2.1 Our general framework

We begin by presenting the assumptions that our Banach space $\mathcal {X}$ must satisfy. For $n\in \mathbb N$ , we define $\chi _n(z)=z^n$ , $z\in \mathbb {D}$ . We assume that $\mathcal {X}$ is a Banach space of analytic functions on $\mathbb {D}$ satisfying the following standard conditions:

1. (H1) For every $\lambda \in \mathbb {D}$ , the evaluation map $E_\lambda : f \in \mathcal {X} \longmapsto f(\lambda ) \in \mathbb {C}$ is continuous.

2. (H2) For every $f\in \mathcal {X}$ , we have $\chi _1 f \in \mathcal {X}$ .

3. (H3) The set of polynomials $\mathcal P= \bigvee (\chi _n : n \ge 0)$ is dense in $\mathcal {X}$ .

The assumption (H1) means that, for every $\lambda \in \mathbb {D}$ , $E_\lambda $ belongs to $\mathcal {X}^*$ , the dual space of $\mathcal {X}$ . In particular, it implies that convergence in $\mathcal {X}$ implies pointwise convergence on $\mathbb {D}$ . We shall denote by $\langle f|\varphi \rangle =\varphi (f)$ the duality bracket between $\varphi \in \mathcal {X}^*$ and $f\in \mathcal {X}$ .

The assumption (H2) means that $\chi _1$ belongs to $\mathfrak {M}(\mathcal {X})$ , the algebra of multipliers of $\mathcal {X}$ defined by

$$ \begin{align*}\mathfrak{M}(\mathcal{X}) := \left \{\varphi \in \mathcal{X} : \varphi f \in \mathcal{X},\,\forall f \in \mathcal{X} \right \}.\end{align*} $$

Using closed graph theorem and (H1) it is easy to see that if $\varphi \in \mathfrak {M}(\mathcal {X})$ , then the multiplication operator by $\varphi $ , $M_\varphi : f \in \mathcal {X} \longmapsto \varphi f \in \mathcal {X}$ , is bounded on $\mathcal {X}$ . In particular, it follows from (H2) that the shift operator acting on $\mathcal {X}$ defined by

$$\begin{align*}S : f \in \mathcal{X} \longmapsto \chi_1 f \in \mathcal{X,} \end{align*}$$

is bounded. Moreover, if we set $\left \Vert \varphi \right \Vert {}_{\mathfrak {M}(\mathcal {X})} = \left \Vert M_\varphi \right \Vert {}_{\mathcal {L}(\mathcal {X})}$ , where $\|\cdot \|_{\mathcal {L}(\mathcal {X})}$ denotes the (operator) norm on $\mathcal {L}(\mathcal {X})$ , the space of linear and bounded operators on $\mathcal {X}$ , then it is well known that $\mathfrak {M}(\mathcal {X})$ is a Banach algebra. Note also that by (H2), we have $\mathcal {P}\subset \mathfrak {M}(\mathcal {X})$ .

We introduce now a (commutative and unital) Banach algebra $\mathcal {A}$ satisfying the following conditions:

1. (H4) either (H4a) $\mathcal {A} \subset \mathfrak {M}(\mathcal {X})$ or (H4b) $\mathcal {A}\subset \mathcal {X}$ and $\mathcal P$ is dense in $\mathcal A$ .

2. (H5) For every $\lambda \in \mathbb {D}$ the evaluation map $f \in \mathcal {A} \longmapsto f(\lambda ) \in \mathbb {C}$ is continuous.

3. (H6) There exists $C>0$ and $A\geq 1$ such that for every $f_1,f_2\in \mathcal {A}$ satisfying

   $$\begin{align*}0 < \delta \le \left\lvert f_1 \right\rvert + \left\lvert f_2 \right\rvert \text{ on } \mathbb{D}, \quad\text{and}\quad \|f_1\|_{\mathcal{A}}+\|f_2\|_{\mathcal{A}}\leq 1, \end{align*}$$

   there exists $g_1,g_2\in \mathcal {A}$ such that $f_1g_1+f_2g_2\equiv 1$ on $\mathbb {D}$ and

   $$\begin{align*}\|g_1\|_{\mathcal{A}},\,\|g_2\|_{\mathcal{A}}\leq \frac{C}{\delta^A}. \end{align*}$$

The assumption (H6) means that the Banach algebra $\mathcal {A}$ satisfies a Corona Theorem with a control on the solutions. It is known to be true in the algebra $H^\infty $ , with any $A>2$ . See [Reference Carleson12, Reference Tolokonnikov36, Reference Uchiyama38]. V. Tolokonnikov [Reference Tolokonnikov37] also proved that (H6) is satisfied for $\mathcal {A}=\mathcal D_\alpha ^p\cap A(\mathbb {D})$ with $1<p\leq \alpha +2$ and $A\geq 4$ . Let us also mention the recent paper [Reference Luo29] of Shuaibing Luo who proved that (H6) holds for $\mathcal {A}=\mathfrak M(\mathcal D(\mu ))$ with $\mu $ a finite positive Borel measure on the closed unit disc $\overline {\mathbb {D}}$ and $A\geq 4$ , and the paper [Reference Fricain, Hartmann, Ross and Timotin21] who studied the case of the algebra of multipliers of some de Branges–Rovnyak spaces.

2.2 Some technical preliminaries

We now present some simple consequences of our assumptions. Let us first remind a standard property for the multiplier algebra $\mathfrak {M}(\mathcal {X})$ . For completeness, we give a proof.

Proof $(1)$ : Let $f \in \mathfrak {M}(\mathcal {X})$ . Since $\chi _0 = 1 \in \mathcal {X}$ , $f = f \chi _0 \in \mathcal {X}$ , whence

$$ \begin{align*}\left\Vert f\right\Vert{}_{\mathcal{X}} = \left\Vert f \chi_0\right\Vert{}_{\mathcal{X}} \le \left\Vert f\right\Vert{}_{\mathfrak{M}(\mathcal{X})} \left\Vert \chi_0\right\Vert{}_{\mathcal{X}}.\end{align*} $$

Note that $M_f^* : \mathcal {X}^* \rightarrow \mathcal {X}^*$ is well defined and, from (H1), $E_\lambda \in \mathcal {X}^*$ for every $\lambda \in \mathbb {D}$ . Then, for every $g \in \mathcal {X}$ , we have

$$ \begin{align*}\langle g,M_f^* E_\lambda\rangle = \langle fg,E_\lambda\rangle = f(\lambda)g(\lambda) = f(\lambda) \langle g,E_\lambda\rangle= \langle g,f(\lambda)E_\lambda\rangle,\end{align*} $$

and we deduce that $M_f^*E_\lambda =f(\lambda )E_\lambda $ . Therefore,

$$ \begin{align*}\left\lvert f(\lambda) \right\rvert\left\Vert E_\lambda\right\Vert{}_{\mathcal{X}^*} \le \left\Vert M_f\right\Vert{}_{\mathcal{L}(\mathcal{X})} \left\Vert E_\lambda\right\Vert{}_{\mathcal{X}^*}.\end{align*} $$

Remark that $\langle \chi _0,E_\lambda \rangle = 1$ , hence $E_\lambda \neq 0$ and dividing the last inequality by $\|E_\lambda \|_{\mathcal {X}^*}$ gives

$$\begin{align*}\left\lvert f(\lambda) \right\rvert \le \left\Vert M_f\right\Vert{}_{\mathcal{L}(\mathcal{X})}. \end{align*}$$

In other words, f is bounded on $\mathbb {D}$ and $\left \Vert f\right \Vert {}_\infty \le \left \Vert f\right \Vert {}_{\mathfrak {M}(\mathcal {X})}$ . Moreover, since $f \in \mathcal {X} \subset \text {Hol}(\mathbb {D})$ , $f \in H^\infty $ and

$$ \begin{align*} \left\Vert f\right\Vert{}_{\mathcal{X}} + \left\Vert f\right\Vert{}_\infty \le \left (\left\Vert \chi_0\right\Vert{}_{\mathcal{X}} + 1 \right )\left\Vert f\right\Vert{}_{\mathfrak{M}(\mathcal{X})}. \end{align*} $$

$(2)$ : Equip $H^\infty \cap \mathcal {X}$ with the following norm

$$\begin{align*}\|f\|_{\infty,\mathcal{X}}=\|f\|_\infty+\|f\|_{\mathcal{X}},\qquad f\in H^\infty\cap\mathcal{X}. \end{align*}$$

It is not difficult to see that $(H^\infty \cap \mathcal {X},\|\cdot \|_{\infty ,\mathcal {X}})$ is a Banach space. Consider now the identity map

$$\begin{align*}\begin{array}{rccl} i:&\mathfrak{M}(\mathcal{X})&\longrightarrow & H^\infty\cap\mathcal{X} \\ &f&\longmapsto & f \end{array}. \end{align*}$$

According to $(1)$ , this map is continuous and the theorem of isomorphism of Banach gives the result. here

The purpose of this paper is to study the cyclicity of $S$ on $\mathcal {X}$ . Let us remind the following notation

$$\begin{align*}[f]_{\mathcal{X}}=\overline{\{pf:p\in\mathcal P\}}^{\mathcal{X}}, \quad\text{for }f\in\mathcal{X}, \end{align*}$$

where $\overline {\{\cdots \}}^{\mathcal {X}}$ denotes the closure of $\{\cdots \}$ in $\mathcal {X}$ . It is clear that f is a cyclic vector of $S$ in $\mathcal {X}$ if and only if

$$\begin{align*}[f]_{\mathcal{X}}=\mathcal{X}. \end{align*}$$

Remark 2.2 It is standard that f is cyclic for $S$ in $\mathcal {X}$ if and only if $1 \in \left [f \right ]_{\mathcal {X}}$ . Indeed, since $\chi _0 = 1 \in \mathcal {X}$ by (H3), if f is cyclic for $S$ in $\mathcal {X}$ , then $1 \in [f]_{\mathcal {X}}$ . Conversely, let $g \in \mathcal {X}$ and $\varepsilon> 0$ . From the assumption (H3), there exists $q \in \mathcal {P}$ such that $\left \Vert q - g\right \Vert {}_{\mathcal {X}} \le \frac {\varepsilon }{2}$ . Moreover, if $1 \in [f]_{\mathcal {X}}$ , then there exists $p \in \mathcal {P}$ such that $\left \Vert p f - 1\right \Vert {}_{\mathcal {X}} \le \frac {\varepsilon }{2\left \Vert q\right \Vert {}_{\mathfrak {M}(\mathcal {X})}}$ . Thus, $pq \in \mathcal {P}$ and

$$\begin{align*}\left\Vert pq f - g\right\Vert{}_{\mathcal{X}} \le \left\Vert pqf - q\right\Vert{}_{\mathcal{X}} + \left\Vert q - g\right\Vert{}_{\mathcal{X}} \le \left\Vert q\right\Vert{}_{\mathfrak{M}(\mathcal{X})} \left\Vert pf - 1\right\Vert{}_{\mathcal{X}} + \frac{\varepsilon}{2} \le \varepsilon, \end{align*}$$

which gives the cyclicity of f.

Proof Let $\varepsilon> 0$ . Since $\varphi $ is cyclic for $S$ in $\mathcal {X}$ , there exists $p \in \mathcal {P}$ such that

$$ \begin{align*}\left\Vert p\varphi - 1\right\Vert{}_{\mathcal{X}} \le \frac{\varepsilon}{2}.\end{align*} $$

Moreover, from (H2), $p \in \mathcal {P} \subset \mathfrak {M}(\mathcal {X}) $ , whence $p\varphi \in \mathfrak {M}(\mathcal {X})$ . Now, since f is also cyclic for $S$ in $\mathcal {X}$ , there exists $q \in \mathcal {P}$ such that

$$ \begin{align*}\left\Vert q f - 1\right\Vert{}_{\mathcal{X}} \le \frac{\varepsilon}{2 \left\Vert p \varphi\right\Vert{}_{\mathfrak{M}(\mathcal{X})}}.\end{align*} $$

Thus,

$$ \begin{align*} \left\Vert f \varphi pq - 1\right\Vert{}_{\mathcal{X}} & \le \left\Vert f \varphi pq - p\varphi\right\Vert{}_{\mathcal{X}} + \left\Vert p\varphi - 1\right\Vert{}_{\mathcal{X}} \\ & \le \left\Vert \varphi p(q f - 1)\right\Vert{}_{\mathcal{X}} + \frac{\varepsilon}{2} \\ & \le \left\Vert p \varphi\right\Vert{}_{\mathfrak{M}(\mathcal{X})}\left\Vert qf - 1\right\Vert{}_{\mathcal{X}} + \frac{\varepsilon}{2} \le \varepsilon, \end{align*} $$

which gives the cyclicity of $f\varphi $ for $S$ in $\mathcal {X}$ .

Let us justify that the assumptions (H1), (H4) and (H5) imply the existence of a constant $c_4>0$ such that, for every $f\in \mathcal {A}$ , we have

(2.1) $$ \begin{align} \left\Vert f\right\Vert{}_{\mathcal{X}} \le c_4 \left\Vert f\right\Vert{}_{\mathcal{A}}. \end{align} $$

Indeed, if (H4a) is satisfied, that is $\mathcal {A}\subset \mathfrak {M}(\mathcal {X})$ , then according to the closed graph theorem applied to the canonical $i : f \in \mathcal {A} \longmapsto f \in \mathfrak {M}(\mathcal {X})$ and (H5), there exists $c_3>0$ such that for every $f \in \mathcal {A}$ , we have

$$\begin{align*}\left\Vert f\right\Vert{}_{\mathfrak{M}(\mathcal{X})} \le c_3 \left\Vert f\right\Vert{}_{\mathcal{A}}, \end{align*}$$

and (2.1) follows from Lemma 2.1.

On the other hand, if (H4b) is satisfied, then $\mathcal {A}\subset \mathcal {X}$ , and (2.1) follows immediately for the closed graph theorem applied to the canonical $i : f \in \mathcal {A} \longmapsto f \in \mathcal {X}$ and the assumptions (H1) and (H5).

For $f \in \mathcal {A}$ , we denote by $\mathcal {I}_f$ the closed ideal of $\mathcal {A}$ generated by f, that is

(2.2) $$ \begin{align} \mathcal{I}_f := \overline{\left \{ gf : g \in \mathcal{A} \right \}}^{\mathcal{A}}.\ \end{align} $$

2.3 Some operator tools

For Theorem C, we introduce now two more assumptions that we shall need in order to use a tauberian result of Atzmon. Apart assumptions (H1),(H2),…,(H6), we also assume that:

1. (H7) The function $\chi _1\in \mathcal {A}$ .

2. (H8) There exists $C>0$ and $p\in \mathbb N$ such that for every $n\geq 0$ , $\|\chi _n\|_{\mathcal {A}}\leq C n^p$ .

In the following, $\sigma (\chi _1)$ denotes the spectrum of $\chi _1$ in the Banach algebra $\mathcal {A}$ .

Proof $(i)$ : According to (H8), we have

(2.3) $$ \begin{align} \|\chi_1^n\|_{\mathcal{A}}=\|\chi_n\|_{\mathcal{A}}\leq C n^p. \end{align} $$

It immediately implies that the spectral radius of $\chi _1$ satisfies $r(\chi _1)\leq 1$ , and then $\sigma (\chi _1)\subset \overline {\mathbb {D}}$ . On the other hand, let $\lambda \in \mathbb {D}$ and assume that $\lambda -\chi _1$ is invertible in $\mathcal {A}$ . It means that there exists $f\in \mathcal {A}$ such that $(\lambda -\chi _1)f=1$ . Evaluating at $\lambda $ gives a contradiction. Hence, $\mathbb {D}\subset \sigma (\chi _1)$ and we conclude by closeness of $\sigma (\chi _1)$ .

$(ii)$ : Let $f\in \text {Hol}(\overline {\mathbb {D}})$ and write

$$\begin{align*}f(z)=\sum_{n=0}^\infty a_n z^n,\qquad z\in\overline{\mathbb{D}}. \end{align*}$$

It is well known that there exists a constant $c>0$ such that $a_n=O(\exp (-cn))$ as $n\to \infty $ . See for instance [Reference Fricain and Mashreghi23, Theorem 5.7]. Thus according to (H8), we have

$$\begin{align*}|a_n|\|\chi_n\|_{\mathcal{A}}\lesssim n^p \exp(-cn).\\[-18pt] \end{align*}$$

Thus, $g=\displaystyle \sum _{n=0}^\infty a_n\chi _n$ defines a function in $\mathcal {A}$ . Now using that convergence in $\mathcal {A}$ implies pointwise convergence, we see that $f=g$ , whence $f\in \mathcal {A}$ .

$(iii)$ : Observe that for $|\lambda |>1$ , the function $z\longmapsto \frac {1}{z-\lambda }$ is in $\text {Hol}(\overline {\mathbb {D}})$ , and we have

$$\begin{align*}\frac{1}{z-\lambda}=-\sum_{n=0}^\infty \frac{z^n}{\lambda^{n+1}}. \end{align*}$$

Thus, using (H8),

$$\begin{align*}\begin{aligned} \|(z-\lambda)^{-1}\|_{\mathcal{A}} &\leq \sum_{n=0}^\infty \frac{\|\chi_n\|_{\mathcal{A}}}{|\lambda|^{n+1}}\\ &\leq C \sum_{n=0}^\infty \frac{n^p}{|\lambda|^n}. \end{aligned} \end{align*}$$

It is easy to check that for every $0<x<1$ and every $p\in \mathbb N$ , we have $\displaystyle \sum _{n=0}^\infty n^px^n\leq \frac {p!}{(1-x)^{p+1}}$ , whence

$$\begin{align*}\|(z-\lambda)^{-1}\|_{\mathcal{A}} \lesssim \frac{p!}{\left(1-\frac{1}{|\lambda|}\right)^{p+1}}=\frac{p!|\lambda|^{p+1}}{(|\lambda|-1)^{p+1}}, \end{align*}$$

which concludes the proof.

We now recall the following famous result of Atzmon [Reference Atzmon4].

Theorem 2.9 (Atzmon)

Let E be a Banach space and let $T \in \mathcal {L}(E)$ whose spectrum $\sigma (T) = \{ \zeta _0\}$ for some $\zeta _0\in \mathbb {T}$ . Suppose that there exists $k \ge 0$ such that

(2.4) $$ \begin{align} \left\Vert T^n\right\Vert \underset{n \rightarrow + \infty}{=} O(n^k) \text{~and~} \log \left\Vert T^{-n}\right\Vert \underset{n \rightarrow + \infty}{=} o(\sqrt{n}). \end{align} $$

Then $(T - \zeta _0 I)^k = 0$ .

Using Cauchy’s inequalities, we can obtain the following version which translates growth assumptions (2.4) on the iterates of T into growth assumptions on the resolvant of T. It appears in [Reference Egueh13] and we refer to it for the proof of this corollary.

Corollary 2.10 Let E be a Banach space and let $T \in \mathcal {L}(E)$ whose spectrum $\sigma (T) = \{ \zeta _0\}$ for some $\zeta _0\in \mathbb {T}$ . Suppose that there exists $k \ge 0$ and $c> 0$ such that

$$\begin{align*}\left\Vert (T-\lambda I)^{-1}\right\Vert \le \frac{c \left\lvert \lambda \right\rvert^k}{(\left\lvert \lambda \right\rvert - 1)^k}, \quad \left\lvert \lambda \right\rvert> 1, \end{align*}$$

and for all $\varepsilon> 0$ , there exists $K_\varepsilon> 0$ such that

$$\begin{align*}\left\Vert (T-\lambda I)^{-1}\right\Vert \le K_\varepsilon \exp\left(\frac{\varepsilon}{1-\left\lvert \lambda \right\rvert}\right), \quad \left\lvert \lambda \right\rvert < 1. \end{align*}$$

Then $(T-\zeta _0I)^k = 0$ .

5.1 De Branges–Rovnyak spaces

To every non-constant function b in the closed unit ball of $H^\infty $ , we associate the de Branges–Rovnyak space $\mathcal H(b)$ defined as the reproducing kernel Hilbert space on $\mathbb {D}$ with positive definite kernel given by

$$\begin{align*}k_\lambda^b(z) = \frac{1 - \overline{b(\lambda)}b(z)}{1 - \overline{\lambda}z},\qquad \lambda,z \in \mathbb{D}. \end{align*}$$

It is well known that $\mathcal H(b)$ is contractively contained into $H^2$ , and moreover, it is invariant with respect to $S$ if and only if $\log (1-|b|)\in L^1(\mathbb {T})$ [Reference Fricain and Mashreghi24, Corollary 20.20].

So from now on, we assume that b is a non-constant function in the closed unit ball of $H^\infty $ which satisfies $\log (1-|b|)\in L^1(\mathbb {T})$ , and we denote by $S_b$ the restriction of the shift operator on $\mathcal H(b)$ . Note that, for every $\lambda \in \mathbb {D}$ , the evaluation map $f\longmapsto f(\lambda )$ is continuous on $\mathcal H(b)$ . It is also known that when $\log (1-|b|)\in L^1(\mathbb {T})$ , the set of polynomials $\mathcal P$ is dense in $\mathcal H(b)$ . Hence, $\mathcal H(b)$ satisfies (H1) to (H3). We refer the reader to [Reference Fricain and Mashreghi24, Reference Sarason33] for an in-depth study of de Branges–Rovnyak spaces and their connections to numerous other topics in operator theory and complex analysis.

Now consider $\mathcal {A}=\mathfrak {M}(\mathcal H(b))$ the Banach algebra of multipliers of $\mathcal H(b)$ . Of course $\mathcal {A}$ satisfies (H4a) (and thus of course (H4)) and also (H5) according to Lemma 2.1. Now we immediately get from Theorem 3.1 and Corollary 3.3 the following result.

The previous result leads to the following question.

With regard to the application of Theorem 4.1, we need to recall the notion of angular derivatives. We say that b has an angular derivative in the sense of Carathéodory at $\zeta \in \mathbb {T}$ if b and $b'$ both have a non-tangential limit at $\zeta $ and $\left \lvert b(\zeta ) \right \rvert = 1$ . We denote by $E_0(b)$ the set of such points. It is known that for $\zeta \in \mathbb {T}$ , every function $f \in \mathcal {H}(b)$ has a non-tangential limit at $\zeta $ if and only if $\zeta \in E_0(b)$ . In particular, if $\zeta \in E_0(b)$ , then there exists $C>0$ such that

(5.1) $$ \begin{align} |f(\zeta)|\leq C \|f\|_{\mathcal H(b)},\qquad f\in\mathcal H(b). \end{align} $$

See [Reference Fricain and Mashreghi24, Theorem 21.1]. It is also known [Reference Fricain and Mashreghi24, Theorem 28.37] that for $\zeta \in \mathbb {T}$ , we have

(5.2) $$ \begin{align} \zeta\text{ is an eigenvalue for }S_b^*\text{ if and only if }\zeta\in E_0(b), \end{align} $$

where here $S_b^*$ denotes the adjoint of $S_b$ in the Banach sense. We then get from Theorem 4.1 the following result.

Proof $(i)\implies (ii)$ : If f is cyclic for $S_b$ in $\mathcal H(b)$ , then since $\mathcal H(b)$ is contractively contained in $H^2$ , the function f is also cyclic for $S$ in $H^2$ . Thus, by Beurling’s theorem, f should be outer. On the other hand, assume that $\zeta _0\in E_0(b)$ . Let $\varepsilon>0$ . There exists $p\in \mathcal P$ such that

$$\begin{align*}\|pf-1\|_{\mathcal H(b)}\leq \varepsilon. \end{align*}$$

Hence, according to (5.1), we get

$$\begin{align*}|p(\zeta_0)f(\zeta_0)-1|\leq C\varepsilon. \end{align*}$$

Since $f(\zeta _0)=0$ , this gives $1\leq C\varepsilon $ and thus a contradiction for sufficiently small $\varepsilon $ . Therefore, $\zeta _0\notin E_0(b)$ .

$(ii)\implies (i)$ : Assume that f is outer and $\zeta _0\notin E_0(b)$ . According to (5.2), we know that $\zeta _0$ is not in the point spectrum of $S_b^*$ . Then it follows from Lemma 4.2 that $z-\zeta _0$ is cyclic for $S_b$ . Since $\mathcal H(b)$ satisfies (H1) to (H3) and $\mathfrak {M}(\mathcal H(b))$ satisfies (H4) to (H8), we can apply Theorem 4.1 to get that f is cyclic for $S_b$ in $\mathcal H(b)$ .

The previous result leads to the following question.

Question 5.5 Let b be a function in the closed unit ball of $H^\infty $ such that $\log (1-|b|)\in L^1(\mathbb {T})$ . Can we characterize those b such that there exists $C>0$ and $p\in \mathbb N$ with

$$\begin{align*}\|\chi_n\|_{\mathfrak{M}(\mathcal H(b))}\leq C n^p,\qquad \text{for every }n\geq 0\,? \end{align*}$$

We can give a positive answer in the case when b is a rational (not inner) function in the closed unit ball of $H^\infty $ .

Proposition 5.6 Let b be a rational (not inner) function in the closed unit ball of $H^\infty $ . Then there exists $C>0$ and $p\in \mathbb N$ such that

$$\begin{align*}\|\chi_n\|_{\mathfrak{M}(\mathcal H(b))}\leq C n^p,\qquad \text{for every }n\geq 0. \end{align*}$$

Proof Since b is a rational (not inner) function in the closed unit ball of $H^\infty $ , then it is known that $\mathfrak {M}(\mathcal H(b))=H^\infty \cap \mathcal H(b)$ . See [Reference Fricain, Hartmann and Ross20]. According to Lemma 2.1, we get that

$$\begin{align*}\|\chi_n\|_{\mathfrak{M}(\mathcal H(b))}\lesssim \|\chi_n\|_\infty+\|\chi_n\|_{\mathcal H(b)}=1+\|\chi_n\|_{\mathcal H(b)}. \end{align*}$$

Thus, it is sufficient to prove that

(5.3) $$ \begin{align} \|\chi_n\|_{\mathcal H(b)}\lesssim n^p, \end{align} $$

for some $p\in \mathbb N$ . When b is a rational (not inner) function in the closed unit ball of $H^\infty $ , we know that there exists a unique rational outer function a such that $a(0)>0$ and

$$\begin{align*}|a|^2+|b|^2=1\qquad \text{on }\mathbb{T}. \end{align*}$$

See [Reference Fricain, Hartmann and Ross19]. We may assume that $\|b\|_\infty =1$ , otherwise $\mathcal H(b)=H^2$ (with equivalent norms) and the result is trivial. Thus, a has at least one zero on $\mathbb {T}$ . Factorize a as

(5.4) $$ \begin{align} a(z)=a_1(z)\prod_{i=1}^s (z-\zeta_i)^{m_i}, \end{align} $$

where $\zeta _i\in \mathbb {T}$ , $m_i\geq 1$ , $s\geq 1$ and $a_1$ is rational function without zeros (and poles) in $\overline {\mathbb {D}}$ . If the series expansion of $b/a\in \text {Hol}(\mathbb {D})$ has the form

$$\begin{align*}\frac{b(z)}{a(z)}=\sum_{j=0}^\infty c_j z^j,\qquad |z|<1, \end{align*}$$

then it is known [Reference Fricain and Mashreghi24, Theorem 24.12] that

(5.5) $$ \begin{align} \|\chi_n\|_{\mathcal H(b)}^2=1+\sum_{j=0}^n |c_j|^2. \end{align} $$

From Cauchy’s inequalities, we have

$$\begin{align*}\left\lvert c_j \right\rvert = \left\lvert \frac{(\frac{b}{a})^{(j)}(0)}{j!} \right\rvert \le \inf_{0 < r < 1} \frac{M(r)}{r^j}, \end{align*}$$

where

$$\begin{align*}M(r) = \sup_{\left\lvert z \right\rvert = r} \left\lvert \frac{b(z)}{a(z)} \right\rvert,\qquad 0<r<1. \end{align*}$$

Using (5.4), for $|z|=r$ , we have

$$\begin{align*}\left\lvert a(z) \right\rvert \gtrsim \prod_{i=1}^s \left\lvert z-\zeta_i \right\rvert^{m_i} \ge \prod_{i=1}^s (\left\lvert \zeta_i \right\rvert - \left\lvert z \right\rvert)^{m_i} = (1-r)^N, \end{align*}$$

where $N=\sum _{i=1}^s m_i$ . We get that $M(r)\leq (1-r)^{-N}$ and thus

$$\begin{align*}\left\lvert c_j \right\rvert \lesssim \inf_{0 < r < 1} (1-r)^{-N} r^{-j}. \end{align*}$$

If we introduce $\varphi (r)=(1-r)^{-N}r^{-j}$ , $0<r<1$ , it is not difficult to check that $\varphi $ has minimum at $r=1-\frac {N}{j+N}=\frac {j}{j+N}$ , which gives that

$$\begin{align*}|c_j|\lesssim \left(1+\frac{j}{N}\right)^N \left(1+\frac{N}{j}\right)^j\lesssim j^N\qquad \text{as }j\to\infty. \end{align*}$$

Thus

$$\begin{align*}\sum_{j=0}^n |c_j|^2\lesssim \sum_{j=0}^n j^{2N}\leq n^{2N+1}. \end{align*}$$

Then (5.3) follows from (5.5), which concludes the proof.

5.2 Besov–Dirichlet spaces

For $p \ge 1$ and $\alpha> -1$ , the Besov–Dirichlet space $\mathcal {D}_\alpha ^p$ consists of functions f holomorphic on $\mathbb {D}$ satisfying

$$\begin{align*}\left\Vert f\right\Vert{}_{\mathcal{D}_\alpha^p}^p := \left\lvert f(0) \right\rvert^p + (1 + \alpha)\int_{\mathbb{D}} \left\lvert f'(z) \right\rvert^p (1 - \left\lvert z \right\rvert^2)^\alpha ~dA(z) < \infty. \end{align*}$$

Let us recall that for $p = 2$ and $\alpha = 1$ , $\mathcal {D}_\alpha ^p = H^2$ the Hardy space of the unit disc, and for $p = 2$ and $\alpha = 0$ , $\mathcal {D}_\alpha ^p = \mathcal {D}$ , the classical Dirichlet space. This example of $\mathcal D_\alpha ^p$ was studied in details by Egueh–Kellay–Zarrabi in [Reference Egueh, Kellay and Zarrabi14], which was a source of inspiration for us. See also [Reference Egueh13].

Let us recall that if $1<p<\alpha +1$ , then $H^p$ is continuously embedded in $\mathcal {D}_\alpha ^p$ . Hence, every outer functions $f\in H^p$ is cyclic for the shift in $\mathcal {D}_\alpha ^p$ . See [Reference Kellay, Le Manach and Zarrabi28, Proposition 3.1]. On the other hand, if $p> \alpha + 2$ , then $\mathcal {D}_\alpha ^p \subset A(\mathbb {D})$ becomes a Banach algebra, and consequently, the only cyclic outer functions are the invertible functions. Thus, a function $f\in \mathcal D_\alpha ^p$ which vanishes at least at one point in $\overline {\mathbb {D}}$ is not cyclic for the shift in $\mathcal {D}_\alpha ^p$ . See [Reference Egueh13, Reference Kellay, Le Manach and Zarrabi28].

We will assume from now on that $\alpha +1\leq p\leq \alpha +2$ .

Lemma 5.7 Let $p>1$ such that $\alpha +1\leq p\leq \alpha +2$ and let $\mathcal {A}=\mathcal D_\alpha ^p\cap A(\mathbb {D})$ endowed with the norm

$$\begin{align*}\left\Vert f\right\Vert{}_{\mathcal{A}}^p := \left\Vert f\right\Vert{}_{\infty}^p + \int_{\mathbb{D}} \left\lvert f'(z) \right\rvert^p (1 - \left\lvert z \right\rvert^2)^\alpha ~dA(z). \end{align*}$$

Then $\mathcal D_\alpha ^p$ satisfies (H1) to (H3) and $\mathcal {A}$ satisfies (H4) to (H8).

Proof It is well known that $\mathcal D_\alpha ^p$ satisfies (H1) to (H3). See [Reference Arazy, Fisher and Peetre3, Reference Hedenmalm, Korenblum and Zhu25, Reference Tolokonnikov37, Reference Zhu40]. It is also known that $\mathcal {A}$ is an algebra. Let us justify that $\mathcal {A}$ satisfied (H4b). We obviously have $\mathcal {A}\subset \mathcal {X}=\mathcal D_\alpha ^p$ . Moreover, for $f\in \mathcal {A}$ , let us consider $\sigma _n(f)$ the nth Fejér mean of f. Since $f\in A(\mathbb D)$ , we have $\|\sigma _n(f)-f\|_\infty \to 0$ as $n\to \infty $ . On the other hand, we also have $\|\sigma _n(f)-f\|_{\mathcal D_\alpha ^p}\to 0$ as $n\to \infty $ (see [Reference Holland and Walsh27]). Hence, we get that $\|\sigma _n(f)-f\|_{\mathcal {A}}\to 0$ as $n\to \infty $ , and we conclude that the polynomials are dense in $\mathcal {A}$ .

It is clear that $\mathcal {A}$ satisfies (H5) because for every $\lambda \in \mathbb {D}$ and every $f\in \mathcal {A}$ , we have

$$\begin{align*}|f(\lambda)|\leq \|f\|_\infty\leq \|f\|_{\mathcal{A}}. \end{align*}$$

The fact that $\mathcal {A}$ satisfies (H6), with constant $A\geq 4$ , is a deep result of Tolokonnikov [Reference Tolokonnikov37].

Clearly $\mathcal {A}$ satisfies (H7). So it remains to check that $\mathcal {A}$ satisfies (H8). We have

$$\begin{align*} \|\chi_n\|^p_{\mathcal{A}}&=\|\chi_n\|_\infty^p+\int_{\mathbb{D}} |\chi'_n(z)|^p (1-|z|^2)^\alpha\,dA(z) \\ &=1+n^p\int_{\mathbb{D}} |z|^{(n-1)p}(1-|z|^2)^\alpha\,dA(z)\\ &\leq 1+n^p\int_0^{2\pi}\int_0^1 (1-r^2)^\alpha r\,dr\,\frac{d\theta}{2\pi}\\ &\leq 1+n^p \int_0^1 (1-r)^\alpha\,dr\\ &\leq 1+\frac{n^p}{\alpha+1}, \end{align*}$$

which gives (H8), and concludes the proof.

Using Theorem 3.1 and Corollary 3.3, we (partially) recover the following result due to Egueh–Kellay–Zarrabi in [Reference Egueh, Kellay and Zarrabi14].

It should be noted that Theorem 5.8 is an extension of a result of Brown-Shields [Reference Brown and Shields10]. See also [Reference Aleman1].

As an application of Theorem 4.1, we recover now the following result of Egueh–Kellay–Zarrabi in [Reference Egueh, Kellay and Zarrabi14].

Note that the case of the classical Dirichlet space $\mathcal D$ was discovered by Hedenmalm and Shields [Reference Hedenmalm and Shields26] and generalized by Richter and Sundberg [Reference Richter and Sundberg31]. Theorem 5.9 was already obtained by Kellay, Lemanach and Zarrabi in [Reference Kellay, Le Manach and Zarrabi28] for $\alpha +1<p\leq \alpha +2$ using technics from [Reference Hedenmalm and Shields26]. Thanks to [Reference Hedenmalm and Shields26, Theorem 3], as observed in [Reference Egueh, Kellay and Zarrabi14], Theorem 5.9 remains true under the assumption that $\mathcal Z(f)$ is a countable set.

5.3 Dirichlet type spaces

Given a finite positive Borel measure $\mu $ on the closed unit disc $\overline {\mathbb D}$ , let

$$\begin{align*}U_\mu(z)=\int_{\mathbb{D}} \left(\log\left|\frac{1-\overline{w}z}{z-w}\right|{}^2\right)\frac{d\mu(w)}{1-|w|^2}+\int_{\mathbb{T}} \frac{1-|z|^2}{|\zeta-z|^2}\,d\mu(\zeta),\qquad z\in\mathbb{D}. \end{align*}$$

The function $U_\mu $ is a positive superharmonic function on $\mathbb {D}$ and we associate to it the Dirichlet type space $\mathcal D(\mu )$ defined as the space of analytic functions f on $\mathbb {D}$ satisfying

$$\begin{align*}\int_{\mathbb{D}} |f'(z)|^2 U_\mu(z)\,dA(z)<\infty, \end{align*}$$

where $dA$ stands the normalized area measure. It is known that $\mathcal D(\mu )\subset H^2$ and if for $f\in \mathcal D(\mu )$ , we define

(5.6) $$ \begin{align} \|f\|^2_{\mathcal D(\mu)}=\|f\|_2^2+\int_{\mathbb{D}} |f'(z)|^2 U_\mu(z)\,dA(z), \end{align} $$

it is known that $\mathcal D(\mu )$ is a reproducing kernel Hilbert space. These Dirichlet type spaces are important in model theory. See [Reference Aleman2]. It turns out that these spaces also enter in our general framework. The key result to check that our assumptions are satisfied is the following deep result of Shuabing Luo [Reference Luo29]. Since the result is not exactly stated like this, we shall explain how to get this following version

Theorem 5.10 (S. Luo)

Let $\mu $ be a finite positive measure on $\overline {\mathbb {D}}$ . There exists $C>0$ such that for every $f_1,f_2\in \mathfrak M({\mathcal D(\mu )})$ satisfying

$$\begin{align*}0 < \delta \le \left\lvert f_1 \right\rvert + \left\lvert f_2 \right\rvert \text{ on } \mathbb{D}, \quad\text{and}\quad \|f_1\|_{\mathfrak M({\mathcal D(\mu)})}+\|f_2\|_{\mathfrak M({\mathcal D(\mu)})}\leq 1, \end{align*}$$

there exists $g_1,g_2\in \mathfrak M({\mathcal D(\mu )})$ such that $f_1g_1+f_2g_2\equiv 1$ on $\mathbb {D}$ and

$$\begin{align*}\|g_1\|_{\mathfrak M(\mathcal D(\mu))},\,\|g_2\|_{\mathfrak M(\mathcal D(\mu))}\leq \frac{C}{\delta^4}. \end{align*}$$

Proof Let $f_1,f_2\in \mathfrak M({\mathcal D(\mu )})$ satisfying

$$\begin{align*}0 < \delta \le \left\lvert f_1 \right\rvert + \left\lvert f_2 \right\rvert \text{ on~} \mathbb{D}, \quad\text{and}\quad \|f_1\|_{\mathfrak M({\mathcal D(\mu)})}+\|f_2\|_{\mathfrak M({\mathcal D(\mu)})}\leq 1. \end{align*}$$

Then it is proved in [Reference Luo29] that, for every $h\in \mathcal D(\mu )$ , there exists $\varphi _1,\varphi _2\in \mathcal D(\mu )$ such that

$$\begin{align*}f_1\varphi_1+f_2\varphi_2\equiv h \text{ on } \mathbb{D}, \end{align*}$$

and for $\ell =1,2$ , we have

$$\begin{align*}\|\varphi_\ell\|_{\mathcal D(\mu)}\lesssim \delta^{-4}\|h\|_{\mathcal D(\mu)}. \end{align*}$$

But is known that $\mathcal D(\mu )$ is a reproducing kernel Hilbert space with a complete Nevanlinna–Pick kernel [Reference Shimorin35]. Hence, it satisfies the Toeplitz Corona Theorem. See [Reference Ball, Trent and Vinnikov5, Reference Wick39]. Thus, it follows that there exists $g_1,g_2\in \mathfrak M(\mathcal D(\mu ))$ such that

$$\begin{align*}f_1g_1+f_2g_2\equiv 1 \text{ on } \mathbb{D}, \end{align*}$$

and for $\ell =1,2$ , we have

$$\begin{align*}\|g_\ell\|_{\mathfrak M(\mathcal D(\mu))}\lesssim \delta^{-4}, \end{align*}$$

which concludes the proof.

Proof It is well known that $\mathcal D(\mu )$ satisfies (H1) to (H3). See [Reference Aleman2]. Moreover, $\mathcal {A}$ satisfies trivially (H4a) and (H7), as well as (H5) according to Lemma 2.1. The hypothesis (H6) follows from Luo’s theorem with $A\geq 4$ . It thus remains to check that $\mathfrak M({\mathcal D(\mu )})$ satisfies (H8). According to [Reference Aleman2, Theorem IV.1.9], for every $g\in \mathcal D(\mu )$ , we have

$$\begin{align*}\int_{\mathbb{D}} |g'(z)|^2 U_\mu(z)\,dA(z)=\int_{\overline{\mathbb{D}}} D_z(g)\,d\mu(z), \end{align*}$$

where

$$\begin{align*}D_z(g)=\int_{\mathbb{T}} \left|\frac{g(z)-g(\zeta)}{z-\zeta}\right|^2\,dm(\zeta), \end{align*}$$

and m is the normalized Lebesgue measure on $\mathbb {T}$ . Let $f\in \mathcal D(\mu )$ . Using (5.6), we get that

(5.7) $$ \begin{align} \|\chi_n f\|_{\mathcal D(\mu)}^2&=\|\chi_n f\|_2^2+\int_{\mathbb{D}}|(\chi_n f)'(z)|^2 U_\mu(z)\,dA(z)\notag\\ &=\|f\|_2^2+\int_{\overline{\mathbb{D}}}D_z(\chi_n f)\,d\mu(z). \end{align} $$

Observe now that

$$\begin{align*}D_z(\chi_n f)=\int_{\mathbb{T}} \left|\frac{z^nf(z)-\zeta^nf(\zeta)}{z-\zeta}\right|^2\,dm(\zeta), \end{align*}$$

and straightforward computations show that

$$\begin{align*}\left|\frac{z^nf(z)-\zeta^nf(\zeta)}{z-\zeta}\right|^2\leq 2n^2|f(z)|^2+2\left|\frac{f(z)-f(\zeta)}{z-\zeta}\right|^2. \end{align*}$$

We then deduce that

$$\begin{align*}D_z(\chi_n f)\leq 2 n^2 \|f\|_2^2+2 D_z(f), \end{align*}$$

whence, according to (5.7), we obtain

$$ \begin{align*} \|\chi_n f\|_{\mathcal D(\mu)}^2 &\leq \|f\|_2^2+2n^2 \mu(\overline{\mathbb{D}}) \|f\|_2^2 +2\int_{\overline{\mathbb{D}}}D_z(f)\,d\mu(z) \\ &\leq 2 \|f\|_{\mathcal D(\mu)}^2+2n^2 \mu(\overline{\mathbb{D}})\|f\|_{\mathcal D(\mu)}^2\\ &\lesssim n^2 \|f\|_{\mathcal D(\mu)}^2. \end{align*} $$

Therefore, we deduce that $\|\chi _n\|_{\mathfrak M(\mathcal D(\mu ))}\lesssim n$ , which proves (H8).

Using Theorem 3.1 and Corollary 3.3, we can recover a partial version of a result of Richter–Sundberg [Reference Richter and Sundberg30, Corollary 5.5] and Aleman [Reference Aleman2].

It should be noted that Theorem 5.12 is proved in [Reference Richter and Sundberg30] with $N=1$ and with the weaker assumption that $f,g\in \mathcal D(\mu )$ but with a measure $\mu $ on $\mathbb {T}$ . The case of a measure on the closed unit disc is obtained in [Reference Aleman2] with $N=1$ . In both papers, the result is obtained using radial approximations technics.

As an application of Theorem 4.1, we get the following result.

It should be noted that in [Reference El-Fallah, Elmadani and Kellay15] (in the case when $\mu $ is a measure on $\mathbb {T}$ ), O. El-Fallah, Y. Elmadani and K. Kellay proved that $\zeta $ is a bounded point evaluation of $\mathcal D(\mu )$ if and only if $c_\mu (\zeta )>0$ , where $c_\mu $ is the Choquet capacity associated to $\mathcal D(\mu )$ . Moreover, they also showed that when $\mu $ has countable support, then a function $f\in \mathcal D(\mu )$ is a cyclic vector for the shift precisely when f is an outer function and $c_\mu (\mathcal Z(f))=0$ .