each n ∈N, with R = 1 for the disc and R = +∞for the complex plane. Assume that the integration operator J : H0 v →H0 v is continuous. Moreover, in the case of the complex plane assume that lim infr→∞ rv(r)(1/v)′(r) −log v(r) > 1. Then every proper closed invariant subspace for J on H0 v is of the form AK (H0 v ) = { f ∈H0 v ; f ( j)(0) = 0, 0 ≤j

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. 9 The Cesàro operator of growth Banach spaces for standard weights The classical Cesàro ope

Theorem 40 Let γ > 0. The Cesàro operator Cγ,0 : A−γ 0 →A−γ 0 is continuous and it has the following properties. (i) σpt(Cγ,0) = { 1 m : m ∈N, m < γ }. (ii) σ(Cγ,0) = σpt(Cγ,0) ∪{λ ∈C: |λ −1 2γ | ≤ 1 2γ }. (iii) If |λ −1 2γ | < 1

Theorem 41 (i) Let γ ≥1. Then ∥Cn γ ∥= ∥Cn γ,0∥= 1 for all n ∈N. (ii) Let 0 < γ < 1. Then ∥Cn γ ∥= ∥Cn γ,0∥= 1/γ n for all n ∈N.

Theorem 42 (i) Let 0 < γ < 1. Both of the operators Cγ and Cγ,0 fail to be power bounded and are not mean ergodic. Moreover, Ker(I −Cγ ) = Ker(I −Cγ,0) = {0}, and Im(I −Cγ ) (resp. Im(I −Cγ,0)) is a proper closed subspace of A−γ (resp. of A−γ 0 ). (ii) Both of the operators C1 and C1,0 are power bounded but not mean ergodic. Moreover, Im(I −C1) (resp. Im(I −C1,0)) is not a closed subspace of A−γ (resp. of A−γ 0 ). (iii) Let γ > 1. Both of the operators Cγ and Cγ,0 are power bounded and uniformly

Proposition 43 The Cesàro operator Cγ,0 is not supercyclic and hence, also not hypercyclic, in each space A−γ 0 , for γ > 0. This follows from the fact that C is not supercyclic on H(D), since A−γ 0 is dense in H(D). A vector space X ⊆H(D) is called a Banach space of analytic functions on D if it contains the polynomials and it is a Banach space relative to a norm for which the natural inclusion of X into H(D) is continuous. Since evaluation at points of D are continuous linear functionals on H(

Theorem 44 Let γ > 0 and ϕ(z) := 1/(1 −z) for z ∈D. The optimal domain [C, A−γ ] of Cγ : A−γ →A−γ is isometrically isomorphic to A−γ and is given by [C, A−γ ] = { f ∈H(D) : f ϕ ∈A−(γ +1)}. (7) Moreover, the norm ∥· ∥[C,A−γ ] is equivalent to the norm f →∥f ϕ∥−(γ +1) and the containment A−γ ⊆[C, A−γ ] is proper. A similar result holds for the optimal domain [C, A−γ 0 ] of Cγ,0 : A−γ 0 →A−γ 0 . The behaviour of the Cesàro operator on the Korenblum space A−∞and related Fréchet and (LB)-spaces of an

Proposition 45 Let g ∈H(D) be an analytic function and let α > 0 and β > 0. (i) The operator Vg : A−α →A−β is continuous if and only if Vg : A−α 0 →A−β 0 is continuous and if and only if sup z∈D (1 −|z|)β−α+1|g′(z)| < ∞. (ii) If α < β, then Vg : A−α →A−β is continuous if and only if Vg : A−α 0 →A−β 0 is

Theorem 46 (Abakumov and Doubtsov [5]) Let g ∈H(D) be a univalent function and let v and w be weight functions. (i) The operator Vg : H∞ w (D) →H∞ v (D) is bounded if and only if sup 0≤θ≤2π sup 0≤r<1 v(r)  r 0 |g′(teiθ| ˜w(t) dt < ∞.

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 Vg : X →X is continuous for some non-constant entire function g such that g(0) = 0. Then {0} ∪{λ ∈C\{0} ; e g λ /∈X} ⊂σ(Vg, X). 123

Theorem 48 (Malman [128]) Let g ∈H(D), g(0) = 0, and let α > 0. (i) If g ∈H∞or g ∈B0, then σ(Vg, A−α) = σ(Vg, A−α 0 ) = {0}. (ii) If g(z) = c log(1/(1 −wz)), z ∈D with c, w ∈C, c ̸= 0, |w| = 1, then Vg : A−α → A−α, α > 0, is continuous and σ(Vg, A−α) = {λ ∈C ; Re( c λ) ≥α}. In the notation of Theorem 48, we understand 0 ∈{λ ∈C ; Re( c λ) ≥α}. With this in mind, this set coincides with the disc {λ ∈C ; |λ −c 2α | ≤|c| 2α }.

Proposition 49 [65] Assume that v(r) = exp(−αr p), α > 0, p > 0. (i) Vg : H∞ v (C) →H∞ v (C) is continuous if and only if g is a polynomial of degree less than or equal to the integer part of p. (ii) Vg : H∞ v (C) →H∞ v (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(Vg, H∞ v (C)) = {0}. (ii) If p = n and g(z) = βzn + k(z), k a polynomial of degree strictly less than n, then σ(Vg, H∞ v (C)) = {λ ∈C ; |λ| ≤|β|/α}. Moreover, we have σ(Vg, H∞ v (C)) = {0} ∪{λ ∈C\{0} ; e g λ /∈H∞ v (C)}. 11 Weighted composition operators Weighted composition operators on variou

Theorem 51 Let v and w be weights on D. (a) The following conditions are equivalent. (1) Wϕ,ψ(H∞ v (D)) ⊂H∞ w (D). (2) Wϕ,ψ : H∞ v (D) →H∞ w (D) is continuous. (3) supz∈D |ψ(z)|w(z) ˜v(ϕ(z)) < ∞. (4) supn∈N0 ∥ψϕn∥v ∥zn∥v

Theorem 52 Let v and w be weights on D. If the operator Wϕ,ψ : H∞ v (D) →H∞ w (D) is continuous, then its essential norm satisfies ∥Wϕ,ψ∥e = lim s→1 sup |ϕ(z)|>s |ψ(z)|w(z) ˜v(ϕ(z)) = lim supn→∞ ∥ψϕn∥v ∥zn∥v . In particular, Wϕ,ψ is compact if and only if

Theorem 53 Let v and w be weights on D. If the operator Wϕ,ψ : H0 v →H0 w is continuous, then its essential norm satisfies ∥Wϕ,ψ∥e = lim sup|z|→1 |ψ(z)|w(z) ˜v(ϕ(z)) . In particular, Wϕ,ψ is compact if and only if lim sup|z|→1 |ψ(z)|w(z) ˜v(ϕ(z)) = 0.

Theorem 54 Let v and w be weights on D. (1) Assume that the operator Wϕ,ψ : H∞ v (D) →H∞ w (D) is continuous. Then Wϕ,ψ is either compact or an isomorphism on a subspace isomorphic to ℓ∞. (2) Assume that the operator Wϕ,ψ : H0 v (D) →H0 w(D) is continuous. Then Wϕ,ψ is either compact or an isomorphism on a subspace isomorphic to c0. In particular, Wϕ,ψ is compact if and only if it is weakly compact in both cases.

Proposition 55 Let ϕ be given. The following holds: 123

Proposition 56 A radial non-increasing weight v satisfies that the operator Cϕ : H∞ v (D) → H∞ v (D) is continuous for every ϕ if and only if the weight ˜v satisfies the condition (L1) sup n ˜v(1 −2−n) ˜v(1 −2−n−1) < ∞. If ϕ(z) = (z + 1)/2, z ∈D, and v(z) = exp(−1/(1 −|z|)), z ∈D, the composition operator Cϕ 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 type

Theorem 57 Let Wϕ,ψ : H0 v (D) →H0 w(D) be continuous. (a) The following conditions are equivalent. (i) Wϕ,ψ : H∞ v (D) →H∞ w (D) is Fredholm. (ii) Wϕ,ψ : H0 v (D) →H0 w(D) is Fredholm. (iii) ψ ∈H∞, there is ε > 0 such that |ψ(z)| > ε for all |z| ≥1 −ε, and ϕ is an automorphism. (b) The following conditions are equivalent. (i) Wϕ,ψ : H∞ v (D) →H∞

Theorem 58 Suppose ϕ is not an automorphism and has fixed point a ∈D. Then σ(Wψ,ϕ, H∞ v (D)) = {λ ∈C : |λ | ≤re,H∞ v (Wψ,ϕ)} ∪{ψ(a)ϕ′(a)n}∞ n=0. A similar result holds for H0 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], if ϕ′(a) = 0, the equation f ◦ϕ = λ f has a non-trivial solution if and only if λ = 1 and the c

Theorem 3.1], σ ∈H∞if and only if re,H∞(Cϕ) = 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 (i) The Koenigs’ eigenfunction σ belongs to H∞ v (D) if and only if the sequence σn := ϕn/ϕ′(0)n, n ∈N, is bounded in H∞ v (D). (ii) Assume that ϕ is univalent. The Koenigs’ eigenfunction σ belongs to H0 v (D) if and only if limn→∞∥σn −σ∥v = 0. Hyvärinen, Lindström, Nieminen and Saukko [115], using ideas of Kamowitz and Gunatil- lake, cal

Theorem 60 (Theorem of Denjoy–Wolff) If ϕ ∈H(D) is a self map on D with no fixed point in D (for example if ϕ ∈Aut(D) is parabolic or hyperbolic), then there is a unique point w in the boundary of D, called the Denjoy–Wolff point of ϕ, such that (ϕn)n converges to w uniformly on the compact subsets of D. If ϕ ∈H(D) is a self map on D that fixes a point p ∈D, and is not a conformal automorphism, then (ϕn)n converges to p uniformly on the compact subsets of D. An elliptic automorphism has one fixed p

Theorem 61 [115] Let vp = (1 −|z|2)p, p > 0 and let Wϕ,ψ : H∞ vp (D) →H∞ vp (D) be continuous. Assume that ψ ∈A(D) is bounded away from zero on D. Then (i) If ϕ is a parabolic automorphism with Denjoy–Wolff point a ∈∂D, then σ(Wϕ,ψ, H∞ vp (D)) = {λ ∈C ; |λ| = |ψ(a)|}. (ii) If ϕ is a hyperbolic automorphism with attractive fixed point a ∈∂D and repulsive fixed point b ∈∂D, such that |ψ(b)/ϕ′(b)p| ≤|ψ(a)/ϕ′(a)p|, then σ(Wϕ,ψ, H∞ vp (D)) =  λ ∈C ; |ψ(b)| ϕ′(b)p ≤|λ| ≤|ψ(a)| ϕ′(a)p

Proposition 62 If Cϕ : H0 v (D) →H0 v (D) 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 Cϕ : H0 v (D) →H0 v (D) is continuous, then Cϕ is hypercyclic. 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 Cϕ : H0 v (D) → H0 v (D) is continuous. If ϕ is a hyperbolic non-automorphism, then Cϕ is hypercyclic.

Proposition 65 Let v(z) = (1 −|z|)1/2 for every z ∈D. If ϕ be a linear fractional trans- formation of D which is a parabolic non-automorphism, then Cϕ : H0 v (D) →H0 v (D) is not hypercyclic. These results were extended to the case of weighted composition operators of the form λCϕ, 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 analyt

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 Cϕ is uniformly mean ergodic on H∞ v (D) and H0 v (D). Moreover, there is k ∈N such that ((Cϕ)[n])n converges to (1/k)(Cϕ + · · · + (Cϕ)k). (ii) If ϕ is equivalent to a irrational rotation, then Cϕ is not uniformly mean ergodic on H∞ v (D) and H0 v (D). (iii) If ϕ is equivalent to a irrational rotation, the

Theorem 67 Let v be a log-convex weight on D satisfying condition (L1). Let ϕ be a self map on D with Denjoy–Wolff point 0. Then the sequence (Cϕn f )n converges to f (0) in H0 v (D) for each f ∈H0 v (D). Tien proves in [158, Theorem 4.8 (b)] that, under some mild assumption, the sequence (Cϕn)n converges to the operator C0( f ) := f (0) in L(H∞ v (D)). Moreover, he gives an example of a weight v and self map ϕ with Denjoy–Wolff point 0 such that Cϕ is power bounded, (Cϕn)n does not converge eve

Theorem 68 Let ϕ be a self map on D with Denjoy-Wolf point z(0) in the boundary of the unit disc. Let v be a log-convex weight on D satisfying condition (L1) such that −log(1 −r) = O(1/v(r)) as r →1−. Then Cϕ is not power bounded and not mean ergodic on H∞ v (D). 11.4 Weighted composition operators on spaces of entire functions Many properties of composition operators on spaces of entire functions have also been inves- tigated. For instance, in the frame of Fock spaces, in 2003, Carswell, MacClu

Lemma 69 (Boyd, Rueda) Let u and v be weights. If the entire function ϕ satisfies that the superposition operator Sϕ maps H∞ u into H∞ v and is bounded, then Sϕ : H∞ u →H∞ v is continuous.

Theorem 70 Let u and v be weights, such that u is strictly decreasing. (a) If the entire function ϕ satisfies the following condition: ∀ε ∈]0, 1[ ∃C > 0 ∃R0 > 0 ∀R ≥R0 : v  u−1  1 εR  max |w|=R|ϕ(w))| ≤C, then the superposition operator Sϕ maps H∞ u (D) into H∞ v (D) and is bounded. 123

Proposition 71 Let u(z) = (1 −|z|)α, α > 0, v(z) = (1 −|z|)β, β > 0. (1) The following conditions are equivalent for an entire function ϕ: (i) ϕ is a polynomial of degree at most the integer part [β/α] of β/α. (ii) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D). (iii) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D) and it is bounded. (2) The following conditions are equivalent for an entire function ϕ: (i) ϕ is a polynomial of degree k less than β/α. (ii) The superposition o

Proposition 72 Let u(z) = (1 −|z|)α, z ∈D and v(z) = exp(− 1 (1−|z|)β ), α, β > 0. Let ϕ be an entire function. The following conditions are equivalent: (i) The function ϕ is of order less than β/α or of order β/α and type zero. (ii) For all 0 < ε < 1 there are C ≥1, R0 > 0 such that |ϕ(z)| ≤C exp(ε|z|β/α) for all z ∈C with |z| ≥R0. (iii) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D). (iv) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D) and it is bounded.

Theorem 73 Let u(r) = (log e 1−r )−α and v(r) = (1 −r)β with α, β > 0. The following statements are equivalent for an entire function ϕ: (i) The function ϕ is of order less than 1/α or of order 1/α and type zero. (ii) For all 0 < ε < 1 there are C > 0, R0 > 0 such that |ϕ(z)| ≤C exp(ε|z|1/α) for all z ∈C with |z| ≥R0. (iii) The superposition operator Sϕ maps H∞ u into H∞ v . (iv) Sϕ is a bounded operator from H∞ u (D) into H∞ v (D). (v) Sϕ is a compact operator from H∞

Proposition 74 Let u(r) = (log e 1−r )α, α > 0, v(r) = (log e 1−r )β, β > 0. (1) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D) and is bounded if and only if ϕ is a polynomial of degree at most [β/α]. (2) The superposition operator Sϕ maps H∞ u (D) into H∞ v (D) and is compact if and only if ϕ is a polynomial of degree less than β/α.

Proposition 75 Let u(r) = exp(−(1 −|z|)−α), α > 0 and let ϕ be an entire function. (1) If there exist C > 0 and R0 > 0 such that |ϕ(w)| ≤C exp((log |w|)γ ) for |w| ≥R0, then for each c > 1 the superposition operator Sϕ maps H∞ u (D) boundedly into the space H∞ vc (D), where vc(r) = exp(− c (1−|z|)αγ ). (2) If the superposition operator Sϕ maps H∞ u (D) into H∞ v (D), v(r) = exp(− 1 (1−|z|)β ), β > 0, then for every c > 1 there exist C > 0 and R0 > 0 such that |ϕ(w)| ≤ C exp(c(log |w|)β/α) for |w

Proposition 75. 123

Theorem 76 There is a bounded harmonic function f : D →C such that T f : H∞ v (D) → H∞ v (D) is not a bounded operator for any weight v on D. As a consequence, the Bergman projection Pv is not a bounded mapping L∞ v →L∞ v for any weight under consideration. Dostanic [94] showed that for the exponential weight v(z) = exp(−1/(1 −|z|)), the orthogonal projection L2 v →A2 v is bounded in L p v if and only if p = 2. Recent research about the continuity of the Bergman projection can be seen in [62, 84

Theorem 77 Let v satisfy condition (B). If the symbol f is radial and continuously differen- tiable on [0, 1], then T f : H∞ v (D) →H∞ v (D) is bounded.

Theorem 78 Let v be a standard weight. (i) If f ∈L1 is radial and satisfies lim sup r→1 | f (r) log(1 −r)| < ∞, then T f : H∞ v (D) →H∞ v (D) is a bounded operator. (ii) If f satisfies lim supr→1| f (r) log(1 −r)| = 0, then T f is compact on H∞ v (D).

Theorem 79 Let v(r) = exp(−α/(1 −r)β) be an exponential weight. (i) Assume that f ∈L1 is radial and lim sup r→1 | f (r)|(1 −r)−1/2−β/4 < ∞. Then, T f : H∞ v (D) →H∞ v (D) is a bounded operator. 123

Theorem 80 Let 0 < α < 1. (i) [124] The exact norm ||H||A−α of H : A−α →A−α is π sin(απ). (ii) [123] If 0 < α ≤2/3, then the exact norm ||H||H∞ α of H : H∞ α →H∞ α is π sin(απ). (iii) [90] There is an exact value α0 with 2/3 < α0 < 1 such that ||H||H∞ α = π

Theorem 81 [139] (i) The Libera operator L acts as a bounded operator from A−α into A−α if and only if 0 < α < 1. (ii) If 1/2 ≤α < 1, then there is g(z) = ∞ n=0 akzk ∈A−α such that |an| ≥c(n + 1)α−1/2 for all n and some c > 0, and L acts as a bounded operator from A−α into A−α. More results about the Libera operator on Banach spaces of analytic functions can be seen in [138, 139] and the references therein. 14.3 Hausdorff operators We gave in [45] a few results about Hausdorff operators on weig

Theorem 82 Let v be a radial weight on D satisfying condition (L1). If the function w ∈ D →K(w)∥Cϕw∥belongs to L1(μ), then the operators HK,μ : H∞ v (D) →H∞ v (D), and HK,μ : H0 v (D) →H0 v (D), are continuous. In this case, we have ∥HK,μ∥≤  D |K(w)| ∥Cϕw∥dμ(w).

Corollary 83 (1) Letv(r) = (1−r2)γ withγ > 0.Ifthefunctionw ∈D →K(w)/(1−|w|)γ belongs to L1(μ), then HK,μ : H∞ v (D) →H∞ v (D) is continuous. (2) Let v(r) = (log e 1−r2 )−α, α > 0. If the function w ∈D →K(w) log(1 −|w|) belongs to L1(μ), then HK,μ : H∞ v (D) →H∞ v (D) is continuous. We now turn our attention to the case of spaces of entire functions. Let μ be a positive measure on (0, ∞). Stylogiannis and Galanopoulos [154] consider formally the Hausdorff operator induced by the measure μ defined

Proposition 84 Let v be a weight on C. (1) If the operator Hμ : H∞ v (C) →H∞ v (C) is continuous, then sup n∈N0  ∞ 0 1 tn+1 dμ(t) ≤∥Hμ∥< ∞, (9) and the operator Hμ : H0 v (C) →H0 v (C) is also continuous. (2) If the operator Hμ : H0

Theorem 85 Let α > 0 and β > 0. Let v be the weight on C defined by v(r) = exp(−βrα). The following conditions are equivalent. (i) Hμ : H∞ v (C) →H∞ v (C) is continuous. (ii) Hμ : H0 v (C) →H0 v (C) is also continuous. (iii) supn∈N  ∞ 0 1 tn+1 dμ(t) < ∞. In this case, we have ∥Hμ∥≤sup

Proposition 86 Let α > 0 and β > 0. Let v be the weight on C defined by v(r) = exp(−βrα). If the sequence   ∞ 0 1 tk dμ(t)  k∈N is in ℓ1, then the operator Hμ is compact on H∞ v (C) and on H0 v (C). Some questions seems to be open concerning the operators mentioned in this last section. Acknowledgements This research was partially supported by the project MCIN PID2020-119457GB-I00- /AEI/10.13039/501100011033. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer