Spectra of Weighted Composition Operators on Analytic Function Spaces

Abstract

Let E be a complex Banach space with open unit ball $B_E.$ For analytic self-maps $\varphi$ of $B_E$ with $\varphi (0) =0,$ we investigate the spectra of weighted composition operators $uC_\varphi$ acting on a large class of spaces of analytic functions. This class contains, for example, weighted Banach spaces of $H^\infty$-type on $B_E$, weighted Bergman spaces $A^p_\alpha ({\mathbb {B}}_N)$ and Hardy spaces $H^p({\mathbb {B}}_N).$ We present a general approach for deducing new information about the spectrum and for estimating the essential spectral radius of $uC_\varphi .$

1 Introduction

Throughout this article, E stands for a complex Banach space of arbitrary dimension and $B_E=\{x\in E: \Vert x\Vert <1\}$ for its open unit ball. Moreover, let $\varphi : B_E \rightarrow B_E$ be an analytic mapping and $u\in H(B_E),$ where $H(B_E)$ is the space of analytic functions on $B_E.$ Recall that a mapping is analytic if it is Fréchet differentiable at every point in its domain. Each such pair $(\varphi , u)$ induces via composition and multiplication a weighted composition operator $uC_\varphi (f) = u (f\circ \varphi )$ which preserves $H(B_E).$ Our object of study is the operator $uC_\varphi$ acting on a Banach space, $X(B_E),$ of analytic functions on $B_E,$ specifically, its spectrum $\sigma (uC_\varphi ).$ This is a topic of current interest; see for instance [4, 5, 7, 11, 13, 16, 25] and other references quoted below. Very little is known about the spectrum of the composition operator $C_\varphi$ acting on classical analytic function spaces for a non-univalent symbol $\varphi$ of the open unit ball ${\mathbb {B}}_N$ in ${\mathbb {C}}^N$.

Some information about the space $X(B_E)$ is unavoidable to obtain interesting results for $uC_\varphi$ acting on $X(B_E)$. Therefore, the space $X(B_E)$ will always satisfy five conditions stated in Sect. 2 that are fulfilled by very natural and common Banach spaces of analytic functions like the weighted Bergman spaces, $A^p_\alpha (\mathbb B_N),$ the Hardy spaces, $H^p({\mathbb {B}}_N)$, $1 \le p < \infty ,$ and, even in the infinite dimensional setting, the weighted spaces of analytic functions $H^\infty _\upsilon (B_E)$ as we show in the examples of the next section.

Our quite general approach allows us to extend earlier results. For instance, the main result (Theorem 15) of Cowen and MacCluer in [9] containing information about the spectrum for (weighted) Hilbert spaces of analytic functions and composition operators with univalent and not unitary on any slice symbol that fixes the origin, is generalized by Corollary 4.14 where it is only required that the norm of the derivative mapping at the fixed point 0 be less than 1 and neither the analytic function spaces are required to be Hilbert. In the context of the weighted spaces of analytic functions $H^\infty _\upsilon (B_E)$ and weighted composition operators, we extend to arbitrary dimensions the description of the spectrum in case $E={\mathbb {C}}$ by Aron and Lindström [2], and the one of Yuan and Zhou [27] given for the case $\upsilon \equiv 1.$ Also the results for weighted Bergman spaces are new.

The main result (Theorem 4.9) provides conditions for the spectrum to contain a disc centered at 0 and all finite products of eigenvalues of the derivative mapping $\varphi '(0)\in {\mathcal {L}}(E),$ the Banach algebra of all bounded operators on E. The radius of such disc is closely related to the essential spectral radius of the operator: actually, equal in the case of $H^\infty _\upsilon ({\mathbb {B}}_N)$ in which we give a complete description of the spectrum. This relationship motivated the estimations in Sect. 3 where we extend Lefèvre’s Theorem 2.5 in [20] to weighted spaces of analytic functions using quite different techniques.

Our standing assumptions are $\varphi (0)=0$ with $\Vert \varphi '(0)\Vert <1$ and that the range of $\varphi$ is a relatively compact set. The core of our results is Lemma 4.8. It is an elaboration on the nice sharpening in [9] of Kamowitz technique [18] that has been further exploited by many other authors [3, 12, 15, 22, 28]. It strongly depends on the existence of (iterated) interpolating sequences that in the infinite dimensional setting was initiated in [12] and developed in [14]. Such suitable interpolating sequences are known to exist when E is a Hilbert space or $E=C_0({\mathcal {X}}),$ ${\mathcal {X}}$ a locally compact Hausdorff topological space. Thus, the mentioned standing assumptions suffice to get the results for both the ball and the polydisc.

2 Conditions and Examples

Recall that $H_b(B_E):=\{f\,{:}~B_E \rightarrow {\mathbb {C}}\,{:}~f \text { analytic and bounded on balls of}~\text {radius less than } 1\}$ is a Fréchet algebra when endowed with the topology of uniform convergence on balls of radius less than 1. By $H^\infty (B_E)$ we denote the subspace of $H_b(B_E)$ of bounded functions endowed with the topology of uniform convergence on $B_E.$

We deal with a vector space $X(B_E)$ of analytic functions on $B_E$ and a norm on it $\Vert \cdot \Vert$ that renders $X(B_E)$ a Banach space. As usual, for each $x\in B_E,$ $\delta _x$ is the evaluation functional defined by $\delta _x(f) = f(x)$ for all $f \in X(B_E)$. We assume that $X(B_E)$ contains the constant functions, so then all $\delta _x$ are non-zero.

The Banach space $X(B_E)$ is assumed to satisfy the following conditions:

(I) For every $x\in B_E$, $\delta _x:X(B_E) \rightarrow {\mathbb {C}}$ is a linear bounded functional, and the closed unit ball $\mathbf{B} =\{f\in X(B_E): ||f||\le 1\}$ of $X(B_E)$ is compact with respect to the compact-open topology $\tau _0.$

In particular, for each $x\in B_E$ there is a $f_x\in X(B_E)$ with $||f_x||\le 1$ such that $||\delta _x||_X = f_x(x).$ Moreover, by the Dixmier–Ng theorem, there is a Banach space $^*X(B_E)$ whose dual space is isometrically isomorphic to $X(B_E)$ and, further, the mapping $x\in B_E \mapsto \delta _x \in {^*X(B_E)}$ is holomorphic because it is weakly holomorphic. Actually, $^*X(B_E)$ is the subspace of $X(B_E)^*$ of the elements that are $\tau _0$-continuous on bounded sets.

(II) For every $g\in H^\infty (B_E) \text { and } f\in X(B_E),$ the function $fg\in X(B_E).$

If both (I) and (II) hold, the multiplication operator $M_g(f)=fg$ is continuous on $X(B_E),$ thanks to the closed graph theorem. A subsequent application of the closed graph theorem shows the existence of a constant $M_X>0$ such that $\Vert M_g\Vert \le M_X \Vert g\Vert _\infty .$

(III) $X(B_E) \subset H_b(B_E).$

This inclusion mapping is a continuous embedding thanks to the closed graph theorem.

Denote by $P_n f$ the n-th term of the Taylor series at 0 of the analytic function $f \in X(B_E)$. For $m\in {\mathbb {N}},$ let

$$\begin{aligned} X_{m} (B_E)=\left\{ f \in X (B_E) : P_n f =0 \text{ for } n = 0, 1, \ldots , m-1 \right\} . \end{aligned}$$

That is, a function in $X(B_E)$ belongs to $X_{m}(B_E)$ if the first m terms of its Taylor series at 0 vanish. Equivalently, $f\in X(B_E)$ belongs to $X_{m}(B_E)$ if, and only if, $\frac{f(x)}{\Vert x\Vert ^m}$ is bounded in some punctured ball centered at 0.

(IV) For each $m\in {\mathbb {N}}$ there is a constant $c(m) > 0$ (depending also on the norm of $X(B_E)$) such that for all $x\in B_E$ we have

$$\begin{aligned} ||\delta _x||_{X_m} \le c(m) ||x||^m ||\delta _x||, \end{aligned}$$

where $X_{m} (B_E)$ is endowed with norm of $X(B_E)$ and $\Vert \delta _x\Vert _{X_m}$ denotes the norm of $\delta _{x}$ restricted to $X_m.$

What can we say about this condition in the complex plane? Now, $B_{{\mathbb {C}}}={\mathbb {D}}$ is the open unit disk. If for a positive integer m we have that $X_m({\mathbb {D}}) = z^mX(\mathcal {{\mathbb {D}}}),$ then there is a constant $c(m) > 0$ such that

$$\begin{aligned} |f(z)| \le c(m)|z|^m\Vert f\Vert _{X}\Vert \delta _z\Vert \end{aligned}$$

(2.1)

for every $f \in X_m({\mathbb {D}})$ and $z \in {\mathbb {D}}$. Indeed, this can be proved following the proof of [3, Proposition 3.3] and for completeness we give the details. The map $f\in X_m({\mathbb {D}}) \mapsto f/z^m\in X({\mathbb {D}})$ is well defined, linear and continuous by the closed graph theorem. Hence there is $c(m) >0$ such that $||f/z^m||_X \le c(m) ||f||_{X_m}$ for each $f\in X_m({\mathbb {D}})$. Now, for $0\ne w\in {\mathbb {D}}$ and $f\in X_m({\mathbb {D}})$, we obtain

$$\begin{aligned} |f(w)| = |w|^m |f(w)/ w^m| \le |w|^m ||f/z^m||_X ||\delta _w|| \le c(m) |w|^m ||f||_{X_m} ||\delta _w||. \end{aligned}$$

Notice that from this result, we can obtain Propositions 2 and 11 in [22].

(V) For every $0<r<1,$ consider $K_r(f)(x)=f(rx).$ The operator $K_r :X(B_E) \rightarrow X(B_E)$ is well defined and $\Vert K_r\Vert \le 1.$ In case $\dim E< \infty ,$ the operator $K_r$ is compact.

The operator $uC_\varphi : X(B_E) \rightarrow X(B_E)$ will be assumed to be bounded. Since $u=uC_\varphi (1),$ we get that $u \in X(B_E).$

Notice that whenever $\varphi (B_E)$ is a relatively compact set strictly inside $B_E,$ $C_\varphi$ is a compact operator: for any net $(f_i)\subset {\mathbf {B}}$ that we may suppose by (II) to be $\tau _0$-convergent to some $g\in {\mathbf {B}},$ we have that $(f_i \circ \varphi )$ is uniformly convergent to $g\circ \varphi$ in $H^\infty (B_E),$ hence convergent in $X(B_E).$

Next, we list a number of spaces satisfying the above conditions to which our main result applies.

2.1 Examples

(a) The weighted space of analytic functions

$$\begin{aligned} H^\infty _\upsilon (B_E):=\left\{ f: B_{E} \rightarrow {\mathbb {C}}: f \text { is analytic and } \Vert f\Vert _\upsilon =\sup _{x \in B_{E}} \upsilon (x) |f(x)|< \infty \right\} \end{aligned}$$

is a Banach space when endowed with the $\Vert \cdot \Vert _\upsilon$ norm. Here, $\upsilon :B_E\rightarrow (0,\infty )$ is a weight, that is, a continuous, bounded and norm non-increasing function, in particular, $\upsilon (x)=\upsilon (y) \text { if } \Vert x\Vert =\Vert y\Vert$. For example, $\upsilon _\alpha (x) = ( 1- ||x||^2)^\alpha$ with $\alpha > 0$ is such a weight. Moreover, the associated weight of $\upsilon$ is defined by $\tilde{\upsilon }(x)=\frac{1}{||\delta _x||}, x \in B_E.$ Notice that for the constant weight $\upsilon (x)=1,$ $H^\infty _\upsilon (B_E)=H^\infty (B_E).$

Using Montel’s theorem [6, Theorem 17.21] it follows that condition (I) holds. Next, we check condition (IV). For given $m \in {\mathbb {N}},$ we need to show that there exists a constant c(m) depending only on m, so that if $f \in H^\infty _{v,m}(B_E)$ and $x \in B_E$, then

$$\begin{aligned} |f(x)| \le c(m)\Vert x \Vert ^m \Vert \delta _{x} \Vert \Vert f \Vert _{v}. \end{aligned}$$

Indeed, for $\xi \in E$ such that $\Vert \xi \Vert =1,$ consider the function $f_\xi : {\mathbb {D}} \rightarrow {\mathbb {C}}$, $f_\xi (z)=f(z\xi )$, where $f\in H^\infty _{v,m}(B_E).$ Moreover, define $w_{\xi }(z) = v(z \xi ).$ By radiality of the weight v,  it follows that $w_\xi (z) = w_{\xi }(|z|).$ Clearly, $f_{\xi } \in H_{w_\xi , m}^\infty ({\mathbb {D}})$ and $\Vert f_{\xi } \Vert _{w_\xi } \le \Vert f \Vert _v.$ Now since $H_{w_\xi , m}^\infty ({\mathbb {D}})=z^m H_{w_\xi }^\infty ({\mathbb {D}}),$ we may apply (2.1) to get for $0\ne x \in B_E$ that

$$\begin{aligned} |f(x)| = |f_{\frac{x}{\Vert x \Vert }}\left( \Vert x \Vert \right) \le c(m) \Vert \delta _{||x||} \Vert \Vert f \Vert _v\Vert x \Vert ^m, \end{aligned}$$

and since $\Vert \delta _{||x||} \Vert \le \Vert \delta _{x} \Vert ,$ the statement follows. Also, the other conditions are easily seen to be satisfied.

(b) The standard weighted Bergman space $A^p_\alpha (\mathbb B_N)$, $\alpha > -1, p\ge 1$, is the set of all analytic functions on ${\mathbb {B}}_N$ such that

$$\begin{aligned} ||f||^p_{A^p_\alpha } = \int _{{\mathbb {B}}_N} |f(z)|^p c_\alpha ( 1 - |z|^2)^\alpha \mathrm{d}v(z) < \infty , \end{aligned}$$

where $\mathrm{d}v(z)$ is the normalized volume measure on ${\mathbb {B}}_N$ and $c_\alpha =\frac{\Gamma (N+\alpha +1)}{N!\Gamma (\alpha +1)}.$ The set of polynomials are dense in $A^p_\alpha ({\mathbb {B}}_N).$ By [26], for $z\in {\mathbb {B}}_N$ we have

$$\begin{aligned} ||\delta _z||= \frac{1}{\left( 1 - |z|^2\right) ^{\frac{N+1+\alpha }{p}}}. \end{aligned}$$

(2.2)

By Montel’s theorem and Fatou’s lemma it can be seen that condition (I) holds. The only condition that we need to verify is (IV), since the other conditions are clearly valid. For $\xi \in {\mathbb {S}}_N$, the map $f\in A^p_\alpha ({\mathbb {B}}_N)\mapsto f_{\xi }\in A^p_{N+\alpha -1}({\mathbb {D}})$ is bounded by Theorem 1.1. in [19], so there is a constant $c(N) > 0$ such that

$$\begin{aligned} ||f_\xi ||_{ A^p_{N+\alpha -1}({\mathbb {D}})}\le c(N) ||f||_{ A^p_\alpha ({\mathbb {B}}_N)}. \end{aligned}$$

If $f\in A^p_{\alpha ,m}({\mathbb {B}}_N),$ then $f_{\xi }\in A^p_{N+\alpha -1,m}({\mathbb {D}})=z^m A^p_{N+\alpha -1}({\mathbb {D}}),$ so by (2.1) we obtain for $0\ne z \in {\mathbb {B}}_N$ that

$$\begin{aligned} |f(z)| = |f_{\frac{z}{| z|}}(|z|) \le c(m)c(N)|z|^m\Vert f\Vert _{A^p_\alpha ({\mathbb {B}}_N)}\Vert \delta _z\Vert . \end{aligned}$$

(c) The Hardy spaces $H^p({\mathbb {B}}_N)$, $1 \le p < \infty$, are defined by

$$\begin{aligned} H^p({\mathbb {B}}_N) = \left\{ f \in H({\mathbb {B}}_N): ||f||^p _{H^p}= \sup _{0<r<1} \int _{{\mathbb {S}}_N} |f(r\zeta )|^p \mathrm{d}\sigma (\zeta ) < \infty \right\} , \end{aligned}$$

where ${\mathbb {S}}_N$ denotes the unit sphere in ${\mathbb {C}}^N$ and $\sigma$ is the normalized surface measure on it. The set of polynomials are dense in $H^p({\mathbb {B}}_N).$ It is known [29] that for $z\in {\mathbb {B}}_N,$ we have

$$\begin{aligned} ||\delta _z||= \frac{1}{(1 - |z|^2)^{\frac{N}{p}}}. \end{aligned}$$

(2.3)

For condition (I) we use Montel’s theorem and Fatou’s lemma. For $\xi \in {\mathbb {S}}_N$, using Theorem 1.1. in [19], we obtain that the map $f\in H^p({\mathbb {B}}_N)\mapsto f_{\xi }\in A^p_{N-2}({\mathbb {D}})$ is bounded. Here, $A^p_{-1}({\mathbb {D}})= H^p({\mathbb {D}}).$ Therefore, there is a constant $c(N) > 0$ such that

$$\begin{aligned} ||f_\xi ||_{ A^p_{N-2}({\mathbb {D}})}\le c(N) ||f||_{ H^p(\mathbb B_N)}. \end{aligned}$$

For $f\in H^p_{m}({\mathbb {B}}_N),$ then $f_{\xi }\in A^p_{N-2,m}({\mathbb {D}})=z^m A^p_{N-2}({\mathbb {D}}),$ so using (2.1) we get for $0\ne z \in {\mathbb {B}}_N$ that

$$\begin{aligned} |f(z)| = |f_{\frac{z}{| z|}}(|z|) \le c(m)c(N)|z|^m\Vert f\Vert _{H^p({\mathbb {B}}_N)}\Vert \delta _z\Vert . \end{aligned}$$

All the other conditions can easily be verified.

(d) The weighted Hardy spaces of bounded type ${\mathcal {H}}({\mathbb {B}}_N)$ introduced by Cowen and MacCluer in [9] also satisfy the above five conditions. See the paper to verify it: Condition (I) is recalled in the bottom line of page 227. Condition (II) is [9, Proposition 1]. Condition (IV) follows from their computations in page 227 for the reproducing kernels, that is, the evaluation functionals:

$$\begin{aligned} \Vert \delta _z\Vert ^2_{X_m}= & {} \sum _{s=m}^\infty \frac{(N-1+s)!|z|^{2s}}{(N-1)!s!}\frac{1}{\tau (s)^2}\\= & {} |z|^{2m}\sum _{j=0}^\infty |z|^{2j}\frac{(N-1+j+m)!}{(N-1)!(j+m)!}\frac{1}{\tau (j+m)^2}\\\le & {} |z|^{2m}\sum _{j=0}^\infty \frac{(N-1+j)!(N-1+j+1)\cdots (N-1+j+m)}{(N-1)!j!(j+1)\cdots (j+m)}\frac{b^{-m}}{\tau (j)^2} \\= & {} |z|^{2m}\sum _{j=0}^\infty \frac{(N-1+j)!}{(N-1)!j!}\left( \left( 1 +\frac{N-1}{j+1}\right) \cdots \left( 1+\frac{N-1}{j+m}\right) \right) \frac{b^{-m}}{\tau (j)^2}\\\le & {} Q(m)|z|^{2m}\Vert \delta _z\Vert ^2b^{-m}, \end{aligned}$$

where b is the assumed constant satisfying $\frac{\tau ^2(s+1)}{\tau ^2(s)}\ge b>0$ and $Q(m)=\prod _{k=1}^m(1+\frac{N-1}{k}).$ Notice also that $||\delta _z||\rightarrow \infty$ when $|z|\rightarrow 1$ by Proposition 2 in [9] and that the set of polynomials is dense in ${\mathcal {H}}({\mathbb {B}}_N).$

3 The Essential Spectral Radius

Recall that for the essential spectral radius of an operator T,  we have that $r_e(T)=\inf _{n} \root n \of {\Vert T^n\Vert _e},$ and by $\varphi _n$ we denote the n-fold iterate of $\varphi$, so that $\varphi _n = \varphi \circ \varphi \circ \cdots \circ \varphi$ (n times). In this section, we obtain estimates for the essential norm and spectral radius of the weighted composition operator $uC_\varphi .$

We first consider the operators $C_\varphi \circ K_r= C_{r\varphi }$ where $0< r < 1.$

Lemma 3.1

Suppose that $\varphi (\lambda B_E)$ is a relatively compact subset of E for all $0<\lambda <1.$ Then every $C_\varphi \circ K_r: \big (H_b(B_E),\tau _0\big ) \rightarrow H_b(B_E)$ defines a linear continuous mapping and $\{C_\varphi \circ K_r: 0<r<1 \}$ is an equicontinuous family.

Proof

The balanced hull, $L_\lambda ,$ of $\varphi (\lambda B_E)\subset \lambda B_E$ is a relatively compact set strictly inside $B_E$ for all $0<\lambda <1.$

Since for all $0<r<1,$

$$\begin{aligned} \Vert (C_\varphi \circ K_r)(f)\Vert _{\lambda B_E}=\sup _{\Vert x\Vert <\lambda }|f(r\varphi (x))|\le \sup _{y\in L_\lambda }|f(y)|=\Vert f\Vert _{L_\lambda }, \end{aligned}$$

$\{C_\varphi \circ K_r: 0<r<1 \}$ is an equicontinuous family. $\square$

The closed unit ball ${\mathbf {B}}$ of $X(B_E)$ is a compact subset of $\big (H_b(B_E),\tau _0\big )$ by condition (I).

Lemma 3.2

Suppose that $\varphi (\lambda B_E)$ is a relatively compact subset of E for all $0<\lambda <1.$ For every $0<\lambda <1,$

$$\begin{aligned} \lim _{r\rightarrow 1} \sup _{f\in {\mathbf {B}}} \sup _{\Vert \varphi (x)\Vert <\lambda }|f(r\varphi (x))-f(\varphi (x))|=0. \end{aligned}$$

Proof

Let us see that for all $f\in H_b(B_E),$ $\lim _{r\rightarrow 1}(C_\varphi \circ K_r)(f)=C_\varphi (f)$ in $H_b(B_E).$ Fix $\epsilon >0$ and $0<\lambda <1.$ The uniform continuity of f in $\lambda B_E$ leads to some $\delta >0$ such that $|f(u)-f(v)|<\epsilon$ if $u,v\in \lambda B_E$ and $\Vert u-v\Vert <\delta .$ Therefore, if $r>1-\delta ,$ one has $\Vert r\varphi (x)-\varphi (x)\Vert <\delta .$ So, for all $x \text { with } \Vert \varphi (x)\Vert < \lambda ,$ we get $|f(r\varphi (x))-f(\varphi (x))|<\epsilon ,$ that is, $\Vert (C_\varphi \circ K_r)(f)-C_\varphi (f)\Vert _{\lambda B_E}<\epsilon .$

Therefore, $\{C_\varphi \circ K_r: 0<r<1 \}$ converges to $C_\varphi$ for the topology of the pointwise convergence, hence also for the topology of uniform convergence on compact subsets of $\big (H(B_E),\tau _0\big )$ since they coincide on compact subsets due to [24, III. 4.5]. So the statement follows because ${\mathbf {B}}$ is $\big (H(B_E),\tau _0\big )$-compact.

$\square$

Proposition 3.3

Assume that $\varphi ( B_E)$ is a relatively compact subset of E. For the weighted composition operator $u C_{\varphi }: X(B_E)\rightarrow H^\infty _v(B_E),$ we have that

$$\begin{aligned} \Vert u C_{\varphi }\Vert _e \le 2 \lim _{s\rightarrow 1} \sup _{\Vert \varphi (x)\Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert _X}{\Vert \delta _x\Vert }. \end{aligned}$$

Proof

First of all, notice that for $0< r <1,$ $\big (C_{\varphi } \circ K_r\big )(f)(x)=f(r\varphi (x)),$ so $C_{\varphi } \circ K_r:X(B_E)\rightarrow H^\infty (B_E)$ is a compact operator as a composition operator whose symbol $r\varphi$ lies in a compact subset of $B_E.$ Since $u\in H^\infty _v(B_E),$ also the operators $uC_{\varphi } \circ K_r:X(B_E)\rightarrow H^\infty _v(B_E)$ are compact.

Next, by using that $v(x) \le {{\tilde{v}}}(x) = \frac{1}{||\delta _x||}$ and $u\in H^\infty _{\tilde{v}}(B_E),$ we estimate $\Vert u C_{\varphi } \circ K_r- u C_{\varphi }\Vert \le$

$$\begin{aligned}&\sup _{f\in {\mathbf {B}}} \sup _{\Vert \varphi (x)\Vert <s}\frac{|u(x)|}{\Vert \delta _x\Vert }\big |f(r\varphi (x))-f(\varphi (x))\big |\nonumber \\&\quad +\sup _{f\in {\mathbf {B}}} \sup _{\Vert \varphi (x)\Vert \ge s}\frac{|u(x)|}{\Vert \delta _x\Vert }\big |f(r\varphi (x))-f(\varphi (x))\big |. \end{aligned}$$

(3.1)

Concerning the first summand, we have

$$\begin{aligned}&\sup _{f\in {\mathbf {B}}} \sup _{\Vert \varphi (x)\Vert<s}\frac{|u(x)|}{\Vert \delta _x\Vert }\big |f(r\varphi (x))-f(\varphi (x))\big |\\&\quad \le \Vert u\Vert \sup _{f\in {\mathbf {B}}}\sup _{\Vert \varphi (x)\Vert <s}\big |f(r\varphi (x))-f(\varphi (x) )\big |. \end{aligned}$$

So, we may apply Lemma 3.2 to conclude that for fixed $0<s<1,$ the first summand tends to 0 whenever $r\rightarrow 1.$

Concerning the second summand, realize that

$$\begin{aligned} f(r\varphi (x))-f(\varphi (x))= K_r(f)(\varphi (x))-f(\varphi (x))=\left( K_r-Id\right) (f)(\varphi (x)). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{\Vert \delta _x\Vert }\big |f(r\varphi (x))-f(\varphi (x))\big |= & {} \frac{1}{\Vert \delta _x\Vert }\Vert \delta _{\varphi (x)}\Vert _X \left| \frac{1}{\Vert \delta _{\varphi (x)}\Vert _X}\big (K_r-Id\big )(f)(\varphi (x)) \right| \\\le & {} \frac{\Vert \delta _{\varphi (x)}\Vert _X}{\Vert \delta _x\Vert }\cdot 2. \end{aligned}$$

So the second summand is bounded from above by $2 \cdot \sup _{\Vert \varphi (x)\Vert \ge s}\frac{|\psi (x)|\Vert \delta _{\varphi (x)}\Vert _X}{\Vert \delta _x\Vert }.$

Therefore, $\Vert u C_{\varphi }\Vert _e \le \liminf _{r\rightarrow 1}\Vert u C_{\varphi } \circ K_r-\psi C_{\varphi }\Vert \le 2 \cdot \sup _{\Vert \varphi (x)\Vert \ge s}\frac{|u(x)|\Vert \delta _{\varphi (x)}\Vert _X}{\Vert \delta _x\Vert }.$ $\square$

Proposition 3.4

Assume that $\varphi ( B_E)$ is a relatively compact subset of E. There exists $M_X >0$ such that for the weighted composition operator $u C_{\varphi }:H^\infty _v(B_E)\rightarrow X(B_E),$ we have

$$\begin{aligned} \Vert u C_{\varphi }\Vert _e \ge M_X^{-1} \lim _{s\rightarrow 1} \sup _{\Vert \varphi (x)\Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert _X}. \end{aligned}$$

Proof

If for some $s<1,\; \{x: \Vert \varphi (x)\Vert >s\}=\emptyset ,$ then the right hand side is 0,  and we are done. So we are left in the case that $\varphi (B_E)$ does not lie strictly inside $B_E.$

We can find a sequence $(x_n)\in B_E$ such that $\lim _n \Vert \varphi (x_n)\Vert =1$ and $\lim _n \frac{ |u(x_n|\Vert \delta _{\varphi (x_n)}\Vert }{\Vert \delta _{x_n}\Vert _X}=\lim _{s\rightarrow 1} \sup _{\Vert \varphi (x)\Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert _X}.$ Without loss of generality, we may assume that also $\big (\varphi (x_n)\big )$ converges to some $a\in B_E$ with $\Vert a\Vert =1.$ Let $l\in E^*,\; \Vert l\Vert =1$ such that $\Vert a\Vert =l(a).$ Thus, $\lim _n l\big (\varphi (x_n)\big )=1.$ Put $z_n=l\big (\varphi (x_n)\big ).$

As shown in the proof of [17, Theorem 3.1] there is a subsequence of $(z_n)$ that we still denote the same, there are functions $f, g_n \in A({\mathbb {D}})$, two sequences of increasing positive integers $(n_k)$ and $(m_k)$, and a sequence of complex numbers $(c_k)$ with $|c_k| <1,$ such that

$$\begin{aligned} \sum _{k=1}^{\infty }{|c_kf^{m_k}(z)g_{n_k}(z)|} \le 1\quad \text { for all } z,\;|z|\le 1 \end{aligned}$$

(3.2)

and

$$\begin{aligned} c_k f^{m_k}(z_k) g_{n_k}(z_k) > 1- \left( \frac{1}{2}\right) ^k \ \text {for all} \ k. \end{aligned}$$

(3.3)

By condition (I), for each $k \in {\mathbb {N}}$ we can also find a function $f_k \in H^\infty _v(B_E)$ such that $\Vert f_k\Vert \le 1$ and that

$$\begin{aligned} \Vert \delta _{\varphi (x_k)}\Vert = f_k(\varphi (x_k)). \end{aligned}$$

(3.4)

Now, consider $F_k :=M_{(c_k f^{m_k} g_{n_k})\circ l}(f_k) .$ According to condition (II), the sequence $(F_{k})\subset H^\infty _v(B_E)$ and

$$\begin{aligned} \big \Vert F_k\big \Vert \le \big \Vert M_{(c_k f^{m_k} g_{n_k})\circ l}\big \Vert \big \Vert f_k\big \Vert \le M_X \big \Vert \left( \left( c_k f^{m_k} g_{n_k}\right) \circ l\right) \big \Vert _{\infty } \big \Vert f_k\big \Vert \le M_X. \end{aligned}$$

Let $T:H^\infty _v(B_E)\rightarrow X(B_E)$ be a compact operator. By (3.2) the map $(\xi _k)_k \mapsto \sum _{k=1}^\infty \xi _k F_k$ is a well-defined, bounded operator from $c_0$ into $H^\infty _v(B_E).$ Consequently, the sequence $(F_{k})$ converges weakly to zero and $||T(F_k)||\rightarrow 0$ in $X(B_E)$. Thus using condition (I), we get

$$\begin{aligned} M_X \ \Vert uC_{\varphi } - T\Vert&\ge \Vert (uC_{\varphi } - T)F_k\Vert \ge \Vert (uC_{\varphi })F_k\Vert -\Vert T(F_k)\Vert \\&\ge \frac{1}{\Vert \delta _{x_k}\Vert _X} |u(x_{k})|\cdot \left\| \delta _{\varphi (x_{k})}\right\| \left( 1- \left( \frac{1}{2}\right) ^k\right) - \Vert T(F_k)\Vert , \end{aligned}$$

and we are done. $\square$

The above two results yield an extension of [20, Theorem 2.5] to the weighted spaces case. We are now able to state the following essential spectral radius result.

Corollary 3.5

Assume that $\varphi ( B_E)$ is a relatively compact subset of E. For the weighted composition operator $u C_\varphi$ acting on $H^\infty _v(B_E),$ we have that

$$\begin{aligned} r_e(u C_\varphi ) = \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} \Vert \varphi _n(x) \Vert \ge s \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert }}. \end{aligned}$$

The proof of the next result is much easier than Proposition 3.4 and the result can be applied to the spaces $H^p({\mathbb {B}}_N)$ and $A^p_\alpha ({\mathbb {B}}_N)$.

Proposition 3.6

Assume that $||\delta _x||\rightarrow \infty$ when $||x||\rightarrow 1$ and that the continuous polynomials are dense in $X(B_E)$. Then for the weighted composition operator $u C_{\varphi }$ acting on $X(B_E),$ we have that

$$\begin{aligned} \Vert u C_{\varphi }\Vert _e \ge \lim _{s\rightarrow 1} \sup _{\Vert x \Vert \ge s} \frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert }. \end{aligned}$$

Proof

Take a sequence $(x_n)\subset B_E$ with $||x_n||\rightarrow 1$ when $n\rightarrow \infty$ such that

$$\begin{aligned} \lim _{s\rightarrow 1} \sup _{\Vert x \Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert }= \lim _{n\rightarrow \infty }|u(x_n)|\frac{||\delta _{\varphi (x_n)}||}{||\delta _{x_n}||}. \end{aligned}$$

Let $l_n:= \frac{\delta _{x_n}}{||\delta _{x_n}||}\in X(B_E)^*,$ then $l_n\rightarrow 0$ weak$^*$ in $X(B_E)^*.$ Indeed, clearly for any continuous polynomial P,  $\lim _n l_n(P)=\lim _n\frac{P(x_n)}{||\delta _{x_n}||}=0,$ and on the closed unit ball of $X(B_E)^*,$ the weak$^*$-topology coincides with that of the pointwise convergence on the total subset of the continuous polynomials.

For an arbitrary compact operator $T: X(B_E) \rightarrow X(B_E),$ it follows now that

$$\begin{aligned} ||uC_\varphi -T||\ge \lim _{n\rightarrow \infty }(||(uC_\varphi )^*(l_n)|| - ||T^*(l_n)||) = \lim _{n\rightarrow \infty }|u(x_n)|\frac{||\delta _{\varphi (x_n)}||}{||\delta _{x_n}||}, \end{aligned}$$

and the statement follows. $\square$

Now, we can deduce the following lower estimate of the essential spectral radius of $uC_\varphi .$

Corollary 3.7

For the weighted composition operator $u C_\varphi$ acting on $H^p({\mathbb {B}}_N)$ and $A^p_\alpha ({\mathbb {B}}_N),$ respectively, for $\alpha > -1, p\ge 1$, we have that

$$\begin{aligned} r_e(u C_\varphi )\ge \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |z |\ge s \end{array}} \frac{\Vert \delta _{\varphi _n(z)} \Vert |u(z)\cdot \cdots \cdot u(\varphi _n(z))|}{\Vert \delta _z \Vert }}. \end{aligned}$$

Here, we recall an upper estimate of the spectral radius of $C_\varphi$ on $A^p_\alpha ({\mathbb {B}}_N).$

Proposition 3.8

Let $\alpha> -1, p > 1.$ Assume that $\varphi (0) =0$ and that the composition operator $C_\varphi$ is bounded on $A^p_\beta ({\mathbb {B}}_N)$ for some $-1< \beta < \alpha$. Then for $C_\varphi$ acting on $A^p_\alpha ({\mathbb {B}}_N),$ we have that

$$\begin{aligned} r_e(C_\varphi )^{\frac{N+1+\alpha }{\alpha - \beta }}\le \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |z |\ge s \end{array}} \frac{\Vert \delta _{\varphi _n(z)} \Vert }{\Vert \delta _z \Vert }}. \end{aligned}$$

In case $N=1$, then $C_\varphi : A^p_\alpha (\mathbb D)\rightarrow A^p_\alpha ({\mathbb {D}})$, $\alpha> -1, p > 1,$ is always bounded and

$$\begin{aligned} r_e(C_\varphi )^{\frac{2+\alpha }{1+\alpha }}\le \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |z |\ge s \end{array}} \frac{\Vert \delta _{\varphi _n(z)} \Vert }{\Vert \delta _z \Vert }}. \end{aligned}$$

Proof

We will use Corollary 4.7 in [24] which gives that

$$\begin{aligned} ||C_\varphi ||_e \le C \limsup _{|z|\rightarrow 1} \left( \frac{1-|z|^2}{1-|\varphi (z)|^2}\right) ^{\frac{\alpha -\beta }{p}}, \end{aligned}$$

where C is an absolute constant. Since $||\delta _z||= \frac{1}{(1 - |z|^2)^{\frac{N+1+\alpha }{p}}},$ we conclude that

$$\begin{aligned} r_e(C_\varphi )\le \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |z |\ge s\\ \end{array}}\left( \frac{\Vert \delta _{\varphi _n(z)} \Vert }{\Vert \delta _z \Vert }\right) ^\frac{\alpha -\beta }{N+1+\alpha }}, \end{aligned}$$

and the first statement follows. For the second statement, we use Corollary 3.9 in [21] (see also page 141 in [10]), that is,

$$\begin{aligned} ||C_\varphi ||_e \le C \limsup _{|z|\rightarrow 1} \left( \frac{1-|z|^2}{1-|\varphi (z)|^2}\right) ^{\frac{\alpha +1}{p}}. \end{aligned}$$

$\square$

The next lemma contains useful information about the essential spectral radius of $uC_\varphi .$

Lemma 3.9

Assume that $\varphi ( B_E)$ is a relatively compact subset of E and that $\varphi (0) =0.$ Suppose that $|u(x)| \ ||\delta _x||^{-1}\rightarrow 0$ as $||x||\rightarrow 1.$ Then,

$$\begin{aligned} \lim _{s\rightarrow 1} \sup _{\Vert x \Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert }=\lim _{s\rightarrow 1} \sup _{\Vert \varphi (x) \Vert \ge s}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert }. \end{aligned}$$

(3.5)

Proof

Since $\Vert \varphi (x)\Vert \le \Vert x\Vert$, the limit on the right hand side is not greater than the one on the left hand side. There is a sequence $(x_n)\subset B_E$ such that $\Vert x_n\Vert \rightarrow 1$ and $\limsup _{\Vert x\Vert \rightarrow 1}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert } = \lim _n\frac{ |u(x_n)|\Vert \delta _{\varphi (x_n)}\Vert }{\Vert \delta _{x_n}\Vert }.$ From the bounded sequence $\big (\varphi (x_n)\big ),$ we get a convergent subsequence, say to $a\in E,$ which we denote the same. If $\Vert a\Vert =1,$ we have $\limsup _{\Vert \varphi (x)\Vert \rightarrow 1}\frac{ |u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert } \ge \lim _n\frac{ |u(x_n)|\Vert \delta _{\varphi (x_n)}\Vert }{\Vert \delta _{x_n}\Vert }$ that leads to the equality in (3.5). While if $\Vert a\Vert <1,$ then the sequence $\big (\delta _{\varphi (x_n)}\big )$ is, by condition (I),  a convergent one in $^*X(B_E),$ and hence bounded. Therefore, according to the assumption $\limsup _{\Vert x\Vert \rightarrow 1}\frac{|u(x)|\Vert \delta _{\varphi (x)}\Vert }{\Vert \delta _x\Vert }=0,$ (3.5) holds as well. $\square$

Remark 3.10

Equality (3.5) also holds if there is a constant $d>0$ such that $\Vert \varphi (x)\Vert \ge d\Vert x\Vert$ for all $x\in B_E$ and $\varphi (0) =0.$ Such is the case of univalent $\varphi :{\mathbb {B}}_N\rightarrow {\mathbb {B}}_N$ with $\varphi (0)=0$ and $\Vert \varphi '(0)\Vert <1,$ as pointed out in [9, page 239].

4 The Spectrum

Define an operator S on a direct sum of Banach spaces $X = X_1 \oplus \cdots \oplus X_m$. Such an operator leaves invariant each direct subsum $X_k \oplus \cdots \oplus X_m$ if and only if it has a lower triangular matrix representation

$$\begin{aligned} S= \begin{pmatrix} S_{11} &{}\quad 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0\\ S_{21} &{}\quad S_{22} &{}\quad 0 &{}\quad \ldots &{}\quad 0\\ \vdots \\ \ldots &{}\quad \ldots &{}\quad \ldots &{}\quad S_{m-1, m-1} &{}\quad 0\\ S_{m1} &{}\quad S_{m2} &{}\quad \ldots &{}\quad S_{m,m-1} &{}\quad S_{mm} \end{pmatrix}, \end{aligned}$$

where $S_{jk} : X_j \rightarrow X_k$. Recall that an operator S is called a Riesz operator if $r_e(S) = 0$.

Throughout this section, we assume that $\varphi (0) = 0$ unless otherwise stated.

Theorem 4.1

[12, Corollary 2.4] Let $X = X_1 \oplus \cdots \oplus X_m$ be a direct sum of Banach spaces, and let S be an operator on X with a lower triangular matrix representation. If X is infinite dimensional, and the operators $S_{11}, \ldots , S_{m-1,m-1}$ are Riesz operators, then $\sigma (S) = \sigma (S_{11}) \cup \cdots \cup \sigma (S_{mm})$.

Let $P_k:=P(^k E)\subset H^\infty (B_E)$ denote the subspace of homogeneous polynomials of degree k on E. The Taylor series expansion at 0 of each element f in $X(B_E)$ yields a direct sum decomposition of $X(B_E)$,

$$\begin{aligned} X(B_E) = P_0 \oplus \cdots \oplus P_{m-1} \oplus X_{m}(B_E), \end{aligned}$$

because the mapping $f\in X(B_E) \mapsto P_k(f)\in P_k$ is a continuous projection of $X(B_E)$ thanks to conditions (II) and (III).

Lemma 4.2

The operator $uC_{\varphi }$ leaves invariant the space $X_{m}(B_E)$.

Proof

Fix $x \in B_E$. It is easy to see, from the Taylor series expansion of $\varphi$ at 0, that the function $g : {\mathbb {D}}\longrightarrow E$ defined by $g(\lambda )= \varphi (\lambda x)$ satisfies $g(\lambda ) = \lambda h_x(\lambda )$ for a particular analytic function $h_x$ which depends on x and $\lambda$. Set $f \in X_{m}(B_E)$. We have that

$$\begin{aligned} u(f \circ \varphi ) (\lambda x)= & {} u(\lambda x)\sum _{n \ge m} P_n f (\varphi (\lambda x))\\= & {} u(\lambda x) \sum _{n \ge m} P_n f (\lambda h_x(\lambda ))\\= & {} u(\lambda x) \sum _{n \ge m} \lambda ^n P_n f ( h_x(\lambda )), \end{aligned}$$

and so there is no non-null term of degree less than m in the series expansion of $u(f \circ \varphi ) (\lambda x).$ Therefore, if $\sum _n Q_n$ is the Taylor series of $uC_\varphi (f)$, there must be no non-null term of degree less than m in

$$\begin{aligned} \sum _n Q_n (\lambda x)= \sum _n \lambda ^n Q_n(x). \end{aligned}$$

$\square$

By Lemma 4.2, the weighted composition operator $u C_{\varphi }$ leaves invariant the spaces $X_{k-1}(B_E)=P_k \oplus \ldots \oplus P_{m-1}\oplus X_{m}(B_E), 0 \le k \le m-1,$ and leaves the space $X_{m}(B_E)$ invariant as well. Consequently, $u C_{\varphi }$ has a lower triangular matrix representation

$$\begin{aligned}u C_{\varphi } = \begin{pmatrix} C_{11} &{}\quad 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0\\ C_{21} &{}\quad C_{22} &{}\quad 0 &{}\quad \ldots &{}\quad 0\\ \vdots \\ \ldots &{}\quad \ldots &{}\quad \ldots &{}\quad C_{m-1,m-1} &{}\quad 0\\ C_{m1} &{}\quad C_{m2} &{}\quad \ldots &{}\quad C_{m,m-1} &{}\quad C_m, \end{pmatrix}, \end{aligned}$$

where the operator $C_m$ is the restriction of $u C_{\varphi }$ to $X_{m}(B_E)$. By Theorem 4.1, we can determine the spectrum of $u C_{\varphi }$ by determining the spectrum of $C_m: X_m(B_E) \rightarrow X_m(B_E)$ and the diagonal elements $C_{kk} : P_k \rightarrow P_k$ as soon as they are Riesz operators.

Next, we proceed to determine the operators $C_{kk}.$ We use the following result:

Lemma 4.3

• (i) For $v:=u-u(0),$ the function $\frac{v(x)}{\Vert x\Vert }$ is bounded in some punctured neighborhood of 0.

• (ii) For every $x\in B_E,$ we have $\Vert \varphi (x)-\varphi '(0)(x)\Vert \le \frac{\Vert x\Vert ^2}{1-\Vert x\Vert }.$

Proof

Since $\lim _{x\rightarrow 0} \frac{v(x)-u'(0)x}{\Vert x\Vert }=\lim _{x\rightarrow 0}\frac{u(z)-u(0)-u'(0)x}{\Vert x\Vert }=0,$ and $\frac{|v(x)|}{\Vert x\Vert }\le \frac{|v(x)-u'(0)x|}{\Vert x\Vert }+ \Vert u'(0)\Vert ,$ the statement i) follows.

To realize ii), consider the Taylor series of $\varphi , \; \sum _{m=1}^\infty P_m\varphi ,$ and recall that according to Cauchy inequalities, $\Vert P_m \varphi \Vert \le 1.$ Then,

$$\begin{aligned} \Vert \varphi (x)-\varphi '(0)(x)\Vert\le & {} \sum _{m=2}^\infty \Vert P_m\varphi (x)\Vert \le \sum _{m=2}^\infty \Vert x\Vert ^m \Vert P_m\varphi \Vert \\\le & {} \sum _{m=2}^\infty \Vert x\Vert ^m= \frac{\Vert x\Vert ^2}{1-\Vert x\Vert }. \end{aligned}$$

$\square$

Proposition 4.4

For every $f\in P_k,$

$$\begin{aligned} C_{kk}(f)(x)=u(0) \hat{f}\left( \varphi '(0)(x),\ldots ,\varphi '(0)(x)\right) , \end{aligned}$$

where $\hat{f}$ is the k-linear symmetric mapping determining f.

Proof

Denote $R(x)=\varphi (x)-\varphi '(0)(x).$ Then we use the binomial formula

$$\begin{aligned} f(\varphi (x))= & {} \hat{f}\left( \varphi '(0)(x)+R(x)\right) \\= & {} \sum _{l=0}^k \left( \, ^k_l \,\right) \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k-l}}\varphi '(0)(x),R(x),{\mathop {\dots }\limits ^{l}}, R(x)\right) . \end{aligned}$$

We claim that if $l>0,$ the corresponding term in this sum belongs to $X_{k+1}:$ Indeed,

$$\begin{aligned}&\frac{\hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k-l}},\varphi '(0)(x),R(x),{\mathop {\dots }\limits ^{l}}, R(x)\right) }{\Vert x\Vert ^{k+1}}\\&\quad = \hat{f}\left( \varphi '(0)\left( \frac{x}{\Vert x\Vert }\right) ,{\mathop {\dots }\limits ^{k-l}}, \varphi '(0)\left( \frac{x}{\Vert x\Vert }\right) , \frac{R(x)}{\Vert x\Vert ^2},{\mathop {\dots }\limits ^{l}}, \frac{R(x)}{\Vert x\Vert }\right) , \end{aligned}$$

where all terms are bounded bearing in mind Lemma 4.3. Moreover, for $l=0$ the term is a k-homogeneous polynomial, and

$$\begin{aligned}&\frac{1}{\Vert x\Vert ^{k+1}} \left| v(x)\hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k}},\varphi '(0)(x)\right) \right| \\&\quad \le \frac{|v(x)|}{\Vert x\Vert } \left| \hat{f}\left( \varphi '(0)\left( \frac{x}{\Vert x\Vert }\right) ,{\mathop {\dots }\limits ^{k}},\varphi '(0)\left( \frac{x}{\Vert x\Vert }\right) \right) \right| \end{aligned}$$

is bounded in a neighborhood of 0,  where $v(x)=u(x) -u(0).$

Now,

$$\begin{aligned}&u(x)f\left( \varphi (x)\right) \\&\quad =\left( u(0)+v(x)\right) \left( \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k}},\varphi '(0)(x)\right. \right. \\&\qquad \left. +\sum _{l=1}^k \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k-l}},\varphi '(0)(x),R(x),{\mathop {\dots }\limits ^{l}}, R(x)\right) \right) \\&\quad =u(0)\left( \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k}},\varphi '(0)(x)\right) \right. \\&\qquad +u(0)\left( \sum _{l=1}^k \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k-l}},\varphi '(0)(x),R(x),{\mathop {\dots }\limits ^{l}}, R(x)\right) \right) \\&\qquad +v(x)\left( \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k}},\varphi '(0)(x)\right) \right. \\&\qquad + v(x)\left( \sum _{l=1}^k \hat{f}\left( \varphi '(0)(x),{\mathop {\dots }\limits ^{k-l}},\varphi '(0)(x),R(x),{\mathop {\dots }\limits ^{l}}, R(x)\right) \right) . \end{aligned}$$

Here, the last three terms belong to $X_k(B_E),$ and so their $k^{th}$ term in the Taylor series vanishes. Therefore, $C_{kk}(f)=u(0)\big (\hat{f}\big ( \varphi '(0)(x),{\mathop {\dots }\limits ^{k}},\varphi '(0)(x)\big )\big ).$

$\square$

Now, we apply Lemma 3.1 in [12] to obtain

Lemma 4.5

$\sigma \big (C_{kk}\big )=\{u(0)\cdot \lambda _1\cdots \lambda _k: \lambda _j\in \sigma (\varphi '(0)), 1\le j\le k\}.$

Definition 4.6

A finite or infinite sequence $\{z_n \} \subset B_E$ is called an iteration sequence for $\varphi$ if $\varphi (z_k) = z_{k+1}$ for $k \ge 0$, and a sequence $\{z_n\} \subset B_E$ is called an interpolating sequence for $H^\infty (B_E)$ if for any bounded sequence $\{a_n\} \subset {\mathbb {C}}$ there exists $f \in H^\infty (B_E)$ such that $f(z_n) = a_n$ for $n \in {\mathbb {N}}$.

Recall the following Schwarz’s lemma type inequality as shown in [12]: Suppose that $\varphi :B_E \longrightarrow B_E$ satisfies $\varphi (0)=0$ and $\Vert \varphi '(0) \Vert <1$. Then for each $s<1$, there exists $a<1$ such that

$$\begin{aligned} \Vert \varphi (x)\Vert \le a\Vert x\Vert ,\quad \text{ for } x \in E, \quad \Vert x\Vert \le s. \end{aligned}$$

(4.1)

Hence, given $0<r<s<1$, there exists $\varepsilon > 0$ such that

$$\begin{aligned} \frac{1-\Vert \varphi (x)\Vert }{1-\Vert x\Vert }\ge 1+\epsilon , \quad x \in B_E, \quad r<\Vert x\Vert <s. \end{aligned}$$

Lemma 4.7

Let E be a complex Banach space and let $\varphi : B_E \rightarrow B_E$ be analytic such that $\varphi (0) = 0$ and $\Vert \varphi '(0) \Vert < 1$. Suppose that there exist $W\subset B_E, \hbox {with} \ \varphi (W) \subset W,$ $\delta > 0$ and $\epsilon > 0$ such that

$$\begin{aligned} \frac{1 - \Vert \varphi (x) \Vert }{1 - \Vert x \Vert } \ge 1 + \epsilon , \text{ for } \text{ all } x \in W \text{ such } \text{ that } \Vert x \Vert \ge \delta \ . \end{aligned}$$

(4.2)

Then, there exists a constant $M \ge 1$ which depends only on $\epsilon$, such that any finite iteration sequence $\{x_0,x_1, \ldots , x_N\}$ satisfying $x_0 \in W$ and $\Vert x_N \Vert \ge \delta$ is an interpolating sequence for $H^\infty (B_E)$ with interpolation constant not greater than M.

The proof of this lemma can be seen in [15]. It relies on the interpolation result [14, Corollary 8]. We will refer to inequalities of the form (4.2) as Julia-type estimates.

Denote

$$\begin{aligned} \gamma (uC_\varphi ;W):=\liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} \Vert \varphi _n(x) \Vert \ge s\\ x \in W \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert _X |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert _X}}. \end{aligned}$$

Lemma 4.8

Consider the weighted composition operator $u C_{\varphi }$ acting on $X(B_E).$ Assume that $\varphi (0)=0,$ $\Vert \varphi '(0)\Vert <1$ and that $\varphi ( B_E)$ is a relatively compact subset of E. Suppose also that $||\varphi _n|| = 1$ for all $n\in {\mathbb {N}}$ and that there exists $W \subseteq B_E$ with $\varphi (W) \subseteq W$ and such that a Julia-type estimate holds for some $\epsilon , \delta > 0.$ If $\lambda \ne 0$ satisfies $|\lambda | < \gamma (uC_\varphi ;W),$ then $\lambda \in \sigma (u C_{\varphi })$.

Proof

We will consider iteration sequences $\{z_k\}_{k= 0}^\infty$ such that $z_0\in W$ and $||z_0||>\delta$. In view of (4.1), the norms of the elements of any such iteration sequence decrease to 0. We define $N=N(z_0)$ to be the largest integer such that $\Vert z_N\Vert > \delta$. The hypothesis guarantees that for all $k\ge 1$, $\varphi _k(B_E)$ is not contained in the ball $\{||z||\le \delta \}$. Consequently, we can find $z_0$ for which $N(z_0)$ is arbitrarily large.

Choose $c<1$ such that

$$\begin{aligned} \Vert \varphi (z)\Vert \le c\Vert z\Vert ,\qquad z\in E,\,\Vert z\Vert \le \sqrt{\delta }. \end{aligned}$$

We can assume that $c>\sqrt{\delta }$. By considering separately the cases $\Vert z_N\Vert \le \sqrt{\delta }$ and $\Vert z_N\Vert > \sqrt{\delta }$, we see also that $\Vert z_{N+1}\Vert \le c\Vert z_N\Vert$. Since $\Vert z_{n+1}\Vert \le c\Vert z_n\Vert$ for $n>N+1$, we obtain by induction that

$$\begin{aligned} \Vert z_{N+k}\Vert \le c^k\Vert z_N\Vert ,\qquad k\ge 0. \end{aligned}$$

Since $u\in H_b(B_E)$ it holds that $0< C:= \max \{\sup _{\Vert z \Vert \le \delta } |u(z)|, \sup _n|u(z_n)|\} <\infty$. Put $D_k(z_0) = |u(z_0)\cdot \cdots \cdot u(z_{k-1})|$.

For any iteration sequence $(z_k)_{k=0}^\infty$ and $m \in {\mathbb {N}},$ let us define $L_{\lambda ,u}$ on $X_{m}(B_E)$ by

$$\begin{aligned} L_{\lambda ,u}(f) = f(z_0) + \sum _{k=1}^\infty u(z_0) {\mathop {\cdots }\limits ^{k}} u(z_{k-1}) f(z_k)\lambda ^{-k}. \end{aligned}$$

By using condition (IV), we get that $L_{\lambda ,u}$ is bounded, because

$$\begin{aligned}&\left| \sum _{k=N+1}^\infty u(z_0) \ldots u(z_{k-1}) f(z_k) \lambda ^{-k}\right| \\&\quad \le D_N(z_0)\sum _{k=N+1}^\infty C^{k-N} |\lambda |^{-k}c(m) \Vert z_k \Vert ^m \Vert \delta _{z_k} \Vert _X \Vert f \Vert \\&\quad \le D_N(z_0) \Vert f \Vert c(m) \sum _{k = N+1}^\infty C^{k-N} |\lambda |^{-k}\left( c^{k-N} \Vert z_N \Vert \right) ^m \Vert \delta _{z_k} \Vert _X \\&\quad \le \Vert f \Vert D_N(z_0) c(m) \sum _{k = N +1}^\infty \Vert \delta _{z_k} \Vert _X \Vert z_N \Vert ^m \left( \frac{C\cdot c^m}{|\lambda |}\right) ^{k-N}\frac{1}{|\lambda |^N}. \end{aligned}$$

Since $(z_k)$ converges to 0 in $B_E,$ also $(\delta _{z_k})$ converges, so

$$\begin{aligned} \Vert f \Vert D_N(z_0) c(m) M_0 \Vert z_N \Vert ^m \frac{1}{|\lambda |^N} \left( \sum _{k = N +1}^\infty \left( \frac{C \cdot c^m}{|\lambda |}\right) ^{k-N}\right) . \end{aligned}$$

So there exists $m_0$ so that if $m \ge m_0,$ then $L_{\lambda , u}$ is bounded, i.e, $\frac{C c^{m_0}}{|\lambda |} < 1$.

Note that $(C_m^* - \lambda I)(L_{\lambda , u}) = - \lambda \delta _{z_0}$, because for any $f \in X_m(B_E)$,

$$\begin{aligned}&\langle (C_m^* - \lambda I)(L_{\lambda , u}), f \rangle = L_{\lambda , u}(u C_{\varphi }(f) - \lambda f) \\&\quad =L_{\lambda , u}(u \cdot f \circ \varphi ) - \lambda L_{\lambda , u}(f) = u(z_0) f(\varphi (z_0))\\&\qquad + \sum _{k = 1}^\infty u_{z_0}\cdot \cdots \cdot u(z_{k-1})u(z_k)f(\varphi (z_k))\lambda ^{-k}\\&\qquad -\lambda f(z_0) - \sum _{k = 1}^\infty u(z_0) \cdot \cdots \cdot u(z_{k-1})\lambda ^{-k+1}f(z_k) = -\lambda f(z_0) \\&\qquad +\sum _{k = 1}^\infty u(z_0) \cdot \cdots \cdot u(z_k) \lambda ^{-k} f(z_{k +1})\\&\qquad - \sum _{k = 2}^\infty u(z_0) \cdot \cdots \cdot u(z_{k -1})\lambda ^{-k+1}f(z_k) = -\lambda f(z_0). \end{aligned}$$

Now, we find a suitable lower bound for $\Vert L_{\lambda , u} \Vert$. For $0 \le K \le N$, pick $l \in E^{*}$, $\Vert l \Vert = 1$ such that $l(z_K) = \Vert z_K \Vert ,$ and by using Lemma 4.7, pick $f \in H^\infty (B_E)$ with $\Vert f \Vert _\infty \le M$ and $f(z_k) = 0$ for all $0 \le k \le N$ except for $k = K,$ in which case $f(z_K) = 1$. By (I), there is $f_0 \in X(B_E)$ such that $\Vert f_0 \Vert \le 1$ and $f_0(z_K) = \Vert \delta _{z_K} \Vert _X$. Then by condition (II), the function $g := l^m \cdot f_0 \cdot f \in X_m(B_E)$ and $\Vert g \Vert \le M_1$.

We now calculate

$$\begin{aligned} L_{\lambda , u}(g)= & {} u(z_0)\cdot \cdots \cdot u(z_{K-1}) \lambda ^{-K} \Vert z_K \Vert ^{m} \Vert \delta _{z_K} \Vert _X + D_N(z_0)\\&\times \,\sum _{k = N + 1}^\infty u(z_N)\cdot \cdots \cdot u(z_{k-1})\lambda ^{-k}f_0(z_k)f(z_k)l^m(z_k). \end{aligned}$$

We assume that $D_K(z_0)\ne 0.$ The first term is bounded above by

$$\begin{aligned} D_K(z_0) |\lambda |^{-K} \Vert z_K \Vert ^m \Vert \delta _{z_K} \Vert _X, \end{aligned}$$

and the second term is bounded above by

$$\begin{aligned}&D_N(z_0) \sum _{k = N+1}^\infty C^{k-N} |\lambda |^{-k}M \Vert \delta _{z_k} \Vert _X \Vert z_k \Vert ^m\\&\quad \le D_N(z_0) M M_0 \left( \sum _{k = N+1}^\infty C^{k-N} |\lambda |^{-k}(c^{k-N} \Vert z_N \Vert )^m\right) \\&\quad =D_N(z_0) M M_0 |\lambda |^{-N}\left( \sum _{k = N+1}^\infty \left( \frac{C}{|\lambda |}c^m\right) ^{k-N}\right) \Vert z_N \Vert ^m\\&\quad \le D_N(z_0) M M_0 |\lambda |^{-N}\left( \sum _{k = N+1}^\infty \left( \frac{C}{|\lambda |}c^m\right) ^{k-N}\right) \Vert z_K \Vert ^m. \end{aligned}$$

Thus,

$$\begin{aligned}&|L_{\lambda , u} (g)| \ge D_K(z_0) |\lambda |^{-K} \Vert z_K \Vert ^m \Vert \delta _{z_K} \Vert _X \\&\quad - D_N(z_0)|\lambda |^{-(N-K)} M M_0 \sum _{k = N +1}^\infty \left( \frac{C}{|\lambda |}c^m\right) ^{k-N} |\lambda |^{-K}||z_K||^m . \end{aligned}$$

There is $m_1 \ge m_0,$ so that if $m \ge m_1$ we have

$$\begin{aligned} \sum _{k = N +1}^\infty \left( \frac{C}{|\lambda |}c^m\right) ^{k-N} < \frac{D_K(z_0)}{D_N(z_0) 2M M_0|\lambda |^{K-N}}\Vert \delta _{z_K} \Vert _X \; \text { if } D_N(z_0)\ne 0. \end{aligned}$$

So,

$$\begin{aligned} |L_{\lambda , u}(g)| \ge \frac{1}{2}D_K(z_0) |\lambda |^{-K} \Vert z_K \Vert ^m \Vert \delta _{z_K} \Vert _X\; \text { regardless the value of } D_N(z_0) . \end{aligned}$$

Consequently,

$$\begin{aligned} \frac{1}{2}D_K(z_0) |\lambda |^{-K} \Vert z_K \Vert ^m \Vert \delta _{z_K} \Vert _X \le \Vert L_{\lambda , u} \Vert \Vert g \Vert _v \le \Vert L_{\lambda , u} \Vert \cdot M_1, \end{aligned}$$

which gives us

$$\begin{aligned} \Vert L_{\lambda , u} \Vert \ge \frac{1}{2}\frac{D_K(z_0)}{M_1} \Vert z_K \Vert ^m \Vert \delta _{z_K} \Vert _X |\lambda |^{-K}, \end{aligned}$$

in the case $|D_K(z_0)|\ne 0.$

If

$$\begin{aligned} |\lambda | < \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{\begin{array}{c} \Vert \varphi _n(x) \Vert \ge s\\ x \in W \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert _X |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert _X}}, \end{aligned}$$

we can pick $\mu > 0$ such that

$$\begin{aligned} |\lambda |< \mu < \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{\begin{array}{c} \Vert \varphi _n(x) \Vert \ge s\\ x \in W \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert _X |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert _X}}. \end{aligned}$$

Hence, there is $n_0$ such that if $K \ge n_0,$ it holds that

$$\begin{aligned} \mu ^K < \lim _{s \rightarrow 1} \sup _{\begin{array}{c} \Vert \varphi _K(x)\Vert \ge s\\ x \in W \end{array}} \frac{\Vert \delta _{\varphi _K(x)} \Vert _X |u(x)\cdot \cdots \cdot u(\varphi _K(x))|}{\Vert \delta _x \Vert _X}. \end{aligned}$$

So for every $K \ge n_0,$ we can find $x \in W$ with $\Vert \varphi _K(x) \Vert > \delta$ such that

$$\begin{aligned} \mu ^K \le \frac{\Vert \delta _{\varphi _K(x)} \Vert _X}{\Vert \delta _x \Vert _X} |u(x)|\cdot \cdots \cdot |u(\varphi _K(x))|, \end{aligned}$$

for which necessarily $D_K(z_0) \ne 0.$

We consider the iteration sequence $\{\varphi _i(x)\}_{i = 0}^\infty$ that satisfies indeed the condition $\Vert \varphi _K(x) \Vert > \delta$. So for $z_0 = x$, we have $N = N(z_0)$ so that $\Vert z_N \Vert > \delta$, $K \le N$ and $z_K = \varphi _K(x)$. Pick $L_{\lambda , u}$ as above.

Now,

$$\begin{aligned} \frac{\Vert (C_m^{*} - \lambda I)(L_{\lambda , u}) \Vert }{\Vert L_{\lambda , u}\Vert }\le & {} \frac{|\lambda | \Vert \delta _{z_0} \Vert _{X_m}}{\frac{D_K(z_0)}{2M_1} \Vert \delta _{\varphi _K(x)}\Vert _X |\lambda |^{-K} \Vert \varphi _K(x)\Vert ^m}\\= & {} \frac{2|\lambda |^{K+1}M_1}{D_K(z_0)} \frac{\Vert \delta _{z_0} \Vert _{X_m}}{\Vert \varphi _K(x) \Vert ^m \Vert \delta _{\varphi _K(x)} \Vert _X}\\\le & {} \frac{2|\lambda |^{K+1}M_1}{D_K(z_0)} \frac{\Vert \delta _{z_0} \Vert _{X_m}}{\delta ^m \Vert \delta _{\varphi _K(x)} \Vert _X}\\\le & {} \frac{2|\lambda |^{K+1}M_1\Vert \delta _{z_0} \Vert _X}{\delta ^m \Vert \delta _{\varphi _K(x)} \Vert _X} \frac{1}{D_K(z_0)}. \end{aligned}$$

Since

$$\begin{aligned} \mu ^K \le \frac{\Vert \delta _{\varphi _K(z_0)} \Vert _X}{\Vert \delta _{z_0} \Vert _X}D_K(z_0), \end{aligned}$$

in combination with the above inequality we now get

$$\begin{aligned} \frac{2|\lambda |^{K+1}M\Vert \delta _{z_0} \Vert _X}{\delta ^m \Vert \delta _{\varphi _K(x)} \Vert _X} \frac{1}{D_K(z_0)}\le & {} \frac{2|\lambda |^{K+1}M\Vert \delta _{z_0} \Vert _X}{\delta ^m \Vert \delta _{z_K} \Vert _X} \frac{\Vert \delta _{\varphi _K(z_0)} \Vert _X}{\Vert \delta _{z_0} \Vert _X} \mu ^{-K}\\\le & {} \frac{2M|\lambda |}{\delta ^m}\left( \frac{|\lambda |}{\mu }\right) ^K. \end{aligned}$$

By choosing $K\ge n_0$ large enough, we see that $C_m^* - \lambda I$ is not bounded from below and consequently $C_m - \lambda I$ is not invertible, i.e., $\lambda \in \sigma (C_m)$. $\square$

We are now ready to formulate the main result.

Theorem 4.9

Consider the weighted composition operator $u C_{\varphi }$ acting on $X(B_E).$ Assume that $\varphi (0)=0,$ $\Vert \varphi '(0)\Vert <1$ and that $\varphi ( B_E)$ is a relatively compact subset of E. Suppose that there exists $W \subseteq B_E$ with $\varphi (W) \subseteq W$ such that a Julia-type estimate holds for some $\epsilon , \delta > 0.$ Then,

$$\begin{aligned}&\left\{ u(0)\right\} \cup \left\{ u(0)\lambda _1\cdots \lambda _k: \lambda _j\in \sigma (\varphi '(0)), \right. \\&\quad \left. 1 \le j\le k,\, k\ge 1\right\} \cup \left\{ \lambda : |\lambda |\le \gamma (uC_\varphi ;W)\right\} \subset \sigma (uC_\varphi ). \end{aligned}$$

Proof

First of all, notice that the linear mapping $\varphi '(0)\in {\mathcal {L}}(E)$ is a compact operator, since by the Cauchy integral formula (see [23, 7.3 Corollary]), $\varphi '(0)(x)=\frac{1}{2\pi i}\int _{|\xi |=1/2}\frac{\varphi (\xi x)}{\xi }\mathrm{d}\xi$ belongs to the closed convex hull of the compact set $2\overline{\varphi (B_E)}.$ Therefore, the mappings $C_{kk}$ in Proposition 4.4 are compact.

In case that $||\varphi _n|| = 1$ for all $n\in {\mathbb {N}},$ the result follows from Theorem 4.1, Lemmas 4.8 and 4.5. If for some $n\in {\mathbb {N}}, ||\varphi _n||< 1,$ then $\gamma (uC_\varphi ;W)=0,$ and the argument is simpler as there is no need of Lemma 4.8. $\square$

From the above result, we get several consequences.

Corollary 4.10

Let E be a Hilbert space or $E=C_0({\mathcal {X}}),$ ${\mathcal {X}}$ a locally compact Hausdorff topological space. Assume that $uC_\varphi : H^\infty _\upsilon (B_E)\rightarrow H^\infty _\upsilon (B_E)$ is bounded with $\varphi (0)=0$ and $\Vert \varphi '(0)\Vert <1.$ Suppose that $\varphi ( B_E)$ is a relatively compact subset of E. Then,

$$\begin{aligned}&\left\{ \lambda \in {\mathbb {C}}: |\lambda | \le \liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} \Vert \varphi _n(x) \Vert \ge s \\ x \in \varphi ( B_E) \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert }}\right\} \\&\quad \cup \sigma _p(uC_\varphi )\subset \sigma (u C_{\varphi }) . \end{aligned}$$

Proof

We need, respectively, [12, Theorem 6.1] and [15, Theorem 2.2]. Each of them guarantees, respectively, that under the current assumptions, $W\equiv \varphi (B_E)$ satisfies the Julia-type estimates (4.2) for some $\epsilon ,\delta >0.$ $\square$

One can find plenty of mappings $\varphi$ which do fulfill the assumptions in Corollary 4.10. Indeed, consider for every pair $(k,m)\in {\mathbb {N}}\times {\mathbb {N}},\;m>1,$ the mapping

$$\begin{aligned} \varphi ^{k,m}:(x_n)\in \ell _2\mapsto \left( x_1^m,\dots ,x_k^m, x_{k+1}^m,\frac{x^m_{k+2}}{2},\frac{x^m_{k+3}}{3},\dots , \frac{x^m_{k+i}}{i},\dots \right) \in \ell _2. \end{aligned}$$

Clearly, $\varphi ^{k,m}(0)=0,$ $(\varphi ^{k,m})'(0)=0,$ since $\varphi ^{k,m}$ is an m-homogeneous polynomial and $\varphi ^{k,m}(B_{\ell _2})\subset B_{\ell _2}$ is relatively compact.

Corollary 4.11

Let $B_E$ be either the n-ball ${\mathbb {B}}_N$ or the n-polydisc $\Delta _N.$ Assume that $uC_\varphi : H^\infty _\upsilon (B_E)\rightarrow H^\infty _\upsilon (B_E)$ is bounded with $\varphi (0)=0$ and $\Vert \varphi '(0)\Vert <1.$ Then,

$$\begin{aligned} \left\{ \lambda \in {\mathbb {C}}: |\lambda | \le r_e(uC_\varphi )\right\} \cup \sigma _p(uC_\varphi )=\sigma (u C_{\varphi }). \end{aligned}$$

Proof

Recall that according to Corollary 3.5,

$$\begin{aligned} r_e(uC_\varphi )=\liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |\varphi _n(x) |\ge s \end{array}} \frac{\Vert \delta _{\varphi _n(x)} \Vert |u(x)\cdot \cdots \cdot u(\varphi _n(x))|}{\Vert \delta _x \Vert }}. \end{aligned}$$

As in the proof of Corollary 4.10 by using [12, Theorem 6.1] and [15, Theorem 2.2] respectively, $W\equiv B_E$ satisfies the Julia-type estimates (4.2) for some $\epsilon ,\delta >0.$ Thus, it is clear from Theorem 4.9 that

$$\begin{aligned} \left\{ \lambda \in {\mathbb {C}}: |\lambda | \le r_e(uC_\varphi ) \right\} \cup \sigma _p(uC_\varphi ) \subset \sigma (u C_{\varphi }). \end{aligned}$$

The converse inclusion follows from the fact that if $\lambda \in \sigma (uC_\varphi )$ and $|\lambda |> r_e(uC_\varphi )$, then $\lambda \in \sigma _p(uC_\varphi )$ by Lemma 7.43 and Theorem 7.44 in [1] or Propositions 2.2 and 3.4 in [5]. $\square$

Remark 4.12

Also for the Hardy space $H^\infty (B_E),\; r_e(uC_\varphi )=\gamma \big (uC_\varphi ;\mathbb \varphi (B_E) \big ).$

Indeed, in this case, we have that $||\delta _x||=1$ and that u is bounded by some $M>0$. Hence,

$$\begin{aligned}&\sup _{||\varphi _n(x)||>s}|u(x) u(\varphi (x))\cdots u(\varphi _n(x)|\\&\quad \le M\sup _{ \begin{array}{c} \Vert \varphi _{n-1}(y) \Vert \ge s\\ y \in \varphi ( B_E) \end{array}} |u(y) u(\varphi (y))\cdots u(\varphi _{n-1}(y))|, \end{aligned}$$

from where we get that $r_e(uC_\varphi )\le \gamma \big (uC_\varphi ;\mathbb \varphi (B_E) \big ),$ as required. This yields the same conclusion as in Corollary 4.11, so we recover the main results concerning the spectrum in [12, 15, 27].

Corollary 4.13

Let $p \ge 1$ and $\alpha > -1.$ If $uC_\varphi$ is a bounded operator on ${\mathcal {H}}({\mathbb {B}}_N)$, $A^p_\alpha ({\mathbb {B}}_N)$ and $H^p({\mathbb {B}}_N),$ respectively, with $\varphi (0)=0$ and $\Vert \varphi '(0)\Vert <1,$ then

$$\begin{aligned}&\{u(0)\} \cup \left\{ u(0)\lambda _1\cdots \lambda _k: \lambda _j\in \sigma (\varphi '(0)), 1\le j\le k,\, k\ge 1\right\} \cup \left\{ \lambda : |\lambda |\right. \\&\quad \left. \le \gamma \left( uC_\varphi ;{\mathbb {B}}_N\right) \right\} \subset \sigma (uC_\varphi ). \end{aligned}$$

Proof

If the range of some iterated of $\varphi$ lies strictly inside ${\mathbb {B}}_N,$ we have $\gamma \big (uC_\varphi ;{\mathbb {B}}_N \big )=0.$ If that wasn’t the case, then $||\varphi _n|| = 1$ for all $n\in {\mathbb {N}}$ and as in the proof of Corollary 4.11, ${\mathbb {B}}_N$ satisfies the Julia-type estimates (4.2) for some $\epsilon ,\delta >0.$ So we may apply Theorem 4.9. $\square$

Corollary 4.14

Let $p \ge 1$ and $\alpha > -1.$ If $C_\varphi$ is a bounded operator on ${\mathcal {H}}({\mathbb {B}}_N)$, $A^p_\alpha ({\mathbb {B}}_N)$ and $H^p({\mathbb {B}}_N),$ respectively with $\varphi (0)=0$ and $\Vert \varphi '(0)\Vert <1,$ then

$$\begin{aligned}&\{1\} \cup \ \left\{ \lambda _1\cdots \lambda _k: \lambda _j\in \sigma (\varphi '(0)), 1\le j\le k,\, k\ge 1\right\} \cup \left\{ \lambda : |\lambda |\right. \\&\quad \left. \le \gamma _0\left( C_\varphi ;{\mathbb {B}}_N\right) \right\} \subset \sigma (C_\varphi ), \end{aligned}$$

where

$$\begin{aligned} \gamma _0(C_\varphi ;{\mathbb {B}}_N)=\liminf _n \root n \of {\lim _{s \rightarrow 1} \sup _{ \begin{array}{c} |z |\ge s \end{array}} \frac{\Vert \delta _{\varphi _n(z)} \Vert }{\Vert \delta _z \Vert }}. \end{aligned}$$

Proof

Notice that Lemma 3.9 applies since $\lim _{\Vert z\Vert \rightarrow 1}\Vert \delta _z\Vert =\infty$ by Proposition 2 in [9], (2.2) and (2.3), respectively. Thus, $\gamma (C_\varphi ;{\mathbb {B}}_N)=\gamma _0(C_\varphi ;{\mathbb {B}}_N).$ Now the statement follows from Corollary 4.13. $\square$

This corollary yields [9, Theorem 15] because any map $\varphi$ with $\varphi (0)=0$ not unitary on any slice does satisfy $\Vert \varphi '(0)\Vert <1,$ as shown in the proof of [9, Lemma 14].

Corollary 4.15

Assume that $\varphi (0) =0$, $||\varphi '(0)||<1$ and that the composition operator $C_\varphi$ is bounded on $A^p_\beta ({\mathbb {B}}_N)$ for some $-1< \beta < \alpha$ and $p > 1.$ Then for $C_\varphi$ acting on $A^p_\alpha ({\mathbb {B}}_N),$ we have

$$\begin{aligned} \{1\}&\cup&\left\{ \lambda _1\cdots \lambda _k: \lambda _j\in \sigma (\varphi '(0)), 1\le j\le k,\, k\ge 1\right\} \\&\cup&\left\{ \lambda \in {\mathbb {C}}: |\lambda |\le r_e(C_\varphi )^{\frac{N+1+\alpha }{\alpha - \beta }}\right\} \subset \sigma (C_\varphi ), \end{aligned}$$

and, when $N=1$, then $C_\varphi : A^p_\alpha ({\mathbb {D}})\rightarrow A^p_\alpha ({\mathbb {D}})$, $\alpha> -1, p > 1,$ is always bounded and

$$\begin{aligned} \left\{ \varphi '(0)^n: n\ge 0\right\} \cup \left\{ \lambda \in {\mathbb {C}}: |\lambda |\le r_e(C_\varphi )^{\frac{2+\alpha }{1+\alpha }}\right\} \subset \sigma (C_\varphi ). \end{aligned}$$

Proof

Both statements follow from Corollary 4.14 and Proposition 3.8. $\square$

Remark 4.16

For every bounded operator $T: E \rightarrow E,$ it holds by the general argument used in the proof of Corollary 4.11 that $\sigma (T)\subset \sigma _p(T) \cup \big \{\lambda \in {\mathbb {C}}: |\lambda |\le r_e(T)\big \}$. Therefore for $C_\varphi : A^p_\alpha ({\mathbb {D}})\rightarrow A^p_\alpha ({\mathbb {D}})$, $\alpha> -1, p > 1,$ with $\varphi (0)=0$ and $|\varphi '(0)| < 1$, we obtain that

$$\begin{aligned}&\left\{ \varphi '(0)^n: n\ge 0\right\} \cup \left\{ \lambda \in {\mathbb {C}}: |\lambda |\le r_e(C_\varphi )^{\frac{2+\alpha }{1+\alpha }}\right\} \\&\quad \subset \sigma (C_\varphi )\subset \left\{ \varphi '(0)^n: n\ge 0\right\} \cup \left\{ \lambda \in {\mathbb {C}}: |\lambda |\le r_e(C_\varphi )\right\} . \end{aligned}$$

The univalent case with $\alpha =0$ was studied in [22]. In fact, it follows also for univalent symbol $\varphi$ with the above assumptions and $\alpha> -1, p > 1,$ that $\sigma (C_\varphi ) =\{\varphi '(0)^n: n\ge 0\} \cup \big \{\lambda \in {\mathbb {C}}: |\lambda |\le r_e(C_\varphi )\big \}.$ Indeed, in this case the essential spectral radius $r_e(C_\varphi )$ can be calculated using that the generalized Nevalinna counting function $N_{\varphi ,2+ \alpha }(z) =\big ( \log \frac{1}{|\varphi ^{-1}(z)|}\big )^{2+\alpha }.$