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\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 $0<r_1<r_2<1$
$$\bigg(\frac{1-r_2}{1-r_1}\bigg)^{t+1}w(r_1)\le w(r_2)\le
w(r_1).$$ From this and since $1-|z|\asymp 1-|z_n|,$ for $z\in
D(z_{n}, 2r),$ we obtain
$$\sigma(z)\asymp \sigma(z_n),\quad z\in D(z_{n}, 2r).$$
Using these facts we obtain
\begin{align}
I_1
& = \int_{|z| \leq 1 - h}|(z^n s'(z) + n z^{n - 1}s)(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag\\
& \leq  \int_{|z| \leq 1 - h} |z^n s'(z)|^2   \mathfrak{N}_{\varphi, \sigma}(z)d m(z) + n^2\int_{|z|\leq 1 - h}|z^{n - 1}s(z)|^2  \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\
& \leq  (1 - h)^{2n}\int_{|z| \leq 1 - h} | s'(z)|^2  \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\ &  + n^2 (1 - h)^{2n - 2}\int_{|z|\leq 1 - h}|s(z)|^2  \mathfrak{N}_{\varphi, \sigma}(z)d m(z).
\end{align}
Thus by Lemma 3.2,  (2) and  (6), we have
\begin{align}
(1 - h)^{2n}\int_{|z| \leq 1 - h}& | s'(z)|^2   \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\
& \leq  (1 - h)^{2n}\|s\|^2_{\mathcal H_\sigma} \bigg(\displaystyle\sum^\infty_{k = 0} k^2 (1 - h)^{2(k - 1)} \sigma_k^{-1}\bigg)\int\mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\
   & \lesssim  (1 - h)^{2n} \sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}}\|r\|^2_{\mathcal{H}_\sigma}\displaystyle\sum^\infty_{k = 0} k^2 (1 - h)^{2(k - 1)} \sigma_k^{-1} \|\varphi\|^2_{\mathcal H_\sigma} \notag \\
   &\lesssim  (1 - h)^{2n} \sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}}\|r\|^2_{\mathcal{H}_\sigma}\displaystyle\sum^\infty_{k = 0} k^2 (1 - h)^{2(k - 1)} \sigma_k^{-1} \|\varphi\|^2_{H^\infty} \notag \\
   & \lesssim  (1 - h)^{2n} \bigg(\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}}\bigg)\displaystyle\sum^\infty_{k = 0} k^2 (1 - h)^{2(k - 1)} \sigma_k^{-1} . \end{align}
Again by Lemma 3.2,  (1) and   (6), we have\begin{align}
(1 - h)^{2n - 2} &\int_{|z| \leq 1 - h}|s(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\
&\lesssim   (1 - h)^{2n - 2} \|s\|^2_{\mathcal H_\sigma} \sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k} \int\mathfrak{N}_{\varphi, \sigma}(z)d m(z)\notag \\
& \lesssim   (1 - h)^{2n - 2}\bigg (\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg )\|r\|^2_{\mathcal{H}_\sigma}\sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k}\|\varphi\|^2_{\mathcal H_\sigma}
\notag \\ & \lesssim   (1 - h)^{2n - 2} \bigg (\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg ) \|r\|^2_{\mathcal H_\sigma}\sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k}\|\varphi\|^2_{H^\infty} \notag \\
& \lesssim   (1 - h)^{2n - 2} \bigg (\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg ) \sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k}.
\end{align}
Combining $(8),\; (9)$ and $(10),$ we have
\begin{align}
I_1   \lesssim  (1 - h)^{2n} &\bigg (\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg ) \sum_{k = 0}^{\infty}\frac{k^2(1 - h)^{2(k - 1)}}{\sigma_k} \notag\\
&+ (1 - h)^{2n - 2} \bigg (\sup_{1\leq  j <\infty}\frac{\sigma_j}{\sigma_{j + n}} \bigg )\sum_{k = 0}^{\infty}\frac{(1 - h)^{2k}}{\sigma_k}.\end{align}
Again
\begin{align}
\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; I_2 & =\int_{1 - h < |z| < 1}|r'(z)|^2 \mathfrak{N}_{\varphi, \sigma}(z)d m(z) \notag \\
& = \int_{\mathbb D}|r'(z)|^2 d\mu_{\sigma,\varphi, h}(z) \notag\\
& \leq \sum^{\infty}_{n=1}\int_{D(z_{n},r)}|r'(z)|^{2}d\mu_{\sigma,\varphi, h}(z) \notag\\
&\leq \sum^{\infty}_{n=1}\mu_{\sigma,\varphi, h}(D(z_{n},r))\sup_{\sigma \in D(z_{n},r)}|r'(\sigma)|^{2}\nonumber\\
& \lesssim\sum^{\infty}_{n=1}\frac{\mu(D(z_{n},r))}{\sigma(z_n)(1 - |z_n|^2)^2}\int_{D(z_{n},2r)}|r'(z)|^{2}\sigma(z)dm(z)\nonumber\\
& \lesssim\sup_{z \in \mathbb D}\frac{\mu_{\sigma,\varphi, h} (D(z, r))}{\sigma(z)(1 - |z|^2)^2}\sum^{\infty}_{n=1}\int_{D(z_{n},2r)}|r'(z)|^{2}\sigma(z)dm(z)\nonumber\\
& \lesssim\sup_{z \in \mathbb D}\frac{\mu_{\sigma,\varphi, h} (D(z, r))}{\sigma(z)(1 - |z|^2)^2}\int_{\mathbb
D}|r'(z)|^{2}\sigma(z)dm(z)\nonumber\\
& \lesssim \sup_{z \in \mathbb D}\frac{\mu_{\sigma,\varphi, h} (D(z, r))}{\sigma(z)(1 - |z|^2)^2}\|r\|^{2}_{H_\sigma} \notag \\
&\lesssim \sup_{z \in \mathbb D}\frac{\mu_{\sigma,\varphi, h} (D(z, r))}{\sigma(z)(1 - |z|^2)^2}.\end{align}
Combining $(7),\; (11)$ and $(12),$ we get the desired upper bound given in  (5).
\end{proof}

{\bf Acknowledgments.} This work  is a part of
the research project sponsored by National Board of Higher
Mathematics (NBHM)/DAE, India (Grant No. 48/4/2009/R\&D-II/426).

\bibliographystyle{amsplain}
\begin{thebibliography}{99}
\bibitem {1} B. Carl, I. Stephani,  Entropy, \textit{Compactness and the Approximation of operators}, { Cambridge Tracts in Mathematics 98, Cambridge University Press} (1990).
\bibitem {2} C.C. Cowen and B.D. MacCluer, \textit{Composition operators on spaces of analytic functions}, { CRC Press Boca Raton, New York,} 1995.
\bibitem {3} K. Kellay and P. Lefevre, \textit{Compact composition operators on weighted Hilbert spaces of analytic functions,} {J. Math. Anal. Appl.} {\bf 386} (2012) 718--727.
\bibitem {4} H. Konig, \textit{Eigenvalue Distribution of Compact operators,} { Operator Theory, Advances and Application Vol. 16, Birkhauser, Basel-Boston-Stuttgart} (1986). 
\bibitem {5} D. Li, H. Queffelec and L. Rodriguez-Piazza, \textit{On approximation numbers of composition operators,} { J. Approx. Theory} {164} (2014) 431--459.
\bibitem {6} P. Lefevre, D. Li , H. Queffelec, and L. Rodriguez-Piazza, \textit{Some examples of compact composition operators on $\mathcal {H}^2$,} { J. Funct. Anal.} {\bf 255}  (2008), 3098--3124.
\bibitem {7}  P. Lefevre, D. Li , H. Queffelec, and L. Rodriguez-Piazza, \textit{Compact composition operators on $\mathcal {H}^2$ and Hardy-Orlicz spaces,} { J. Math. Anal. Appl.} {\bf 354}  (2009), 360--371.
\bibitem {8} P. Lefevre, D. Li , H. Queffelec, and L. Rodriguez-Piazza, \textit{Nevanlinna counting function and Carleson function of analytic maps,} {Math Ann.} {\bf 351}  (2011), 305--326.
\bibitem {9} P. Lefevre, D. Li , H. Queffelec, and L. Rodriguez-Piazza, \textit{Some revisited results about composition operators on Hardy spaces,}  {Rev. Mat. Iberoam.}  {\bf 28}  (2012), 57--76
\bibitem{10} {\sc J.H. Shapiro},  \textit{Composition operators and classical function theory,} Springer-Verlag, New York  1993.
\bibitem{11}	A.K. Sharma, \textit{Essential norm of generalized composition operators on weighted Hardy spaces,} Operators and Matrices {\bf 8} (2014), 399-409.
\bibitem{12}	A.K. Sharma, \textit{Weighted composition operators from Cauchy integral transforms to logarithmic weighted-type spaces,} Ann. Funct. Anal. {\bf 4}  (2013), 163–174.
\bibitem{13} { A.K. Sharma} and { S. Ueki},  \textit{Composition operators from Nevanlinna type spaces to Bloch type spaces,} {Banach J. Math. Anal.} {\bf 6} (2012),  112--123. 
\bibitem{14} {A. Sharma} and {A.K. Sharma}, \textit{Carleson measures and a class of generalized integration operators on the Bergman space,} { Rocky Mountain J. Math.}   {\bf 41} (2011), 1711--1724.
\bibitem {15} {S. Stevi\'c} and  {A.K. Sharma}, \textit{ Essential norm of composition operators between weighted Hardy spaces,} {Appl. Math. Comput.} {\bf 217} (2011), 6192--6197.
\end{thebibliography}

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