%------------------------------------------------------------------------------ % Here please write the date of submission of paper or its revisions: %------------------------------------------------------------------------------ % \documentclass[12pt, reqno]{amsart} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} \hypersetup{colorlinks=true,linkcolor=red, anchorcolor=green, citecolor=cyan, urlcolor=red, filecolor=magenta, pdftoolbar=true} %\usepackage{draftwatermark} \usepackage{lineno} \textheight 22.5truecm \textwidth 14.5truecm \setlength{\oddsidemargin}{0.35in}\setlength{\evensidemargin}{0.35in} \setlength{\topmargin}{-.5cm} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{summary}[theorem]{Summary} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{problem}[theorem]{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{question} \newtheorem{ques}[thm]{Question} \theoremstyle{example} \newtheorem{exm}[thm]{Example} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} %\numberwithin{equation}{section} % MATH ----------------------------------------------------------- \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Real}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}} %\setcounter{page}{1} \newcommand{\RR}{{\mathbb R}} \newcommand{\DD}{{\mathbb D}} \newcommand{\BB}{{\mathbb B}} \newcommand{\CC}{{\mathbb C}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\QQ}{{\mathbb Q}} \newcommand{\NN}{{\mathbb N}} \newcommand{\bsk}{{\bigskip}} \def\a{\alpha} \def\vp{\varphi} \def\msk{\medskip} \def\ol{\overline} \def\bege{\begin{equation}} \def\ende{\end{equation}} \def\bsk{\bigskip} \def\cent{\centerline} \def\a{\alpha} \def\om{\omega} \def\b{\beta} \def\g{\gamma}\def\ol{\overline} \def\ve{\varepsilon} \def\d{\delta} \def\pt{\partial} \def\bin{\atopwithdelims} \def\begr{\begin{eqnarray}} \def\endr{\end{eqnarray}} \def\qand{\quad\mbox{ and }\quad} \def\qfor{\quad\mbox{ for }\quad} \def\qie{\quad\mbox{ i.e. }\quad} \def\bege{\begin{equation}} \def\ende{\end{equation}} \def\begr{\begin{eqnarray}} \def\endr{\end{eqnarray}} \def\bnum{\begin{enumerate}} \def\enum{\end{enumerate}} \usepackage{amscd} %\SetWatermarkText{Galley Proof}\SetWatermarkScale{4} \begin{document} %\linenumbers \setcounter{page}{71} \begin{center}{\footnotesize Khayyam J. Math. 1 (2015), no. 1, 71--81}\\\end{center} \noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=0.24]{KJM.jpg}} \vspace{0.5cm} \title[COMPOSITION OPERATORS]{APPROXIMATION NUMBERS OF COMPOSITION OPERATORS ON WEIGHTED HARDY SPACES } \author[A.K. Sharma, A. Bhat]{Ajay K. Sharma$^{1*}$ and Ambika Bhat$^2$} \address{$^{1}$ School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra-182320, J\& K, India.} \email{aksju\_76@yahoo.com} \address{$^{2}$ Ambika Bhat, School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra-182320, J\& K, India.} \email{ambikabhat.20@gmail.com} \dedicatory{{\rm Communicated by S. Hejazian}} \subjclass[2010]{Primary 47B33, 46E10; Secondary 30D55.} \keywords{Composition operator, weighted Hardy space, approximation number.} \date{Received: 30 July 2014; Revised: 27 November 2014; Accepted: 2 December 2014. \newline \indent $^{*}$ Corresponding author} \begin{abstract} In this paper we find upper and lower bounds for approximation numbers of compact composition operators on the weighted Hardy spaces $\mathcal H_\sigma$ under some conditions on the weight function $\sigma.$ \\ %\textbf{still under review by the journal}. \end{abstract} \maketitle \section{Introduction and preliminaries} Let $\mathbb D$ be the open unit disk in the complex plane $\mathbb C$, $H(\DD)$ the class of all holomorphic functions on $\DD$ and $H^\infty(\DD)$ the space of all bounded analytic function on $\DD$ with the norm $||f||_\infty = \sup_{z \in \DD}|f(z)|.$ For $z \in \DD,$ let $$\beta_z(w)=\frac{z-w}{1-\bar z w},\quad z, w\in\DD,$$ that is, the involutive automorphism of $\DD$ interchanging points $z$ and $0$. Let $\sigma$ be a positive integrable function on $[0 , 1)$. We extend $\sigma$ on $\DD$ defining $\sigma(z) = \sigma(|z|)$ for all $ z \in \DD$ and call it a weight or a weight function. By $\mathcal{H}_\sigma$ we denote the weighted Hardy space consisting of all $f \in H(\DD)$ such that $$||f||^2_{\mathcal{H}_\sigma} = |f(0)|^2 + \int_\DD |f'(z)|^2 \sigma(z) dA(z) < \infty,$$ where $\displaystyle dA(z) = \frac{1}{\pi} dx dy = \frac{1}{\pi}r dr d \theta$ is the normalized area measure on $\mathbb D$. A simple computation shows that a function $f(z) = \displaystyle\sum^\infty_{n = 0}a_n z^n$ belongs to $\mathcal{H}_\sigma$ if and only if $$\displaystyle\sum^\infty_{n = 0}|a_n|^2 \sigma_n < \infty,$$ where $\sigma_0 = 1$ and $$\sigma_n = \sigma(n) = 2n^2 \int^1_0 r^{2n - 1}w(r) dr, \;\; n\in \mathbb{N}.$$ The sequence $(\sigma_n)_{n \in \mathbb{N}_0}$ is called the weight sequence of the weighted Hardy space $\mathcal{H}_\sigma.$ The properties of the weighted Hardy space with the weight sequence $(\sigma_n)_{n \in \mathbb{N}_0},$ clearly depends upon $\sigma_n.$ \\ Let $\mathcal{H}_\sigma$ be a weighted Hardy space with weight sequence $\{\sigma_n\}.$ Then for each $\lambda \in \mathbb{D},$ the evaluation functional in $\mathcal{H}_\sigma$ at $\lambda$ is a bounded linear functional and for $f \in \mathcal{H}_\sigma,\;\;\; f(\lambda) = \langle f, K_\lambda \rangle,$ where $$K_\lambda(z) = \displaystyle\sum^\infty_{k = 0} \frac{(\overline{\lambda} z)^k}{\sigma(k)} \; \; \mbox{and} \; \; ||K_\lambda||^2_{\mathcal{H}_\sigma} = \displaystyle\sum^\infty_{k = 0} \frac{|\lambda|^{2k}}{\sigma(k)}.$$ Moreover, \begin{equation}|f(z)| \leq || f||_{\mathcal{H}_\sigma }\Big(\displaystyle\sum_{k = 0}^\infty r^{2k}(\sigma_k)^{-1}\Big)^{1/2}\end{equation} \begin{equation}|f'(z)| \leq || f||_{\mathcal{H}_\sigma }\Big(\displaystyle\sum_{k = 0}^\infty k^2 r^{2(k-1)}(\sigma_k)^{-1}\Big)^{1/2}\end{equation} for $|z|\leq r$ where $\sigma(k) = ||z^k||^2_{\mathcal{H}_\sigma},$ see Theorem 2.10 in \cite{2}.\\ For more about weighted Hardy spaces and some related topics, see \cite{2}, \cite{3} and \cite{15}.\\ Throughout the paper, a weight $\sigma$ will satisfy the following properties: \begin{itemize} \item [$(W_1)$] $\sigma$ is non-increasing; \item[$(W_2)$] $\frac{\sigma(r)}{(1 - r)^{1 + \delta}}$ is non-decreasing for some $\delta > 0;$ \item[$(W_3)$] $\lim_{r \to 1}\sigma(r) = 0$. \end{itemize} We also assume that $\sigma$ will satisfy one of the following properties: \begin{itemize} \item[$(W_4)$] $\sigma$ is convex and $\lim_{r \to 1}\sigma(r) = 0$; or \item[$(W_5)$] $\sigma$ is concave. \end{itemize} Such a weight function is called {\it admissible} (see \cite{3}). If $\sigma$ satisfies condition $(W_1),$ $(W_2),$ $(W_3)$ and $(W_4),$ then it is said that $\sigma$ is $I$-{\it admissible}. If $\sigma$ satisfies condition $(W_1),$ $(W_2),$ $(W_3)$ and $(W_5),$ then it is said that $\sigma$ is $II$-{\it admissible}. $I$-admissibility corresponds to the case $\mathcal{H}^2 \subseteq \mathcal{H}_\sigma \subset \mathcal{A}^2_\alpha$ for some $\alpha > -1,$ whereas $II$-admissibility corresponds to the case $\mathcal{D} \subseteq \mathcal{H}_\sigma \subset \mathcal{H}^2.$ If we say that a weight is admissible it means that it is $I$-admissible or $II$-admissible.\\ Recall that for $z$ and $w$ in $\mathbb D,$ the pseudohyperbolic distance $d$ between $z$ and $w$ is defined by $$d(z,w) = |\beta_{z}(w)|.$$ For $r\in(0,1)$ and $z \in \mathbb D,$ denote by $D(z,r),$ the pseudohyperbolic disk whose pseudohyperbolic center is $z$ and whose pseudohyperbolic radius is $r$, that is $$D(z,r) = \big \{w \in \mathbb D : d(z,w) < r \big\}.$$ We need Carleson type Theorem for weighted Hardy spaces, see \cite{11} \\ \begin{theorem} Let $\sigma$ be an admissible weight, $r\in (0, 1)$ fixed and $\mu$ be a positive Borel measure on $\mathbb D.$ Then the following statements are equivalent: \begin{enumerate} \item The following quantity is bounded $$C_1:=\sup_{z\in\DD}\frac{\mu(D(z , r))}{\sigma(z)(1 - |z|^2)^2};$$ \item There is a constant $C_{2} > 0$ such that, for every $f\in H_\sigma,$ $$\int_{\mathbb D}|f'(w)|^2 d\mu(w) \leq C_{2}\|f\|^2_{H_\sigma};$$ \item The following quantity is bounded $$C_3:=\sup_{z\in \DD}\int_{\mathbb D}\frac{(1 - |z|^2)^{2 + 2\gamma}}{\sigma(z)|1 - \bar{z}w|^{4 + 2\gamma}} d\mu(w).$$ \end{enumerate} Moreover, the following asymptotic relationships hold $$ C_{1}\asymp C_2\asymp C_3.$$ The generalized Nevanlinna counting function shall play a key role in our work. The generalized Nevanlinna counting function associated to a weight function $\omega$ is defined for every $z \in \mathbb{D}\setminus\{\varphi(0)\}$ by $$\mathfrak{N}_{\varphi, \sigma}(z) = \displaystyle\sum_{\varphi(\lambda) = z} \sigma(\lambda),$$ where $\mathfrak{N}_{\varphi, \sigma}(z) = 0$ when $z \notin \varphi(\mathbb{D}).$ By convention, we define $\mathfrak{N}_{\varphi, \sigma}(z) = 0$ when $z = \varphi(0).$ When $ \sigma(r) = \sigma_0(r) \asymp \log 1/r,\;\;\; \mathfrak{N}_{\varphi, \sigma_0} = N_{\varphi},$ the usual Nevanlinna counting function associated to $\varphi.$ \\ For more about generalized and classical Nevanlinna counting functions, see \cite{2} and \cite{3}. The generalized Nevanlinna counting function $\mathfrak{N}_{\varphi, \sigma}$ provides the following non-univalent change of variable formula (see \cite{2}, Theorem 2.32). \end{theorem} \begin{lemma} If $g$ and $\sigma$ are positive measurable function on $\mathbb D$ and $\varphi$ a holomorphic self-map of $\mathbb D,$ then \begin{equation*}\displaystyle\int_{\mathbb D} ( g \circ \varphi)(z)|\varphi'(z)|^2 \sigma(z) dA(z) = \displaystyle\int_{\mathbb D} g(z) \mathfrak{N}_{\varphi, \sigma}(z) dA(z). \end{equation*} Recall that the essential norm $||T||_e$ of a bounded linear operator on a Banach space $X$ is given by $$||T||_e = \inf \{ ||T - K||: K \mbox { is compact on } X \}.$$ It provides a measure of non-compactness of $T.$ Clearly, $T$ is compact if and only if $||T||_e = 0.$ \\ Let $\varphi$ be a non-constant analytic self-map (a so called Schur function) of $\DD$ and let $C_\varphi : \mathcal{H}_\omega \to H(\DD)$ the associated composition operator: $$C_\varphi f = f \circ \varphi.$$ For more about composition operators on weighted Hardy spaces, see \cite{3}, \cite{11} and \cite{15}.\\ The next theorem can be found in \cite{15}. \end{lemma} \begin{theorem} Let $\sigma_1$ and $\sigma_2$ be two admissible weights $((I)$-admissible or $(II)$-admissible$)$ and $\varphi$ be a holomorphic self-map of $\mathbb{D}$. Then $C_\varphi : \mathcal{H}_{\sigma_1}\rightarrow \mathcal{H}_{\sigma_2}$ is bounded if and only if $$\sup_{|z| < 1} \frac{\mathfrak{N}_{\varphi, \sigma_2}(z)}{\sigma_1(z)} < \infty. $$ Moreover, if $C_\varphi : \mathcal{H}_{\sigma_1}\rightarrow \mathcal{H}_{\sigma_2}$ is bounded, then $$||C_\varphi||^2_{\mathcal{H}_{\sigma_1}\rightarrow \mathcal{H}_{\sigma_2}}\asymp \sup_{|z| < 1}\frac{\mathfrak{N}_{\varphi, \sigma_2}(z)}{\sigma_1(z)}.$$ \end{theorem} As in \cite{5}, we first introduce the following notations. If $$\varphi^\sharp(z) = \displaystyle\lim_{w \to z}\frac{\rho(\varphi(w), \varphi(z))}{\rho(w, z)} = \frac{|\varphi'(z)|(1 - |z|^2)}{1 - |\varphi(z)|^2}$$ is the pseudo-hyperbolic derivative of $\varphi,$ we set: $$[\varphi] = \displaystyle\sup_{z \in \mathbb{D}}\varphi^\sharp(z) = ||\varphi^\sharp||_\infty.$$\\ Also recall that the approximation (or singular) numbers $a_n(T)$ of an operator $T \in \mathcal{L}(H_1, H_2),$ between two Hilbert spaces $H_1$ and $H_2$ are defined by: $$a_n(T) = \inf\{||T - R||; \;\;rank(R) < n\},\;\; n = 1,2, \cdots.$$ We have $$a_n(T) = c_n(T) = d_n(T),$$ where the numbers $c_n$(resp. $d_n$) are the Gelfand (resp. Kolmogorov) numbers of $T$ (\cite{1}, page 59 and page 51 respectively). In the sequel we shall need the following quantity: $$\tau(T) = \liminf_{n \to \infty}[a_n(T)]^{1/n}.$$ These approximation numbers form a non-increasing sequence such that $$a_1(T) = ||T||, \;\; a_n(T) = \sqrt{a_n(T^*T)}$$ are verify the so-called ``ideal" and ``subadditivity" properties (\cite{4}, see page 57 and page 68): $$a_n(ATB) \leq ||A|| a_n(T) ||B||; \;\; a_{n + m - 1}(S + T) \leq a_n(S) + a_m(T).$$ Moreover, the sequence $(a_n(T))$ tends to $0$ if and only if $T$ is compact. If for some $p,$ $1 \leq p < \infty,$ $(a_n(T))\in l_p,$ where $$l_p = \Big\{a = \{a_n\}^\infty_{n = 1} : ||a||_p = \Big(\displaystyle\sum^\infty_{n = 1}|a_n|^p\Big)^{1/p} < \infty\Big\},$$ then we say that $T$ belongs to the Schatten class $S_p.$\\ The upper and lower bounds for approximation numbers of composition operators on the Hardy space were computed by Li, Queffelec and Rodriguez-Piazza in \cite{5}. In this paper, we generalized some of the results concerning upper and lower bounds for approximation numbers of composition operators to weighted Hardy spaces $\mathcal H_\sigma$ under some conditions on the weight function $\sigma.$ \\ Throughout the paper constants are denoted by $C,$ they are positive and not necessarily the same at each occurrence. The notation $A \lesssim B$ means that there is a positive constant $C$ such that $\leq CB.$ When $A \lesssim B$ and $B \lesssim A$, we write $A \asymp B.$ \section{Lower Bound} We first show that, each M\"{o}bius transformations $\beta_z$ always induce a bounded composition operator on $\mathcal H_\sigma$. This property ensures that, we may consider the operator $C_\varphi$ under the assumption $\varphi(0) = 0.$\\ \begin{proposition} { Let $\sigma$ be an admissible weight. Then for each $z \in \mathbb D$, $C_{\beta_z}$ is bounded on $\mathcal{H}_\sigma.$} \end{proposition} \begin{proof} By the change of variable formula, we have \begin{align} \|C_{\beta_z}f\|^2_{\mathcal{H}_\sigma} & = |f(\beta_z(0))|^2 + \int_{\mathbb D} |f'(\beta_z(w))|^2 |\beta_z'(w)|^2 \sigma(w) dm(w) \notag\\ & = |f(z)|^2 + \int_{\mathbb D} |f'(w)|^2 |\beta_z'(\xi_a(w))|^2 \sigma(\beta_z(w)) |\beta_z'(w)|^2 dm(w) \notag\\ & = |f(z)|^2 + \int_{\mathbb D} |f'(w)|^2 |(\beta_z \circ \beta_z)'(w)|^2 \sigma(\beta_z(w)) dm(w) \notag\\ & = |f(z)|^2 + \int_{\mathbb D} |f'(w)|^2 \sigma(\beta_z(w)) dm(z).(3) \end{align} By Lemma 2.1 of \cite{3}, we have $$\sigma(\beta_z(w)) \asymp \sigma(w). \;\;\;\;\eqno(4)$$ From $(3)$ and $(4),$ we have $$\|C_{\beta_z}f\|^2_{\mathcal{H}_\sigma} \lesssim |f(z)|^2 + \|f\|^2_{\mathcal{H}_\sigma}$$ for each $f \in \mathcal{H}_\sigma.$ This implies that $C_{\beta_z}(\mathcal{H}_\sigma) \subset \mathcal{H}_\sigma.$ Thus by closed graph theorem, $C_{\beta_z}$ is bounded on $\mathcal{H}_\sigma$. \end{proof} \begin{proposition} For each $z \in \mathbb D$, $C_{\beta_z}$ is invertible. \end{proposition} \begin{proof} By Proposition 1, $C_{\beta_z}$ is bounded. Now the proof is an easy consequence of Theorem 1.6 in \cite{2}.\\ In the following result, we show that if $\sigma$ is $II$-admissible, or $\sigma$ is $I$-admissible and $C_\varphi$ is compact on $\mathcal H_\sigma$, then the approximation numbers of $C_\varphi$ on $\mathcal H_\sigma$ cannot supersede a geometric speed. \end{proof} \begin{theorem} { Let $\sigma$ be an admissible weight and $\varphi$ be a Schur function such that for $C_\varphi : \mathcal{H}_\sigma \to \mathcal{H}_\sigma$ is bounded. Suppose that $C_\varphi$ is compact on $\mathcal H_\sigma$, whenever $\omega$ is $I$-admissible. Then there exist positive constant $C > 0$ and $0 < r < 1$ such that $$a_n(C_\varphi)\geq Cr^n, \;\;\;\;\; n = 1,2,\cdots.$$ More precisely, one has $\beta (C_\varphi)\geq [\varphi]^2$ and hence for each $k < [\varphi]$ there exist a constant $C_k > 0$ such that $$a_n(C_\varphi)\geq C_k k^{2n}.$$}\\ For the proof we need the following lemma (see \cite{5}). \end{theorem} \begin{lemma} { Let $T : H \to H$ be a compact operator. Suppose that $(\lambda_n)_{n \geq 1}$ the sequence of eigenvalues of $T$ rearranged in non-increasing order satisfies for some $\delta > 0$ and $r \in (0, 1)$ $$|\lambda_n| \geq \delta r^n, \;\;\; n = 1,2, \cdots.$$ Then there exist $\delta_1 > 0$ such that $$a_n(T) \geq \delta_1 r^{2n},\;\;\; n = 1,2,\cdots.$$ In particular $\beta(T) \geq r^2.$} \end{lemma} \begin{proposition} {Let $\omega$ be an admissible weight and $\varphi$ be a Schur function such that for $C_\varphi : \mathcal{H}_\omega \to \mathcal{H}_\omega$ is compact. Then $\tau(C_\varphi)\geq [\phi]^2.$} \end{proposition} \begin{proof} The proof follows on same lines as the proof of Proposition 3.3 in \cite{5}. We include it for completeness. For every $z \in \mathbb{D},$ let $\beta_z$ be the involutive automorphism of $\mathbb D.$ Then we have $$\beta_z(z) = 0, \; \beta_z(0) = z, \; \beta'_z(z) = \frac{1}{|z|^2 - 1}, \; \beta'_z(0) = |a|^2 - 1.$$ Let $\psi = \beta_{\varphi(z)}\circ \varphi\circ \beta_z.$ Then $0$ is a fixed point of $\psi,$ whose derivative by the chain rule is $$\psi'(0) = \beta'_{\varphi(z)}(\phi(z)) \varphi'(z)\beta'_z(0) = \frac{\varphi'(z)(1 - |z|^2)}{1 - |\varphi(z)|^2} = \varphi^\sharp (z).$$ By Schwarz's lemma $$\frac{(1 - |z|^2)}{1 - |\varphi(z)|^2} |\varphi'(z)| = |\psi'(0)| \leq 1.$$ Let us first assume that, the composition operator $C_\varphi$ is compact on $\mathcal H_\sigma$. Then so is $C_\psi,$ since we have $$C_\psi = C_{\beta_z}\circ C_\varphi\circ C_{\beta_{\varphi(z)}}.$$ If $\psi'(0)\neq 0,$ the sequence of eigenvalues of $C_\psi$ the Hardy space $H^2$ is $([\psi'(0)]^n)_{n \geq 0}$ (see \cite{2}, page 96). Since $II$-admissibility corresponds to the case $\mathcal{H}_\sigma \subset H^2$, so the result given for $H^2$ holds for $\mathcal{H}_\sigma$ and would also holds for any space of analytic functions in $\mathbb D$ on which $C_\psi$ is compact. By Lemma $2.4,$ we have $$\tau(C_\psi)\geq |\psi'(0)| = |\varphi^\sharp(z)|^2 \geq 0.$$ This trivially still holds if $\psi'(0) = 0.$ Now since $C_{\beta_z}$ and $C_{\beta_{\varphi(z)}}$ are invertible operators, we have that $\tau(C_\varphi) = \tau(C_\psi)$ and therefore, we have $$\tau(C_\varphi) = [\varphi]^2$$ for all $ z\in \mathbb{D}.$ By passing to the supremum on $z\in \mathbb{D},$ we end the proof of Proposition $2.5$ and that of Theorem $2.3$ in the compact case. If $C_\varphi$ is not compact, the proposition trivially holds. Indeed, in this case, we have $\tau (C_\varphi) = 1 \geq [\varphi]^2.$ \end{proof} \section{ Upper Bound} \begin{theorem} { Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$ such that $\varphi(0) = 0.$ Let $\sigma$ be an admissible weight. Assume that $\sup \frac{\sigma(k)}{\sigma (k + n)} < \infty$ and $r \in (0 , 1)$ is fixed. Then the approximation number of $C_\varphi : \mathcal{H}_{\sigma_1} \to \mathcal{H}_{\sigma_2}$ has the upper bound \begin{align} a_n(C_\varphi) \lesssim \displaystyle\inf_{0 < h < 1} &\bigg [(1 - h)^{2n} \sum_{k = 0}^{\infty}\frac{k^2(1 - h)^{2(k - 1)}}{\sigma_k} + (1 - h)^{2n - 2} \sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k}\bigg] \notag\\ &\bigg (\sup_{1\leq j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg ) + \displaystyle\sup_{z \in \mathbb D}\frac{\mu_{\sigma, \varphi, h}(D(z, r))}{\sigma(z)(1 - |z|^2)^2}.\end{align}} To prove the theorem, we need the following lemma.\\ \noindent{\bf Lemma 3.2.} {\it Let $f(z) = \displaystyle\sum_{k = n}^{\infty}a_k z^k$ and $g(z) = z^n f(z)$. Then $$\|g\|^2_{\mathcal{H}_\sigma} \leq \sup_{1\leq j <\infty}\frac{\sigma_j}{\sigma_{j + n}}\|f\|^2_{\mathcal{H}_\sigma}.$$} \end{theorem} \begin{proof} $ \displaystyle\|g\|^2_{\mathcal{H}_\sigma} = \sum_{k = 0}^{\infty}|a_{k + n}| \sigma_k = \sum_{k = 0}^{\infty}|a_{k + n}| \sigma_{k + n} \frac{\sigma_k}{\sigma_{k + n}} \leq \sup_{1\leq k <\infty}\frac{\sigma_k}{\sigma_{k + n}}\|f\|^2_{\mathcal{H}_\sigma}.$ \end{proof} \begin{proof} We denote by $P_n$ the projection operator defined by $$P_n f = \displaystyle\sum^{n - 1}_{k = 0} \hat{f}(k) z^k$$ and we take $R = C_\varphi \circ P_n$, that is, if we have $f(z) = \displaystyle\sum^\infty_{k = 0} \hat{f}(k) z^k \in \mathcal{H}_\sigma$ then $$R(f) = \displaystyle\sum^{n - 1}_{k = 0} \hat{f}(k) \varphi^k$$ so that $(C_\varphi - R)f = C_\varphi(r).$ Then, we have $$ r(z) = \displaystyle\sum^\infty_{k = n} \hat{f}(k) z^k = z^n s(z),$$ where \begin{align} \;\; ||s||^2_{\mathcal{H}_\sigma} \leq C \sup \frac{\sigma(j)}{\sigma (j + k)}||r||^2_{\mathcal{H}_\sigma},\mbox{and} ||r||_{\mathcal{H}_\sigma}\leq ||f||_{\mathcal{H}_\sigma}.\end{align} Assume that $||f||_{\mathcal{H}_\sigma} \leq 1$ and $dm_{\varphi, \sigma} = \mathfrak{N}_{\varphi, \sigma}(z)d m(z).$ Fix $0 < h < 1.$ Let $$\mu_{\varphi, \sigma}(z) = (m_{\varphi, \sigma}\circ \varphi^{-1})(z)$$ and $\mu_{\varphi, \sigma, h}$ be the restriction of the measure $\mu_{\varphi, \sigma}(z) $ to the annulus $1 - h < |z| \leq 1.$ Then we have \begin{align} ||(C_\varphi - R)f ||^2_{\mathcal{H}_\sigma} & = ||C_\varphi(r)||^2_{\mathcal{H}_\sigma} \notag \\ & = |r(\varphi(0))|^2 + \int_\mathbb{D}|r'(\varphi(z))|^2 |\varphi'(z)|^2 \sigma(z)d m(z) \notag \\ & = \int_\mathbb{D}|r'(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag\\ & \leq \int_{|z| \leq 1 - h}|r'(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) + \int_{1 - h\leq |z| \leq 1}|r'(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\ & = I_1 + I_2.\end{align} Let $(z_{n})_{n \in \NN}$ be a sequence with a positive separation constant such that $$ \displaystyle\cup_{n = 1}^{\infty}D(z_{n},r) = \mathbb D$$ and every point in $ \mathbb D$ belongs to at most $M$ sets in the family $\{D(z_{n}, 2r)\}_{n\in \mathbb {N}}.$ Since $\sigma$ is an almost standard weight we have that for $0