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 $v : \mathbb{D} \to \mathbb{R}$ on the unit disc such that $\lim_{r\to 1^-} v(r) = 0$ : - (i) v is essential. - (ii) v is equivalent to a log-convex radial weight w on $\mathbb{D}$ . - (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 approximable from below by monomials. It follows from [36, Example 3.3] that there exist log-convex, radial, non-increasing, continuous, strictly positive weights $v: \mathbb{C} \to \mathbb{R}$ on the complex plane with $\lim_{r\to\infty} r^n v(r) = 0$ for each $n \in \mathbb{N}$ , 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 $v : \mathbb{C} \to \mathbb{R}$ on the complex plane such that $\lim_{r\to\infty} r^n v(r) = 0$ for each $n \in \mathbb{N}$ .  184 Page 6 of 40 J. Bonet - (i) v is essential. - (ii) v is 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 $\mathbb{C}$ . The following example is taken from [3, Example 1].

Proposition 4. Let $\alpha > 1$ and let $v_{\alpha}$ be a weight on the complex plane such that $v_{\alpha}(r) := \exp(-(\log r)^{\alpha})$ , $e \le r < \infty$ . The weight $v_{\alpha}$ is essential if $\alpha \ge 2$ and not essential if $1 < \alpha < 2$ .

Proposition 5. Let $v: G \to [0, \infty[$ be a weight on a planar domain G. Then the space $H_v^\infty(G)$ is normed if and only if $E_v$ is not a discrete set (that is, it has a limit point in G). Moreover, if $H_v^\infty(G)$ is normed, then the inclusion map $J: H_v^\infty(G) \to (H(G), \tau_{co})$ has closed graph.

Proposition 6. Assume that the space $H_v^{\infty}(G)$ is normed. The following conditions are equivalent: - (i) The space $H_v^{\infty}(G)$ is a Banach space. - (ii) The inclusion map $J: H_v^\infty(G) \to (H(G), \tau_{co})$ is continuous. (Equivalently, every sequence in $H_v^\infty(G)$ bounded in the norm is a normal family.) - (iii) The closed unit ball $B_v^{\infty}$ of $H_v^{\infty}(G)$ is bounded in $(H(G), \tau_{co})$ . - (iv) For each $z \in G$ , the point evaluation functional $\delta_z(f) := f(z), z \in G$ , satisfies $\delta_z \in H_v^\infty(G)'$ and, moreover, $\sup_{z \in K} \|\delta_z\|_v' < \infty$ for each compact subset K of G.

Theorem 7. Let $v: G \to [0, \infty[$ be a bounded weight on a planar domain G such that the space $H_v^\infty(G)$ is normed. The space $H_v^\infty(G)$ is complete if and only if there is a bounded, continuous, strictly positive weight $\tilde{v}$ on G such that $H_v^\infty(G) = H_{\tilde{v}}^\infty(G)$ . 

Proposition 8. Let $v: G \to [0, \infty[$ be a weight on a planar domain G. If $H_v^\infty(G)$ is a Banach space containing non-zero functions, then the boundary $\partial G$ is contained in the closure $\overline{E}_v$ of $E_v$ in $\mathbb C$ .

Corollary 9. Let $v: G \to [0, \infty[$ be a weight on a planar domain G (other than the plane itself) such that $H_v^{\infty}(G)$ is normed and it contains non-zero functions. - (1) If $E_v$ is contained in a convex closed proper subset A of G, then $H_v^{\infty}(G)$ is not a Banach space. - (2) If the closure of $E_v$ in $\mathbb{C}$ is a compact subset of G, then $H_v^{\infty}(G)$ is not a Banach space. As a consequence of Corollary 9 one easily deduces the following example: the weight $v(z) = \max\{0, \operatorname{Re} z\}$ is continuous on the unit disc $\mathbb{D}$ , vanishes in the left-hand half of the disc, it is strictly positive in the remaining open right semi-disc, and $H_v^{\infty}(\mathbb{D})$ is not a Banach space.

Proposition 10. Let $v: G \to [0, \infty[$ be a weight on a planar domain G. Suppose that for each $z \in G$ there is a bounded open set $U \subset G$ such that $z \in U$ , $\partial U \subset E_v$ , and v is bounded away from 0 on $\partial U$ . Then $H_v^{\infty}(G)$ is a Banach space

Corollary 11. Let $v: G \to [0, \infty[$ be a continuous weight on a planar domain G such that $G \setminus E_v$ is discrete, i.e., the zeros of v are isolated. Then $H_v^\infty(G)$ is a Banach space. As a consequence of Corollary 11, if $F \in H(G)$ is a non-zero analytic function on a planar domain G and $v(z) := |F(z)|, z \in G$ , then $H_v^{\infty}(G)$ is a Banach space.

Corollary 12. Let $v: G \to [0, \infty[$ be a weight on a planar domain G such that $\overline{G \setminus E_v}$ is a compact subset of G. If $\inf_{z \in K} v(z) > 0$ for each compact subset $K \subset G \setminus (\overline{G \setminus E_v})$ , then $H_v^\infty(G)$ is a Banach space.

Corollary 13. Let $G = \mathbb{D}$ (resp. $G = \mathbb{C}$ ). Let v be a bounded radial weight on G. The space $H_v^{\infty}(G)$ is a Banach space if and only if $E_v$ is not compact in G or, equivalently, if and only if there is an increasing sequence $(r_k)_k$ in ]0, 1[ tending to 1 (resp. $(r_k)_k$ in $]0, \infty[$ tending to $\infty$ ) such that $v(r_k) > 0$ for each $k \in \mathbb{N}$ . In particular, if $v(z) := |F(|z|)|, z \in G$ , for a non-zero function $F \in H(G)$ , then $H_v^{\infty}(G)$ is a Banach space. Let v be the weight on $\mathbb{D}$ defined by $v(z) := a_n > 0$ if |z| = 1 - (1/n), and v(z) = 0 otherwise. Then $H_v^{\infty}(G)$ is a Banach space by Corollary 13. Observe that the sequence $(a_n)_n \subset ]0, \infty[$ need not be bounded. Similar examples can be obtained by replacing $\mathbb{D}$ by $\mathbb{C}$ and 1 - (1/n) by $n, n \in \mathbb{N}$ .

Proposition 14. Let $F \in H(G)$ 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 $F(z) \neq 0$ . Then $H_v^{\infty}(G)$ is a Banach space that coincides with the set of all $f \in H(G)$ such that there is C = C(f) > 0 with $|f(z)| \leq C|F(z)|$ for each $z \in G$ . Example 15 (1) Let q > 0 and $v(z) = |\operatorname{Re} z|^q$ , $z \in \mathbb{D}$ . Then the normed space $H_v^{\infty}(\mathbb{D})$ is complete. - (2) Let v be a weight on $\mathbb{D}$ 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 $v(r_n e^{i\theta}) \ge a_n$ almost everywhere in $[0, 2\pi]$ . Then the normed space $H_v^{\infty}(\mathbb{D})$ is complete. - (3) Let w be a continuous weight on G such that $H_w^\infty(\mathbb{D})$ is a Banach space. Let v be a weight on G such that $\{z \in G : v(z) = w(z)\}$ is dense in G (note that this implies that $E_v$ is not discrete, so that $H_v^\infty(\mathbb{D})$ is normed). Then $H_v^\infty(\mathbb{D})$ is actually a Banach space.  184 Page 8 of 40 J. Bonet

Theorem 16. (Kislyakov) There is C > 0 such that for each $(b_j)_{j=n}^m$ , 0 < n < m, there is a polynomial $P(z) = \sum_{j=n}^m c_j z^j$ such that $|b_j| \le |c_j|$ , j = n, ..., m and $||P||_{\infty} \le C(\sum_{j=n}^m |b_j|^2)^{1/2}$ . The solid hull $S(H^p)$ for $1 \le p < 2$ seems to be unknown. The solid core of the Hardy spaces $s(H^p) = H^2$ , $1 \le p \le 2$ , is known. Moreover $s(H^\infty) = \ell_1$ . In particular, the space $H^\infty$ is not solid. The disc algebra $A(\mathbb{D})$ is also not solid. It is an open problem to describe the solid core $s(H^p)$ for 2 . Bennet, Stegenga and Timoney in their paper [34] determined the solid hull and the solid core of the weighted spaces $H_v^\infty(\mathbb{D})$ when the weight v is doubling. Recall that the weight v on $\mathbb{D}$ is doubling if there is M>0 such that $v(1-r)\leq Mv(1-(r/2))$ for each 0< r<1. Exponential weights $v(r)=\exp(-a/(1-r)^b)$ with a,b>0 are not doubling. 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 $v_{\alpha}(z)=(1-|z|^2)^{\alpha}$ , where $\alpha\geq 0$ , we recall that $A^{-\alpha}=H^{\infty}_{n_{\alpha}}(\mathbb{D})$ . The solid hull of $A^{-\alpha}$ is known: $$S(A^{-\alpha}) = \left\{ (b_m)_{m=0}^{\infty} : \sup_{n \in \mathbb{N}_0} \left( \sum_{m=2^n}^{2^{n+1}-1} |b_m|^2 (m+1)^{-2\alpha} \right)^{1/2} < \infty \right\}.$$ This is Theorem 8.2.1 of [117]. Moreover, the solid core $s(A^{-\alpha})$ can also be characterized, see Theorem 8.3.4 of [117]: $$s(A^{-\alpha}) = \left\{ (b_m)_{m=0}^{\infty} : \sup_{n \in \mathbb{N}_0} \left( \sum_{m=2^n}^{2^{n+1}-1} |b_m|(m+1)^{-\alpha} \right) < \infty \right\}.$$  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 $H_n^{\infty}$ , 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 $(e_n)_{n=1}^{\infty}$ of elements of a separable Banach space X is a Schauder basis, if every element $f \in X$ can be presented as a convergent sum $$f = \sum_{n=1}^{\infty} f_n e_n,$$ where the numbers $f_n \in \mathbb{K}$ are unique for $f \in X$ . We denote by $\Lambda = \{z^k : k = 0, 1, 2, ...\}$ the sequence of monomials.

Theorem 17. If $S(H_v^{\infty}) = H_v^{\infty}$ , then $\Lambda$ is a Schauder basis of $H_v^0$ . By a theorem of Lusky [126], $\Lambda$ is never a basis for $H_n^0(\mathbb{D})$ . This implies the following consequence.

Corollary 18. In the case of analytic functions on the disc $\mathbb{D}$ , one always has $S(H_{\nu}^{\infty}(\mathbb{D})) \neq$ $H_v^{\infty}(\mathbb{D})$ and $s(H_v^{\infty}(\mathbb{D})) \neq H_v^{\infty}(\mathbb{D})$ . Lusky [126] proved that the monomials $\Lambda = \{z^k : k = 0, 1, 2, ...\}$ are a basis of $H_n^0(\mathbb{C})$ 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 $H_v^{\infty}$ is solid if and only if $\Lambda$ is a Schauder basis of $H_v^0$ . Now we present the solid hull and solid core of weighted Banach spaces $H_n^{\infty}$ for concrete weights on the disc and on the complex plane.

Theorem 20. For $v(r) = \exp(-1/(1-r))$ the solid hull of $H_n^{\infty}(\mathbb{D})$ is $$\left\{ (b_m)_{m=0}^{\infty} : \sup_{n} \exp(-2n^2) \sum_{m=n^4+1}^{(n+1)^4} |b_m|^2 \left(1 - \frac{1}{n^2}\right)^{2m} < \infty \right\},\,$$ and the solid core is $$\left\{ (b_m)_{m=0}^{\infty} : \sup_{n} \exp(-n^2) \sum_{m=n^4+1}^{(n+1)^4} |b_m| \left(1 - \frac{1}{n^2}\right)^m < \infty \right\},\,$$

Theorem 21. Let v be the weight $v(r) = \exp(-ar^p)$ on $\mathbb{C}$ , where a > 0 and p > 0 are constants. Then, the solid hull of $H_n^{\infty}(\mathbb{C})$ is $$\left\{ (b_m)_{m=0}^{\infty} : \sup_{n \in \mathbb{N}} \sum_{pn^2+1 < m < p(n+1)^2} |b_m|^2 e^{-2n^2} n^{4m/p} (ap)^{-m/p} < \infty \right\}.$$ and the solid core is $$\left\{ (b_m)_{m=0}^{\infty} : \sup_{n \in \mathbb{N}} \sum_{pn^2+1 < m < p(n+1)^2} |b_m| e^{-n^2} n^{2m/p} (ap)^{-m/2p} < \infty \right\}.$$  184 Page 10 of 40 J. Bonet These results are a consequence of a general theorem. To state it, we need a few definitions. Let $r_m \in ]0$ , 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 (b) if there exist numbers b > 2, K > b and $0 < m_1 < m_2 < \ldots$ with $\lim_{n \to \infty} m_n = \infty$ such that $$b \leq \left(\frac{r_{m_n}}{r_{m_{n+1}}}\right)^{m_n} \frac{v(r_{m_n})}{v(r_{m_{n+1}})}, \ \left(\frac{r_{m_{n+1}}}{r_{m_n}}\right)^{m_{n+1}} \frac{v(r_{m_{n+1}})}{v(r_{m_n})} \leq K.$$

Theorem 22. If the weight v satisfies condition (b), we have $$S(H_v^{\infty}) = \left\{ (b_m)_{m=0}^{\infty} : \sup_n v(r_{m_n}) \left( \sum_{m_n < m \le m_{n+1}} |b_m|^2 r_{m_n}^{2m} \right)^{1/2} < \infty \right\},\,$$ and $$s(H_v^{\infty}) = \left\{ (b_m)_{m=0}^{\infty} : \sup_n v(r_{m_n}) \left( \sum_{m_n < m \le m_{n+1}} |b_m| r_{m_n}^m \right) < \infty \right\}.$$ 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 $h(z) = \sum_{k=0}^{\infty} b_k z^k$ , and put $M_{\infty}(h, r) = \sup_{|z|=r} |h(z)|$ . For the numbers $m_n$ as in condition (b) set $$(R_n h)(z) = \sum_{k=0}^{m_{n-1}} b_k z^k + \sum_{m_{n-1} < k \le m_n} \frac{[m_n] - k}{[m_n] - [m_{n-1}]} b_k z^k.$$ Define the operators $V_n = R_n - R_{n-1}$ for $n \in \mathbb{N}$ . Theorem 23 (Lusky) The norms $$\|h\|_v$$ and $\sup_{n} \sup_{r_{m_n-1} \le r \le r_{m_n+1}} M_{\infty}(V_n h, r) v(r)$ are equivalent. Moreover, the operators $V_n$ are uniformly bounded on $H_v^{\infty}$ and there are $c_1 > 0$ and $c_2 > 0$ with $$c_1 \sup_n \|V_n h\|_v \le \|h\|_v \le c_2 \sup_n \|V_n h\|_v \quad \text{for all } h \in H_v^{\infty}.$$ 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. - (1) For the complex plane. Let $v(r) = \exp(-ar^p)$ , $r \in [0, \infty[, a > 0, p > 0 \text{ and let } b > 2$ . The weight v satisfies condition (b) for the sequence $m_n := p(\log b)n^2$ with $K = b^5$ . - (2) For the unit disc. Let $v(z) = \exp\left(-\frac{a}{(1-r)^b}\right)$ . The weight v satisfies condition (b) for the sequence $m_n = b^{1+1/b}a^{-1/b}n^{2+2/b} bn^2$ . In particular, for $v(z) = \exp\left(-\frac{1}{(1-r)}\right)$ , 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 $H^{\infty}$ 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.

Theorem 24. For every $f \in H_v^{\infty}$ there is $g \in H_v^0$ with $$d(f, H_v^0) = \|f - g\|_v = \limsup_{r \to R} M(f, r)v(r).$$ 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 $\mathbb D$ or $\mathbb C$ , we denote by $\sigma_n f$ 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 $M(\sigma_n f, r) \leq M(f, r)$ for each 0 < r < R.

Lemma 25. Let $f \in H_v^{\infty}$ and assume that there is $0 < \tau < 1$ with $$\tau \|f\|_{v} \le \limsup_{r \to R} M(f, r)v(r).$$ Then, for each $\varepsilon > 0$ and $m \in \mathbb{N}$ there is $n \in \mathbb{N}$ , n > m, such that with $\rho = (1 - \tau)/(1 + \tau)$ we have $$\left(\frac{1+\tau}{2(1+\varepsilon)}\right)\|f-\rho\sigma_n f\|_v \leq \limsup_{r\to R} M(f,r)v(r) = \limsup_{r\to R} M(f-\rho\sigma_n f,r)v(r).$$ The following simple examples show that the distance $d(f, H_v^0)$ can be attained at many points of $H_v^0$ for a given function $f \in H_v^\infty$ . (1) Consider the weight $v(r) = e^{-r}$ , $r \in [0, \infty[$ , on the complex plane and the analytic function $f(z) = e^z$ , $z \in \mathbb{C}$ . Clearly $f \in H_v^\infty$ and $||f||_v = 1$ . Set $P_n(z) = \sum_{k=0}^n \frac{z^k}{k!}$ for each $n \in \mathbb{N}$ . We have, for each n, $P_n \in H_v^0$ and $$||f - P_n||_v = \sup_{r>0} e^{-r} \sum_{k=n+1}^{\infty} \frac{r^k}{k!} = 1 = d(f, H_v^0).$$  184 Page 12 of 40 J. Bonet (2) Now define the weight v(r) = 1 - r, $r \in [0, 1[$ , on the unit disc. The function $f(z) = \frac{1}{1-z} = \sum_{k=0}^{\infty} z^k$ belongs to $H_v^{\infty}$ and $||f||_v = 1$ . Set $P_n(z) = \sum_{k=0}^n z^k$ for each $n \in \mathbb{N}$ . We have, for each n, $P_n \in H_v^0$ and $$M(f - P_n, r) = \sum_{k=n+1}^{\infty} r^k = \frac{r^{n+1}}{1 - r}.$$ Therefore $$||f - P_n||_v = \sup_{r \in [0,1[} (1-r)M(f - P_n, r) = 1 = d(f, H_v^0).$$

Theorem 26. (Abanin and Tien [6, 8]) Let v be a radial, log-convex weight on $\mathbb{C}$ . - (a) The following conditions are equivalent. - (i) The operator $D: H_v^{\infty}(\mathbb{C}) \to H_v^{\infty}(\mathbb{C})$ is continuous. - (ii) $-\log v(r) = O(r)$ as $r \to \infty$ . - (iii) $\limsup_{r\to\infty} v(r)(1/v)'(r) < \infty$ . - (b) The following conditions are equivalent. - (i) The operator $D: H_v^{\infty}(\mathbb{C}) \to H_v^{\infty}(\mathbb{C})$ is compact. - (ii) $-\log v(r) = o(r) \text{ as } r \to \infty.$ - (iii) $\lim_{r \to \infty} v(r)(1/v)'(r) = 0$ .  184 Page 14 of 40 J. Bonet The differentiation operator D is continuous and surjective on $H_v^{\infty}(\mathbb{C})$ for v(r) = $e^{-\alpha r}$ , $\alpha > 0$ , 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 $\mathbb{D}$ . - (i) The operator $D: H_v^{\infty}(\mathbb{D}) \to H_w^{\infty}(\mathbb{D})$ , with w(r) = (1-r)v(r) is continuous. - (ii) $v(r)(1-r)^{-\alpha}$ is increasing on $[r_0, 1)$ for some $\alpha > 0$ and some $r_0 > 0$ . - (iii) $\limsup_{r \to 1} \left( -\frac{(1-r)v'(r)}{v(r)} \right) < \infty.$ (iv) $\sup_{n} \frac{v(1-2^{-n})}{v(1-2^{-n+1})} < \infty.$ If these equivalent conditions hold, then the continuous operator $D: H_n^{\infty}(\mathbb{D}) \to H_n^{\infty}(\mathbb{D})$ is not compact. Theorem 27 can be considered as an extension of a classical result of Hardy and Littlewood [97, Theorem 5.5]: $f \in H(\mathbb{D})$ satisfies $\sup_{z \in \mathbb{D}} (1-|z|)^{\gamma} |f(z)| < \infty$ if and only if $\sup_{z \in \mathbb{D}} (1-|z|)^{\gamma} |f(z)| < \infty$ $|z|)^{\gamma+1}|f'(z)|<\infty.$ The integration operator $Jf(z) = \int_0^z f(\zeta)d\zeta$ , $z \in G$ , is also well-defined and continuous on the Fréchet space H(G) for $G = \mathbb{D}$ and $G = \mathbb{C}$ . The continuity of J on spaces of type $H_{,,}^{\infty}$ was also investigated in [6, 110]. We recall some results.

Theorem 28. Let v be a radial, log-convex weight on $\mathbb{C}$ . - (a) The following conditions are equivalent. - (i) The operator $J: H_v^\infty(\mathbb{C}) \to H_v^\infty(\mathbb{C})$ is continuous. (ii) $\limsup_{r \to \infty} v(r) \int_0^r \frac{1}{v(t)} dt < \infty$ . - (iii) $\lim \inf_{r \to \infty} v(r) (1/v)'(r) > 0$ . - (b) The following conditions are equivalent. - (i) The operator $J: H_v^\infty(\mathbb{C}) \to H_v^\infty(\mathbb{C})$ is compact. (ii) $\lim_{r \to \infty} v(r) \int_0^r \frac{1}{v(t)} dt = 0$ . - (iii) $\lim_{r\to\infty} v(r)(1/v)'(r) = \infty$ . As a consequence, the integration operator J is continuous on $H_v^{\infty}(\mathbb{C})$ for v(r) = $e^{-\alpha r}, \alpha > 0.$

Corollary 29. Let v be a radial, log-convex weight on $\mathbb{C}$ . The following conditions are equivalent. - (i) The operator $D: H_v^{\infty}(\mathbb{C}) \to H_v^{\infty}(\mathbb{C})$ is continuous and surjective. - (ii) $0 < \liminf_{r \to \infty} v(r)(1/v)'(r) \le \limsup_{r \to \infty} v(r)(1/v)'(r) < \infty$ . - (iii) There are $A, C \ge 1$ such that, for all $0 \le r < \infty$ , $$\frac{1}{A}e^{-Cr} \le v(r) \le Ae^{-r/C}.$$

Theorem 30. Let v be a radial, log-convex weight on $\mathbb{D}$ and w(r) := v(r)/(1-r), $0 \le r < 1$ . The following conditions are equivalent. (i) The integration operator $J: H_w^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is continuous.  - (ii) $v(r)(1-r)^{\alpha}$ is increasing on $[r_0, 1)$ for some $\alpha > 0$ and some $r_0 > 0$ . - (iii) $\lim \inf_{r \to 1} \left( -\frac{(1-r)v'(r)}{v(r)} \right) > 0.$ - (iv) There is $k \in \mathbb{N}$ such that $\limsup_{n \to \infty} \frac{v(1-2^{-n-k})}{v(1-2^{-n})} < 1$ . (v) There is $\gamma > 1$ such that $\limsup_{r \to 1} \frac{v(r^{\gamma})}{v(r)} < 1$ .

Corollary 31. Let v be a radial, log-convex weight on $\mathbb{C}$ and w(r) := v(r)/(1-r), $0 \le r < 1$ . The differentiation operator $D: H_v^\infty(\mathbb{D}) \to H_v^\infty(\mathbb{D})$ is continuous and surjective if and only if $$0< \lim\inf_{r\to 1}\left(-\frac{(1-r)v'(r)}{v(r)}\right) \leq \lim\sup_{r\to 1}\left(-\frac{(1-r)v'(r)}{v(r)}\right) < \infty.$$ Remark 32 The integration and differentiation operators are continuous on $H_n^{\infty}$ if and only if they are continuous on $H_v^0$ . See e.g. [30, Lemma 2.1]. 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].

Theorem 33. Assume that the differentiation operator $D: H_v^0(\mathbb{C}) \to H_v^0(\mathbb{C})$ 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) $\lim_{r\to\infty} v(r)e^r = 0$ .

Theorem 34. Assume that the differentiation operator $D: H_v^0(\mathbb{C}) \to H_v^0(\mathbb{C})$ is continuous. The following conditions are equivalent: - (1) D is hypercyclic on $H_v^0(\mathbb{C})$ . (2) $\liminf_{n\to\infty} \frac{\|z^n\|_v}{n!} = 0$ .

Proposition 35. The operator J is never hypercyclic on $H_n^0(\mathbb{C})$ and it has no periodic points different from 0 in $H_n^{\infty}(\mathbb{C})$ .

Corollary 36. Let v be a weight such that the differentiation operator $D: H^0_n(\mathbb{C}) \to H^0_n(\mathbb{C})$ is continuous. Then (1) If there is A > 0 such that $1/v(r) \le Ar^{-1/2}e^r$ , r > 0, then D is not hypercyclic on $H_n^0(\mathbb{C})$ . In particular D is not hypercyclic for $v(r) = \exp(-\log^2 r)$ or for v(r) = $\exp(-\alpha r)$ , $0 < \alpha < 1$ .  184 Page 16 of 40 J. Bonet (2) If there are B > 0, α ≥ 1, r₀ > 0 such that v(r) ≤ B exp(-αr) for each r ≥ r₀, then D is hypercyclic on H_v⁰(ℂ). In particular, D is hypercyclic for v(r) = exp(-αr), α ≥ 1. (3) The differentiation operator on H_α⁰(ℂ) is not hypercyclic and has no periodic point (3) The differentiation operator on $H^0_{\alpha}(\mathbb{C})$ is not hypercyclic and has no periodic point different from 0 if $\alpha < 1$ , it is hypercyclic and has a dense set of periodic points if $\alpha > 1$ and it is hypercyclic but has no periodic point different from 0 if $\alpha = 1$ .

Theorem 37. (1) The differentiation operator D satisfies (i) $$||D^n||_{\alpha} = n! \left(\frac{e\alpha}{n}\right)^n, \quad n \in \mathbb{N}.$$ - (ii) It is power bounded if and only if $\alpha < 1$ . - (iii) The spectrum of D is the closed disc of radius $\alpha$ . - (iv) It is uniformly mean ergodic on $H^{\infty}_{\alpha}(\mathbb{C})$ and $H^{0}_{\alpha}(\mathbb{C})$ if $\alpha < 1$ , not mean ergodic if $\alpha > 1$ , and it is not mean ergodic on $H^{\infty}_{1}(\mathbb{C})$ and not uniformly mean ergodic on $H^{0}_{1}(\mathbb{C})$ . - (2) The integration operator J satisfies - (i) $||J^n||_{\alpha} = 1/\alpha^n$ , $n \in \mathbb{N}$ . - (ii) It is power bounded if and only if $\alpha \geq 1$ . - (iii) It is never hypercyclic on $H^0_{\alpha}(\mathbb{C})$ . - (iv) The spectrum of J is the closed disc of radius $1/\alpha$ . - (v) If $\alpha > 1$ , then J is uniformly mean ergodic on $H_{\alpha}^{\infty}(\mathbb{C})$ and $H_{\alpha}^{0}(\mathbb{C})$ , and it is not mean ergodic on these spaces if $\alpha < 1$ . - (vi) If $\alpha = 1$ , then J is not mean ergodic on $H_1^{\infty}(\mathbb{C})$ , and mean ergodic but not uniformly mean ergodic on $H_1^0(\mathbb{C})$ . 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.

Lemma 38. Let E be a Banach space of analytic functions on the open domain $G = \mathbb{D}$ or $G = \mathbb{C}$ , such that the inclusion map $E \subset H(G)$ is continuous and the polynomials are contained and dense in E. For each $N \in \mathbb{N}$ we have $$A_N(E) = \overline{\operatorname{span}(\{z^j \; ; \; j \ge N\})}.$$

Theorem 39. (Abanin, Tien [7]) Let v be a radial, continuous, non-increasing, log-convex weight $v:[0,R[\to(0,\infty[$ , which satisfies $\lim_{r\to R} r^n v(r)=0$ for each $n\in\mathbb{N}$ , with R=1 for the disc and $R=+\infty$ for the complex plane. Assume that the integration operator $J:H_v^0\to H_v^0$ is continuous. Moreover, in the case of the complex plane assume that $$\lim\inf_{r\to\infty}\frac{rv(r)(1/v)'(r)}{-\log v(r)}>1.$$ Then every proper closed invariant subspace for J on $H_v^0$ is of the form $$A_K(H_v^0) = \{ f \in H_v^0 ; f^{(j)}(0) = 0, 0 \le j < K \}$$ for some $K \in \mathbb{N}$ . The weight $v(z) := \exp(-|z|^{\alpha})$ , $\alpha > 1$ , 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.

Theorem 40. Let $\gamma > 0$ . The Cesàro operator $C_{\gamma,0}: A_0^{-\gamma} \to A_0^{-\gamma}$ is continuous and it has the following properties. - (i) $\sigma_{pt}(C_{\gamma,0}) = \{\frac{1}{m} : m \in \mathbb{N}, m < \gamma\}.$ - (ii) $\sigma(C_{\gamma,0}) = \sigma_{pt}(C_{\gamma,0}) \cup \{\lambda \in \mathbb{C} : |\lambda \frac{1}{2\gamma}| \le \frac{1}{2\gamma}\}.$ - (iii) If $|\lambda \frac{1}{2\gamma}| < \frac{1}{2\gamma}$ (equivalently Re $(\frac{1}{\lambda}) > \gamma$ ), then $\text{Im}(\lambda I C_{\gamma,0})$ is a closed subspace of $A_0^{-\gamma}$ and has codimension 1. Moreover, the Cesàro operator $C_{\gamma}: A^{-\gamma} \to A^{-\gamma}$ is continuous and it satisfies - (iv) $\sigma_{pt}(C_{\gamma}) = \{\frac{1}{m} : m \in \mathbb{N}, m \leq \gamma\}, and$ - (v) $\sigma(C_{\nu}) = \sigma(C_{\nu,0})$ . Now we present several results obtained in collaboration with Albanese and Ricker [9].

Theorem 41. (i) Let $\gamma \geq 1$ . Then $\|C_{\gamma}^n\| = \|C_{\gamma,0}^n\| = 1$ for all $n \in \mathbb{N}$ . (ii) Let $0 < \gamma < 1$ . Then $\|C_{\gamma}^n\| = \|C_{\gamma,0}^n\| = 1/\gamma^n$ for all $n \in \mathbb{N}$ .

Theorem 42. (i) Let $0 < \gamma < 1$ . Both of the operators $C_{\gamma}$ and $C_{\gamma,0}$ fail to be power bounded and are not mean ergodic. Moreover, $$Ker(I - C_{\gamma}) = Ker(I - C_{\gamma,0}) = \{0\},\$$ and $\text{Im}(I-C_{\gamma})$ (resp. $\text{Im}(I-C_{\gamma,0})$ ) is a proper closed subspace of $A^{-\gamma}$ (resp. of $A_0^{-\gamma}$ ). (ii) Both of the operators $C_1$ and $C_{1,0}$ are power bounded but not mean ergodic. Moreover, $\operatorname{Im}(I-C_1)$ (resp. $\operatorname{Im}(I-C_{1,0})$ ) is not a closed subspace of $A^{-\gamma}$ (resp. of $A_0^{-\gamma}$ ). (iii) Let $\gamma > 1$ . Both of the operators $C_{\gamma}$ and $C_{\gamma,0}$ are power bounded and uniformly mean ergodic. Moreover, $\text{Im}(I - C_{\gamma})$ (resp. $\text{Im}(I - C_{\gamma,0})$ ) is a proper closed subspace of $A^{-\gamma}$ (resp. of $A_0^{-\gamma}$ ). In addition, $$Im(I - C_{\gamma}) = \{ h \in A^{-\gamma} : h(0) = 0 \}.$$ (5) Moreover, with $\varphi(z) := 1/(1-z)$ , for $z \in \mathbb{D}$ , the linear projection operator $P_{\nu} : A^{-\gamma} \to \mathbb{D}$ $A^{-\gamma}$ given by $$P_{\gamma}(f) := f(0)\varphi, \quad f \in A^{-\gamma},$$ is continuous and satisfies $\lim_{n\to\infty} (C_{\gamma})_{[n]} = P_{\gamma}$ in the operator norm.

Proposition 43. The Cesàro operator $C_{\gamma,0}$ is not supercyclic and hence, also not hypercyclic, in each space $A_0^{-\gamma}$ , for $\gamma > 0$ . This follows from the fact that C is not supercyclic on $H(\mathbb{D})$ , since $A_0^{-\gamma}$ is dense in $H(\mathbb{D})$ . A vector space $X \subseteq H(\mathbb{D})$ is called a Banach space of analytic functions on $\mathbb{D}$ if it contains the polynomials and it is a Banach space relative to a norm for which the natural inclusion of X into $H(\mathbb{D})$ is continuous. Since evaluation at points of $\mathbb{D}$ are continuous linear functionals on $H(\mathbb{D})$ , this is equivalent to each evaluation functional $\delta_z: f \mapsto f(z)$ at a point $z \in \mathbb{D}$ being an element of the dual Banach space X' of X.  The optimal domain of the Cesàro operator C when it acts on a Banach space of analytic functions X on $\mathbb D$ is defined by $$[\mathsf{C}, X] := \{ f \in H(\mathbb{D}) : \mathsf{C}(f) \in X \},\$$ which is a Banach space for the norm $$||f||_{\mathsf{IC}} x_1 := ||\mathsf{C}(f)||_X, \quad f \in [\mathsf{C}, X].$$ (6) If C acts on X, then $X \subseteq [C, X]$ and the natural inclusion map is continuous. Moreover, [C, X] is the largest of all Banach spaces of analytic functions Y on $\mathbb D$ that C maps continuously into X.

Theorem 44. Let $\gamma > 0$ and $\varphi(z) := 1/(1-z)$ for $z \in \mathbb{D}$ . The optimal domain $[C, A^{-\gamma}]$ of $C_{\gamma}: A^{-\gamma} \to A^{-\gamma}$ is isometrically isomorphic to $A^{-\gamma}$ and is given by $$[C, A^{-\gamma}] = \{ f \in H(\mathbb{D}) : f\varphi \in A^{-(\gamma+1)} \}.$$ (7) Moreover, the norm $\|\cdot\|_{[C,A^{-\gamma}]}$ is equivalent to the norm $f \to \|f\varphi\|_{-(\gamma+1)}$ and the containment $A^{-\gamma} \subseteq [C,A^{-\gamma}]$ is proper. A similar result holds for the optimal domain $[C, A_0^{-\gamma}]$ of $C_{\gamma,0}: A_0^{-\gamma} \to A_0^{-\gamma}$ . The behaviour of the Cesàro operator on the Korenblum space $A^{-\infty}$ 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.

Proposition 45. Let $g \in H(\mathbb{D})$ be an analytic function and let $\alpha > 0$ and $\beta > 0$ . (i) The operator $V_g:A^{-\alpha}\to A^{-\beta}$ is continuous if and only if $V_g:A_0^{-\alpha}\to A_0^{-\beta}$ is continuous and if and only if $$\sup_{z\in\mathbb{D}}(1-|z|)^{\beta-\alpha+1}|g'(z)|<\infty.$$ - (ii) If $\alpha < \beta$ , then $V_g: A^{-\alpha} \to A^{-\beta}$ is continuous if and only if $V_g: A_0^{-\alpha} \to A_0^{-\beta}$ is continuous and if and only if $g \in A^{-(\beta-\alpha)}$ . - (iii) The operator $V_g: A^{-\alpha} \to A^{-\beta}$ is compact if and only if it is weakly compact if and only if $V_g: A_0^{-\alpha} \to A_0^{-\beta}$ is compact and if and only if it is weakly compact. Moreover, these conditions are equivalent to $$\lim_{|z| \to 1} (1 - |z|)^{\beta - \alpha + 1} |g'(z)| = 0.$$

Theorem 46. (Abakumov and Doubtsov [5]) Let $g \in H(\mathbb{D})$ be a univalent function and let v and w be weight functions. (i) The operator $V_g: H_w^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is bounded if and only if $$\sup_{0 \le \theta \le 2\pi} \sup_{0 \le r < 1} v(r) \int_0^r \frac{|g'(te^{i\theta})|}{\tilde{w}(t)} dt < \infty.$$ (i) The operator $V_g: H_w^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is compact $$\lim_{\rho \to 1} \sup_{r > \rho} \sup_{0 < \theta < 2\pi} v(r) \int_0^r \frac{|g'(te^{i\theta})|}{\tilde{w}(t)} dt = 0.$$ The proof of Theorem 46 uses methods of [152] and Theorem 2.

Lemma 47. Let $X \subset H(\mathbb{D})$ be a Banach space that contains the constants and such that the inclusion $X \subset H(\mathbb{D})$ is continuous. Assume that $V_g : X \to X$ is continuous for some non-constant entire function g such that g(0) = 0. Then $$\{0\} \cup \overline{\{\lambda \in \mathbb{C} \setminus \{0\} \; ; \; e^{\frac{g}{\lambda}} \notin X\}} \subset \sigma(V_g, X).$$  The corresponding result is valid for Banach spaces $X \subset H(\mathbb{C})$ containing the constants and such that the inclusion $X \subset H(\mathbb{C})$ is continuous.

Theorem 48. (Malman [128]) Let $g \in H(\mathbb{D})$ , g(0) = 0, and let $\alpha > 0$ . - (i) If $g \in H^{\infty}$ or $g \in \mathcal{B}_0$ , then $\sigma(V_g, A^{-\alpha}) = \sigma(V_g, A_0^{-\alpha}) = \{0\}.$ - (ii) If $g(z) = c \log(1/(1 \overline{w}z))$ , $z \in \mathbb{D}$ with $c, w \in \mathbb{C}$ , $c \neq 0$ , |w| = 1, then $V_g : A^{-\alpha} \to A^{-\alpha}$ , $\alpha > 0$ , is continuous and $\sigma(V_g, A^{-\alpha}) = \{\lambda \in \mathbb{C} : \operatorname{Re}(\frac{c}{\lambda}) \geq \alpha\}$ . In the notation of Theorem 48, we understand $0 \in \{\lambda \in \mathbb{C} ; \operatorname{Re}(\frac{c}{\lambda}) \geq \alpha\}$ . With this in mind, this set coincides with the disc $\{\lambda \in \mathbb{C} ; |\lambda - \frac{c}{2\alpha}| \leq \frac{|c|}{2\alpha}\}$ .

Proposition 49. [65] Assume that $v(r) = \exp(-\alpha r^p)$ , $\alpha > 0$ , p > 0. - (i) $V_g: H_v^{\infty}(\mathbb{C}) \to H_v^{\infty}(\mathbb{C})$ is continuous if and only if g is a polynomial of degree less than or equal to the integer part of p. - (ii) $V_g: H_v^{\infty}(\mathbb{C}) \to H_v^{\infty}(\mathbb{C})$ is compact if and only if g is a polynomial of degree strictly less than p.

Theorem 50. [43] Assume that $v(r) = \exp(-\alpha r^p)$ , $\alpha > 0$ , p > 0. Let g be a polynomial of degree $n \in \mathbb{N}$ 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 $\sigma(V_g, H_v^{\infty}(\mathbb{C})) = \{0\}$ . - (ii) If p = n and $g(z) = \beta z^n + k(z)$ , k a polynomial of degree strictly less than n, then $\sigma(V_g, H_v^{\infty}(\mathbb{C})) = \{\lambda \in \mathbb{C} ; |\lambda| \leq |\beta|/\alpha\}.$ Moreover, we have $\sigma(V_g, H_v^{\infty}(\mathbb{C})) = \{0\} \cup \overline{\{\lambda \in \mathbb{C} \setminus \{0\} ; e^{\frac{g}{\lambda}} \notin H_v^{\infty}(\mathbb{C})\}}.$

Theorem 51. Let v and w be weights on $\mathbb{D}$ . - (a) The following conditions are equivalent. - $(1) \ W_{\varphi,\psi}(H_v^{\infty}(\mathbb{D})) \subset H_w^{\infty}(\mathbb{D}).$ - $(2) \ W_{\varphi,\psi}: H_{v}^{\infty}(\mathbb{D}) \to H_{w}^{\infty}(\mathbb{D}).$ $(3) \ \sup_{z \in \mathbb{D}} \frac{|\psi(z)|w(z)}{\tilde{v}(\varphi(z))} < \infty.$ $(4) \ \sup_{n \in \mathbb{N}_{0}} \frac{\|\psi\varphi^{n}\|_{v}}{\|z^{n}\|_{v}} < \infty.$ In this case, $\psi \in H_w^{\infty}$ and $\|W_{\varphi,\psi}\| = \sup_{z \in \mathbb{D}} \frac{|\psi(z)| w(z)}{\tilde{v}(\varphi(z))}$ . Moreover, this norm is comparable with the convergence of $\tilde{v}(z)$ . rable with the supremum which appears in (4). - (b) The following conditions are equivalent. - $\begin{array}{ll} (1) \ \ W_{\varphi,\psi}(H^0_v(\mathbb{D})) \subset H^0_w(\mathbb{D}). \\ (2) \ \ W_{\varphi,\psi}: H^0_v(\mathbb{D}) \to H^0_w(\mathbb{D}) \ is \ continuous. \\ (3) \ \ \psi \in H^0_w \ and \ \sup_{z \in \mathbb{D}} \frac{|\psi(z)|w(z)}{\tilde{\nu}(\varphi(z))} < \infty. \end{array}$ In this case, $\|W_{\varphi,\psi}\| = \sup_{z \in \mathbb{D}} \frac{|\psi(z)|w(z)}{\tilde{\eta}(\varphi(z))}$ .

Theorem 52. Let v and w be weights on $\mathbb{D}$ . If the operator $W_{\varphi,\psi}: H_n^{\infty}(\mathbb{D}) \to H_m^{\infty}(\mathbb{D})$ is continuous, then its essential norm satisfies $$\|W_{\varphi,\psi}\|_e = \lim_{s \to 1} \sup_{|\varphi(z)| > s} \frac{|\psi(z)| w(z)}{\tilde{v}(\varphi(z))} = \lim \sup_{n \to \infty} \frac{\|\psi\varphi^n\|_v}{\|z^n\|_v}.$$ In particular, $W_{\varphi,\psi}$ is compact if and only if $$\lim_{s \to 1} \sup_{|\varphi(z)| > s} \frac{|\psi(z)| w(z)}{\tilde{v}(\varphi(z))} = \lim_{n \to \infty} \frac{\|\psi\varphi^n\|_v}{\|z^n\|_v} = 0.$$

Theorem 53. Let v and w be weights on $\mathbb{D}$ . If the operator $W_{\varphi,\psi}: H_v^0 \to H_w^0$ is continuous, then its essential norm satisfies $$\|W_{\varphi,\psi}\|_e = \lim \sup_{|z| \to 1} \frac{|\psi(z)|w(z)}{\tilde{v}(\varphi(z))}.$$ In particular, $W_{\varphi,\psi}$ is compact if and only if $$\lim \sup_{|z| \to 1} \frac{|\psi(z)|w(z)}{\tilde{v}(\varphi(z))} = 0.$$

Theorem 54. Let v and w be weights on $\mathbb{D}$ . - (1) Assume that the operator $W_{\varphi,\psi}: H_{\nu}^{\infty}(\mathbb{D}) \to H_{\nu}^{\infty}(\mathbb{D})$ is continuous. Then $W_{\varphi,\psi}$ is either compact or an isomorphism on a subspace isomorphic to $\ell_{\infty}$ . - (2) Assume that the operator $W_{\varphi,\psi}: H_v^0(\mathbb{D}) \to H_w^0(\mathbb{D})$ is continuous. Then $W_{\varphi,\psi}$ is either compact or an isomorphism on a subspace isomorphic to $c_0$ . In particular, $W_{\varphi,\psi}$ is compact if and only if it is weakly compact in both cases.

Proposition 55. Let $\varphi$ be given. The following holds:  - (1) $C_{\varphi}: H_{v}^{\infty}(\mathbb{D}) \to H_{v}^{\infty}(\mathbb{D})$ is continuous for all radial non-increasing weights v if and only if there is 0 < s < 1 such that $|\varphi(z)| \le |z|$ for all $|z| \ge s$ . - (2) $C_{\varphi}: H_{\eta}^{\infty}(\mathbb{D}) \to H_{\eta}^{\infty}(\mathbb{D})$ is compact for all radial non-increasing weights v if and only if $\varphi(\mathbb{D}) \subset s\mathbb{D}$ for some 0 < s < 1.

Proposition 56. A radial non-increasing weight v satisfies that the operator $C_{\omega}: H_{\nu}^{\infty}(\mathbb{D}) \to$ $H_n^{\infty}(\mathbb{D})$ is continuous for every $\varphi$ if and only if the weight $\tilde{v}$ satisfies the condition (L1) $$\sup_{n} \frac{\tilde{v}(1-2^{-n})}{\tilde{v}(1-2^{-n-1})} < \infty.$$ If $\varphi(z) = (z+1)/2$ , $z \in \mathbb{D}$ , and $v(z) = \exp(-1/(1-|z|))$ , $z \in \mathbb{D}$ , the composition operator $C_{\varphi}$ is not continuous on $H_{v}^{\infty}(\mathbb{D})$ . We refer to Lusky [125, 127] for the relevance of condition (L1) in connection with the isomorphic classification of spaces $H_{\nu}^{\infty}$ . 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 $W_{\varphi,\psi}: H_v^0(\mathbb{D}) \to$ $H_w^0(\mathbb{D})$ 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 $H_n^{\infty}$ and $H_n^0$ were investigated in [102, 105, 115].

Theorem 57. Let $W_{\varphi,\psi}: H_n^0(\mathbb{D}) \to H_m^0(\mathbb{D})$ be continuous. - (a) The following conditions are equivalent. - (i) $W_{\varphi,\psi}: H_v^{\infty}(\mathbb{D}) \to H_w^{\infty}(\mathbb{D})$ is Fredholm. (ii) $W_{\varphi,\psi}: H_v^{0}(\mathbb{D}) \to H_w^{0}(\mathbb{D})$ is Fredholm. - (iii) $\psi \in H^{\infty}$ , there is $\varepsilon > 0$ such that $|\psi(z)| > \varepsilon$ for all $|z| \ge 1 \varepsilon$ , and $\varphi$ is an automorphism. - (b) The following conditions are equivalent. - (i) $W_{\varphi,\psi}: H_v^\infty(\mathbb{D}) \to H_w^\infty(\mathbb{D})$ is invertible. (ii) $W_{\varphi,\psi}: H_v^0(\mathbb{D}) \to H_w^0(\mathbb{D})$ is invertible. (iii) $\psi$ and $1/\psi$ belong to $H^\infty$ , and $\varphi$ is an automorphism. As a consequence a continuous composition operator $C_{\varphi}: H_{v}^{0} \to H_{w}^{0}$ is Fredholm if and only if it is invertible and if and only if $\varphi$ is an automorphism. These conditions are also equivalent to $C_{\varphi}: H_{v}^{\infty}(\mathbb{D}) \to H_{w}^{\infty}(\mathbb{D})$ being invertible or Fredholm. 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 $H_v^{\infty}(\mathbb{D})$ . As a consequence a composition operator $C_{\varphi}$ of $H_{v_n}^{\infty}(\mathbb{D})$ for a standard weight $v_p(z) = (1-|z|)^p$ , p > 0, is an isometry if and only if $\varphi$ 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.  184 Page 24 of 40 J. Bonet Boyd and Rueda have obtained interesting related results on isometries on weighted Banach spaces $H_v^\infty(U)$ of holomorphic functions defined on an open subset U of $\mathbb{C}^n$ . 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.

Theorem 58. Suppose $\varphi$ is not an automorphism and has fixed point $a \in \mathbb{D}$ . Then $$\sigma(W_{\psi,\varphi},H_v^\infty(\mathbb{D})) = \{\lambda \in \mathbb{C} : |\lambda| \le r_{e,H_v^\infty}(W_{\psi,\varphi})\} \cup \{\psi(a)\varphi'(a)^n\}_{n=0}^\infty.$$ A similar result holds for $H_v^0(\mathbb{D})$ . Further extensions and related results can be seen in [98, 99, 103, 104]. Let $\varphi$ be an analytic self map on $\mathbb D$ which is not an automorphism and has a (necessarily unique) fixed point $a \in \mathbb D$ . By Koenigs' Theorem [150, Chapter 6], if $\varphi'(a) = 0$ , the equation $f \circ \varphi = \lambda f$ has a non-trivial solution if and only if $\lambda = 1$ and the constant functions are the only solutions. On the other hand, if $\varphi'(a) \neq 0$ , then (i) the equation $f \circ \varphi = \lambda f$ has a non-trivial solution if and only if $\lambda = \varphi'(a)^n$ , $n \in \mathbb N_0$ , (ii) there is a unique function $\sigma \in H(\mathbb D)$ , called the Koenigs' eigenfunction of $\varphi$ , such that $\sigma \circ \varphi = \varphi'(w)\sigma$ , $\sigma'(w) = 1$ , and (iii) a function $f \in H(\mathbb D)$ satisfies $f \circ \varphi = \varphi'(a)^n f$ for some $n \in \mathbb N_0$ if and only if $f = c\sigma^n$ for some $c \in \mathbb C$ . As a consequence of Schwarz Lemma, an analytic self map $\varphi$ on $\mathbb D$ which is not an automorphism and has a fixed point $a \in \mathbb D$ satisfies $\varphi'(a) \neq 0$ if and only if $0 < |\varphi'(a)| < 1$ . Moreover, in this case, the Koenigs' eigenfunction of $\varphi$ can be obtained as the limit of the sequence $(\sigma_n)_n$ with $\sigma_n := \varphi_n/\varphi'(a)^n$ , $n \in \mathbb N$ , which converges to $\sigma$ uniformly on the compact subsets of $\mathbb D$ . Here $\varphi_n = \varphi \circ \cdots \circ \varphi$ is the n-fold composition of $\varphi$ . The spectrum and essential spectrum of the composition operator $C_{\varphi}: H(\mathbb{D}) \to H(\mathbb{D})$ for an analytic self map $\varphi$ on $\mathbb{D}$ which is not an automorphism and has a fixed point $a \in \mathbb{D}$ with $0 \le |\varphi'(a)| < 1$ have been determined recently in [20]: $\sigma(C_{\varphi}, H(\mathbb{D})) = \{0\} \cup \{\varphi'(a)^n : n \in \mathbb{N}_0\}$ , 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 $H(\mathbb{D})$ 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 $C_{\varphi}$ on both $H_v^{\infty}(\mathbb{D})$ and $H_v^0(\mathbb{D})$ determines whether the Koenigs eigenfunction $\sigma$ of $\varphi$ belongs to $H_v^{\infty}(\mathbb{D})$ and $H_v^0$ respectively. Let $\varphi$ be an analytic self map on $\mathbb{D}$ which is not an automorphism such that $\varphi(0)=0$ and $0<|\varphi'(0)|<1$ . Bourdon [71] proved that $\sigma\in H_{v_p}^0(\mathbb{D})$ , for $v_p(z)=(1-|z|)^p$ , p>0, if and only if $|\varphi'(0)|>r_{e,H_{v_p}^0}$ . 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], $\sigma \in H^{\infty}$ if and only if $r_{e,H^{\infty}}(C_{\varphi}) = 0$ .

Proposition 59. [98] Let $\varphi$ be an analytic self map on $\mathbb{D}$ which is not an automorphism such that $\varphi(0) = 0$ and $0 < |\varphi'(0)| < 1$ . Let v be a radial weight on $\mathbb{D}$ . Then - (i) The Koenigs' eigenfunction $\sigma$ belongs to $H_v^{\infty}(\mathbb{D})$ if and only if the sequence $\sigma_n := \varphi_n/\varphi'(0)^n$ , $n \in \mathbb{N}$ , is bounded in $H_v^{\infty}(\mathbb{D})$ . - (ii) Assume that $\varphi$ is univalent. The Koenigs' eigenfunction $\sigma$ belongs to $H_v^0(\mathbb{D})$ if and only if $\lim_{n\to\infty} \|\sigma_n \sigma\|_v = 0$ . Hyvärinen, Lindström, Nieminen and Saukko [115], using ideas of Kamowitz and Gunatillake, calculated the spectrum of the invertible weighted composition operator $W_{\varphi,\psi}$ for an automorphic symbol $\varphi$ on a wide class of analytic function spaces; in particular for spaces of type $H_v^\infty(\mathbb{D})$ . The analysis of the spectral behaviour depends on the type of the symbol $\varphi$ , that is, if it is an elliptic, parabolic or hyperbolic automorphism. An automorphism $\varphi \in Aut(\mathbb{D})$ is an injective analytic function on $\mathbb{D}$ such that $\varphi(\mathbb{D}) = \mathbb{D}$ . As a consequence of Schwarz Lemma, we get $\varphi \in Aut(\mathbb{D})$ if and only if there are $a \in \mathbb{D}$ and $\lambda \in \mathbb{C}$ , $|\lambda| = 1$ such that $\varphi(z) = \lambda \frac{a-z}{1-\overline{a}z}$ . For $z, a \in \mathbb{D}$ , $\varphi_a(z) := \frac{a-z}{1-\overline{a}z}$ is the automorphism of $\mathbb{D}$ which interchanges 0 and a. An automorphism $\varphi \in Aut(\mathbb{D})$ is called - elliptic if it has one fixed point in $\mathbb{D}$ and one outside $\overline{\mathbb{D}}$ , which could be in the infinity point of the Riemann sphere. - Example: $\varphi(z) = iz$ . Fixed points: 0 and $\infty$ . - hyperbolic if it has two different fixed points in the boundary $\mathbb{D}$ . Example: $\varphi(z) = \frac{z+0.5}{1+0.5z}$ . Fixed points: 1 and -1. - parabolic if it has one fixed point in the boundary of $\mathbb{D}$ with multiplicity 2. Example: $\varphi(z) = \frac{(1+i)z-i}{iz+1-i}$ . Fixed point: 1.

Theorem 60. (Theorem of Denjoy–Wolff) If $\varphi \in H(\mathbb{D})$ is a self map on $\mathbb{D}$ with no fixed point in $\mathbb{D}$ (for example if $\varphi \in Aut(\mathbb{D})$ is parabolic or hyperbolic), then there is a unique point w in the boundary of $\mathbb{D}$ , called the Denjoy–Wolff point of $\varphi$ , such that $(\varphi_n)_n$ converges to w uniformly on the compact subsets of $\mathbb{D}$ . If $\varphi \in H(\mathbb{D})$ is a self map on $\mathbb{D}$ that fixes a point $p \in \mathbb{D}$ , and is not a conformal automorphism, then $(\varphi_n)_n$ converges to p uniformly on the compact subsets of $\mathbb{D}$ . An elliptic automorphism has one fixed point in $\mathbb{D}$ . A hyperbolic automorphism has two fixed points in the boundary $\partial \mathbb{D}$ of $\mathbb{D}$ , one is attractive and the other one repulsive, and a parabolic automorphism has one fixed point in the boundary of $\mathbb{D}$ with multiplicity 2. See more details in [79], [89] and [150]. Recall that the disc algebra $A(\mathbb{D})$ is the space of continuous functions on the closed unit disc $\overline{\mathbb{D}}$ which are analytic on $\mathbb{D}$ . It is a Banach space endowed with the supremum norm on $\mathbb{D}$ .

Theorem 61. [115] Let $v_p = (1-|z|^2)^p$ , p > 0 and let $W_{\varphi,\psi}: H_{v_p}^{\infty}(\mathbb{D}) \to H_{v_p}^{\infty}(\mathbb{D})$ be continuous. Assume that $\psi \in A(\mathbb{D})$ is bounded away from zero on $\mathbb{D}$ . Then (i) If $\varphi$ is a parabolic automorphism with Denjoy–Wolff point $a \in \partial \mathbb{D}$ , then $$\sigma(W_{\varphi,\psi}, H_{v_n}^{\infty}(\mathbb{D})) = \{\lambda \in \mathbb{C} ; |\lambda| = |\psi(a)|\}.$$ (ii) If $\varphi$ is a hyperbolic automorphism with attractive fixed point $a \in \partial \mathbb{D}$ and repulsive fixed point $b \in \partial \mathbb{D}$ , such that $|\psi(b)/\varphi'(b)^p| \leq |\psi(a)/\varphi'(a)^p|$ , then $$\sigma(W_{\varphi,\psi}, H_{v_p}^{\infty}(\mathbb{D})) = \left\{ \lambda \in \mathbb{C} \; ; \; \frac{|\psi(b)|}{\varphi'(b)^p} \le |\lambda| \le \frac{|\psi(a)|}{\varphi'(a)^p} \right\}.$$  184 Page 26 of 40 J. Bonet (iii) If $\varphi$ is an automorphism such that there is a $j \in \mathbb{N}$ such that $\varphi_j(z) = z$ for each $z \in \mathbb{D}$ , then, for the smaller such $n \in \mathbb{N}$ , we have $$\sigma(W_{\varphi,\psi},H^{\infty}_{v_p}(\mathbb{D})) = \overline{\left\{\lambda \in \mathbb{C} \; ; \; \lambda^n = \prod_{m=0}^{n-1} (\psi \circ \varphi_m)(z), \; z \in \mathbb{D}\right\}}.$$ (iv) If $\varphi$ is an automorphism such that $\varphi_n$ does not coincide with the identity function id(z) = z for each $n \in \mathbb{N}$ , and $a \in \mathbb{D}$ is the unique fixed point of $\varphi$ , then $\sigma(W_{\varphi,\psi}, H_{v_p}^{\infty}(\mathbb{D})) = \{\lambda \in \mathbb{C} \; ; \; |\lambda| = |\psi(a)|\}.$ Further results can be seen in [103, 104]. Many questions remain open about the spectrum of weighted composition operators.

Proposition 62. If $C_{\varphi}: H_{v}^{0}(\mathbb{D}) \to H_{v}^{0}(\mathbb{D})$ is a continuous hypercyclic operator, then $\varphi$ has no fixed point in $\mathbb{D}$ and it is injective.

Theorem 63. If $\varphi$ is an automorphism which fixes no point in $\mathbb{D}$ and $C_{\varphi}: H_{v}^{0}(\mathbb{D}) \to H_{v}^{0}(\mathbb{D})$ is continuous, then $C_{\varphi}$ is hypercyclic. We refer to the book of Shapiro [150] for linear fractional transformations of the unit disc.

Theorem 64. Let $\varphi$ be a linear fractional transformation of $\mathbb{D}$ such that $C_{\varphi}: H_{v}^{0}(\mathbb{D}) \to H_{v}^{0}(\mathbb{D})$ is continuous. If $\varphi$ is a hyperbolic non-automorphism, then $C_{\varphi}$ is hypercyclic.

Proposition 65. Let $v(z) = (1 - |z|)^{1/2}$ for every $z \in \mathbb{D}$ . If $\varphi$ be a linear fractional transformation of $\mathbb{D}$ which is a parabolic non-automorphism, then $C_{\varphi}: H_v^0(\mathbb{D}) \to H_v^0(\mathbb{D})$ is not hypercyclic. These results were extended to the case of weighted composition operators of the form $\lambda C_{\varphi}$ , $\lambda \in \mathbb{C}$ , 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 $\mathbb{C}$ , 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 $H^{\infty}$ and the disc algebra $A(\mathbb{D})$ . 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 $C_{\varphi}$ on $H_{\nu}^{\infty}(\mathbb{D})$ and $H_{\nu}^{0}(\mathbb{D})$ . 

Proposition 66. Let v be a log-convex weight on $\mathbb{D}$ satisfying condition (L1). Let $\varphi$ be an elliptic automorphism with fixed point $z(0) \in \mathbb{D}$ . Then - (i) If $\varphi$ is equivalent to a rational rotation, then $C_{\varphi}$ is uniformly mean ergodic on $H_{v}^{\infty}(\mathbb{D})$ and $H_{v}^{0}(\mathbb{D})$ . Moreover, there is $k \in \mathbb{N}$ such that $((C_{\varphi})_{[n]})_{n}$ converges to $(1/k)(C_{\varphi} + \cdots + (C_{\varphi})^{k})$ . - (ii) If $\varphi$ is equivalent to a irrational rotation, then $C_{\varphi}$ is not uniformly mean ergodic on $H_{\nu}^{\infty}(\mathbb{D})$ and $H_{\nu}^{0}(\mathbb{D})$ . - (iii) If $\varphi$ is equivalent to a irrational rotation, then $C_{\varphi}$ is mean ergodic on $H_v^0(\mathbb{D})$ , and $((C_{\varphi})_{[n]}f)$ converges to $C_{z(0)}f=f(z(0))$ for each $f\in H_v^0(\mathbb{D})$ .

Theorem 67. Let v be a log-convex weight on $\mathbb{D}$ satisfying condition (L1). Let $\varphi$ be a self map on $\mathbb{D}$ with Denjoy-Wolff point 0. Then the sequence $(C_{\varphi_n} f)_n$ converges to f(0) in $H_v^0(\mathbb{D})$ for each $f \in H_v^0(\mathbb{D})$ . Tien proves in [158, Theorem 4.8 (b)] that, under some mild assumption, the sequence $(C_{\varphi_n})_n$ converges to the operator $C_0(f) := f(0)$ in $\mathcal{L}(H_v^\infty(\mathbb{D}))$ . Moreover, he gives an example of a weight v and self map $\varphi$ with Denjoy-Wolff point 0 such that $C_{\varphi}$ is power bounded, $(C_{\varphi_n})_n$ does not converge even in the weak operator topology on $H_v^\infty(\mathbb{D})$ , but it converges in the strong operator topology on $H_v^0(\mathbb{D})$ . This operator $C_{\varphi}$ is mean ergodic and not uniformly mean ergodic on $H_v^0(\mathbb{D})$ , and not mean ergodic on $H_v^\infty(\mathbb{D})$ .

Theorem 68. Let $\varphi$ be a self map on $\mathbb{D}$ with Denjoy-Wolf point z(0) in the boundary of the unit disc. Let v be a log-convex weight on $\mathbb{D}$ satisfying condition (L1) such that $-\log(1-r) = O(1/v(r))$ as $r \to 1-$ . Then $C_{\varphi}$ is not power bounded and not mean ergodic on $H_{v}^{\infty}(\mathbb{D})$ .

Lemma 69. (Boyd, Rueda) Let u and v be weights. If the entire function $\varphi$ satisfies that the superposition operator $S_{\varphi}$ maps $H_u^{\infty}$ into $H_v^{\infty}$ and is bounded, then $S_{\varphi}: H_u^{\infty} \to H_v^{\infty}$ is continuous.

Theorem 70. Let u and v be weights, such that u is strictly decreasing. (a) If the entire function $\varphi$ satisfies the following condition: $$\forall \varepsilon \in ]0, 1[ \exists C > 0 \exists R_0 > 0 \forall R \ge R_0 :$$ $$v\left(u^{-1}\left(\frac{1}{\varepsilon R}\right)\right) \max_{|w|=R} |\varphi(w)| \le C,$$ then the superposition operator $S_{\varphi}$ maps $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ and is bounded.  (b) If the entire function $\varphi$ satisfies the following condition: $$\lim_{R \to \infty} v\left(u^{-1}\left(\frac{k}{R}\right)\right) M(\varphi, R) = 0$$ for each $k \in \mathbb{N}$ , then $S_{\varphi}: H_{u}^{\infty}(\mathbb{D}) \to H_{v}^{\infty}(\mathbb{D})$ is compact.

Proposition 71. Let $u(z) = (1 - |z|)^{\alpha}$ , $\alpha > 0$ , $v(z) = (1 - |z|)^{\beta}$ , $\beta > 0$ . - (1) The following conditions are equivalent for an entire function $\varphi$ : - (i) $\varphi$ is a polynomial of degree at most the integer part $[\beta/\alpha]$ of $\beta/\alpha$ . - (ii) The superposition operator $S_{\varphi}$ maps $H_{\eta}^{\infty}(\mathbb{D})$ into $H_{\eta}^{\infty}(\mathbb{D})$ . - (iii) The superposition operator $S_{\omega}$ maps $H_{\nu}^{\infty}(\mathbb{D})$ into $H_{\nu}^{\infty}(\mathbb{D})$ and it is bounded. - (2) The following conditions are equivalent for an entire function $\varphi$ : - (i) $\varphi$ is a polynomial of degree k less than $\beta/\alpha$ . - (ii) The superposition operator $S_{\varphi}$ maps $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ and it is compact.

Proposition 72. Let $u(z) = (1-|z|)^{\alpha}$ , $z \in \mathbb{D}$ and $v(z) = \exp(-\frac{1}{(1-|z|)^{\beta}})$ , $\alpha, \beta > 0$ . Let $\varphi$ be an entire function. The following conditions are equivalent: - (i) The function $\varphi$ is of order less than $\beta/\alpha$ or of order $\beta/\alpha$ and type zero. - (ii) For all $0 < \varepsilon < 1$ there are $C \ge 1$ , $R_0 > 0$ such that $|\varphi(z)| \le C \exp(\varepsilon |z|^{\beta/\alpha})$ for all $z \in \mathbb{C}$ with $|z| > R_0$ . - (iii) The superposition operator $S_{\varphi}$ maps $H_{u}^{\infty}(\mathbb{D})$ into $H_{v}^{\infty}(\mathbb{D})$ . (iv) The superposition operator $S_{\varphi}$ maps $H_{u}^{\infty}(\mathbb{D})$ into $H_{v}^{\infty}(\mathbb{D})$ and it is bounded.

Theorem 73. Let $u(r) = (\log \frac{e}{1-r})^{-\alpha}$ and $v(r) = (1-r)^{\beta}$ with $\alpha, \beta > 0$ . The following statements are equivalent for an entire function $\varphi$ : - (i) The function $\varphi$ is of order less than $1/\alpha$ or of order $1/\alpha$ and type zero. - (ii) For all $0 < \varepsilon < 1$ there are C > 0, $R_0 > 0$ such that $|\varphi(z)| \le C \exp(\varepsilon |z|^{1/\alpha})$ for all $z \in \mathbb{C}$ with $|z| \geq R_0$ . - (iii) The superposition operator $S_{\omega}$ maps $H_{\nu}^{\infty}$ into $H_{\nu}^{\infty}$ . - (iv) $S_{\varphi}$ is a bounded operator from $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ . (v) $S_{\varphi}$ is a compact operator from $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ .

Proposition 74. Let $u(r) = (\log \frac{e}{1-r})^{\alpha}$ , $\alpha > 0$ , $v(r) = (\log \frac{e}{1-r})^{\beta}$ , $\beta > 0$ . - (1) The superposition operator $S_{\varphi}$ maps $H_{\nu}^{\infty}(\mathbb{D})$ into $H_{\nu}^{\infty}(\mathbb{D})$ and is bounded if and only if $\varphi$ is a polynomial of degree at most $[\beta/\alpha]$ . - (2) The superposition operator $S_{\varphi}$ maps $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ and is compact if and only if $\varphi$ is a polynomial of degree less than $\beta/\alpha$ .

Proposition 75. Let $u(r) = \exp(-(1-|z|)^{-\alpha})$ , $\alpha > 0$ and let $\varphi$ be an entire function. - (1) If there exist C > 0 and $R_0 > 0$ such that $|\varphi(w)| \le C \exp((\log |w|)^{\gamma})$ for $|w| \ge R_0$ , then for each c>1 the superposition operator $S_{\varphi}$ maps $H_u^{\infty}(\mathbb{D})$ boundedly into the space $H_{v_c}^{\infty}(\mathbb{D})$ , where $v_c(r) = \exp(-\frac{c}{(1-|z|)^{\alpha\gamma}})$ . - (2) If the superposition operator $S_{\varphi}$ maps $H_u^{\infty}(\mathbb{D})$ into $H_v^{\infty}(\mathbb{D})$ , $v(r) = \exp(-\frac{1}{(1-|r|)^{\beta}})$ , $\beta > 0$ , then for every c > 1 there exist C > 0 and $R_0 > 0$ such that $|\varphi(w)| \le$ $C \exp(c(\log |w|)^{\beta/\alpha})$ for $|w| > R_0$ . If $\gamma > 1$ , there exist entire transcendental functions $\varphi$ satisfying the assumptions of Proposition 75.  184 Page 30 of 40 J. Bonet

Theorem 76. There is a bounded harmonic function $f: \mathbb{D} \to \mathbb{C}$ such that $T_f: H_v^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is not a bounded operator for any weight v on $\mathbb{D}$ . As a consequence, the Bergman projection $P_v$ is not a bounded mapping $L_v^{\infty} \to L_v^{\infty}$ for any weight under consideration. Dostanic [94] showed that for the exponential weight $v(z) = \exp(-1/(1-|z|))$ , the orthogonal projection $L_v^2 \to A_v^2$ is bounded in $L_v^p$ if and only if p = 2. Recent research about the continuity of the Bergman projection can be seen in [62, 84, 114, 140]. A general, abstract necessary and sufficient condition for the boundedness and for the compactness of the Toeplitz operator $T_f: H_v^\infty(\mathbb{D}) \to H_v^\infty(\mathbb{D})$ was presented in [60, Theorem 3.6], when f is a radial symbol in $L^1$ , when the weight v satisfies a certain condition (B), which was introduced by Lusky in [127]. This condition (B) is closely related to condition (B) described in Sect. 5 and it is satisfied by standard and exponential weights, but not by logarithmic weights. The proof of [60, Theorem 3.6] is based on the deep work of Lusky. Since the conditions for boundedness of $T_f$ might be hard to evaluate, in [61] we prove sufficient conditions, which are easier to formulate and control, for the boundedness and compactness of Toeplitz operators $T_f$ on $H_v^\infty(\mathbb{D})$ when the weights v and symbol f are assumed to be radial functions on $\mathbb{D}$ , and v has condition (B) of Lusky. Here are some results in this directions.

Theorem 77. Let v satisfy condition (B). If the symbol f is radial and continuously differentiable on [0, 1], then $T_f : H_v^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is bounded.

Theorem 78. Let v be a standard weight. (i) If $f \in L^1$ is radial and satisfies $$\limsup_{r \to 1} |f(r)\log(1-r)| < \infty,$$ then $T_f: H_v^\infty(\mathbb{D}) \to H_v^\infty(\mathbb{D})$ is a bounded operator. (ii) If f satisfies $\limsup_{r\to 1} |f(r)\log(1-r)| = 0$ , then $T_f$ is compact on $H_v^\infty(\mathbb{D})$ .

Theorem 79. Let $v(r) = \exp(-\alpha/(1-r)^{\beta})$ be an exponential weight. (i) Assume that $f \in L^1$ is radial and $$\limsup_{r \to 1} |f(r)| (1-r)^{-1/2-\beta/4} < \infty.$$ Then, $T_f: H_n^{\infty}(\mathbb{D}) \to H_n^{\infty}(\mathbb{D})$ is a bounded operator.  Page 32 of 40 J. Bonet (ii) If $$\lim_{r \to 1} \sup |f(r)| (1-r)^{-1/2-\beta/4} = 0,$$ then $T_f$ is compact on $H_n^{\infty}(\mathbb{D})$ .

Theorem 82. Let v be a radial weight on $\mathbb{D}$ satisfying condition (L1). If the function $w \in$ $\mathbb{D} \to K(w) \|C_{\varphi_m}\|$ belongs to $L^1(\mu)$ , then the operators $$\mathcal{H}_{K,\mu}: H_v^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D}),$$ and $$\mathcal{H}_{K,\mu}: H_v^0(\mathbb{D}) \to H_v^0(\mathbb{D}),$$ are continuous. In this case, we have $$\|\mathcal{H}_{K,\mu}\| \leq \int_{\mathbb{D}} |K(w)| \|C_{\varphi_w}\| d\mu(w).$$

Corollary 83. (1) Let $v(r) = (1-r^2)^{\gamma}$ with $\gamma > 0$ . If the function $w \in \mathbb{D} \to K(w)/(1-|w|)^{\gamma}$ belongs to $L^1(\mu)$ , then $\mathcal{H}_{K,\mu}: H_v^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is continuous. (2) Let $v(r) = (\log \frac{e}{1-r^2})^{-\alpha}$ , $\alpha > 0$ . If the function $w \in \mathbb{D} \to K(w) \log(1-|w|)$ belongs to $L^1(\mu)$ , then $\mathcal{H}_{K,\mu}: H_v^{\infty}(\mathbb{D}) \to H_v^{\infty}(\mathbb{D})$ is continuous. We now turn our attention to the case of spaces of entire functions. Let $\mu$ be a positive measure on $(0, \infty)$ . Stylogiannis and Galanopoulos [154] consider formally the Hausdorff operator induced by the measure $\mu$ defined by $$\mathcal{H}_{\mu}(f)(z) := \int_{0}^{\infty} \frac{1}{t} f\left(\frac{z}{t}\right) d\mu(t), \quad z \in \mathsf{C},$$ where $f \in H(\mathbb{C})$ is an entire function. The operator $\mathcal{H}_{\mu}$ is studied in [154] on Fock spaces $\mathcal{F}_{\alpha}^{p}$ , $1 \leq p \leq \infty$ , $\alpha > 0$ .

Proposition 84. Let v be a weight on $\mathbb{C}$ . (1) If the operator $\mathcal{H}_{\mu}: H_{\nu}^{\infty}(\mathbb{C}) \to H_{\nu}^{\infty}(\mathbb{C})$ is continuous, then $$\sup_{n \in \mathbb{N}_0} \int_0^\infty \frac{1}{t^{n+1}} d\mu(t) \le \|\mathcal{H}_\mu\| < \infty, \tag{9}$$ - and the operator $\mathcal{H}_{\mu}: H^0_v(\mathbb{C}) \to H^0_v(\mathbb{C})$ is also continuous. (2) If the operator $\mathcal{H}_{\mu}: H^0_v(\mathbb{C}) \to H^0_v(\mathbb{C})$ is continuous, then (9) holds. - (3) If the operator $\mathcal{H}_{\mu}: H_{\nu}^{\infty}(\mathbb{C}) \to H_{\nu}^{\infty}(\mathbb{C})$ is compact, then $$\lim_{k \to \infty} \int_0^\infty \frac{1}{t^k} d\mu(t) = 0.$$

Theorem 85. Let $\alpha > 0$ and $\beta > 0$ . Let v be the weight on $\mathbb{C}$ defined by $v(r) = \exp(-\beta r^{\alpha})$ . The following conditions are equivalent. - (i) $\mathcal{H}_{\mu}: H^{\infty}_{v}(\mathbb{C}) \to H^{\infty}_{v}(\mathbb{C})$ is continuous. (ii) $\mathcal{H}_{\mu}: H^{\infty}_{v}(\mathbb{C}) \to H^{0}_{v}(\mathbb{C})$ is also continuous. (iii) $\sup_{n \in \mathbb{N}} \int_{0}^{\infty} \frac{1}{t^{n+1}} d\mu(t) < \infty$ . In this case, we have $$\|\mathcal{H}_{\mu}\| \leq \sup_{n \in \mathbb{N}} \int_0^{\infty} \frac{1}{t^{n\alpha+1}} d\mu(t).$$ 

Proposition 86. Let $\alpha > 0$ and $\beta > 0$ . Let v be the weight on $\mathbb{C}$ defined by $v(r) = \exp(-\beta r^{\alpha})$ . If the sequence $\left(\int_0^{\infty} \frac{1}{t^k} d\mu(t)\right)_{k \in \mathbb{N}}$ is in $\ell_1$ , then the operator $\mathcal{H}_{\mu}$ is compact on $H_v^{\infty}(\mathbb{C})$ and on $H_v^{\infty}(\mathbb{C})$ . 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 Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit <a href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</a>.