# Nuclear Volterra composition operators between Bloch and weighted type
spaces

Aakriti Sharma  Department of Mathematics, Central University of Jammu,
Central University of Jammu, (Bagla) Rahya-Suchani, Samba-181143, J&K, India.
<EMAIL_ADDRESS>and  Ajay K. Sharma  Department of Mathematics,
Central University of Jammu, Central University of Jammu, (Bagla) Raya-
Suchani, Samba-181143, J&K, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we completely characterize nuclear Volterra composition
operators
$T^{\phi}_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
and
$S^{\phi}_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
acting between weighted type spaces in terms of the symbols $g$ and $\phi$ of
$T^{\phi}_{g}$ and $S^{\phi}_{g}$ and weights $\nu$ and $\mu$, when the
weights $\nu$ and $\mu$ are normal weights in the sense of Shields and
Williams. Moreover, nuclear Volterra composition operators acting between
little weighted type spaces and Bloch spaces of order $\beta$ are also
characterized.

###### Key words and phrases:

Nuclear operator; Volterra composition operator, weighted Banach space,
Bergman space, weighted Bloch space, little weighted Banach space, little
weighted Bloch space.

###### 2010 Mathematics Subject Classification:

47B33, 47B38, 46E10, 32A37.

The first author is thankful to CSIR(India), Grant Number (File no.
09/1231(0001)/2019-EMR-I).
The second author is thankful to DST(SERB) for the project grant under MATRICS
scheme. Grant No. MTR/2018/000479.

##  1\. Introduction

Denote by ${H}(\mathbb{D})$, the class of all analytic functions on
$\mathbb{D}$, where $\mathbb{D}$ is the open unit disk in the complex plane
$\mathbb{C}$. For $g\in H(\mathbb{D}),$ Ch. Pommerenke [28] introduced an
integral operator $T_{g}:H(\mathbb{D})\to H(\mathbb{D})$ which maps every
$f\in H(\mathbb{D})$ to a function vanishing at $0,$ defined as

$T_{g}f(z)=\int_{0}^{z}f(\zeta)dg(\zeta)=\int_{0}^{z}f(\zeta)g^{\prime}(z)d\zeta,\;\;\;z\in\mathbb{D}.$

Another integral operator, generally known as a companion operator of $T_{g}$
was defined by Yoneda [31] as

$S_{g}f(z)=\int_{0}^{z}f^{\prime}(\zeta)g(\zeta)d\zeta,\;\;\;z\in\mathbb{D}.$

The operators $T_{g}$ and $S_{g}$ can be viewed as cousins of Hankel and
Toeplitz operators and they appeared in the literature under various names
such as Volterra operators, Cesaro operators, Riemann Stieltjes operators and
integration operators. For any $g\in{H}(\mathbb{D})$ and $\phi$ an analytic
self-map of $\mathbb{D}$, the Volterra composition operators $T^{\phi}_{g}$
and $S^{\phi}_{g}$ are defined, respectively on ${H}(\mathbb{D})$ by

$(T^{\phi}_{g}f)(z)=\int_{0}^{z}f(\phi(\zeta))g^{\prime}(\zeta)d\zeta,\;\;\;z\in\mathbb{D}$

and

$(S^{\phi}_{g}f)(z)=\int_{0}^{z}f^{\prime}(\phi(\zeta))g(\zeta)d\zeta,\;\;\;z\in\mathbb{D}.$

Recently, these type of operators are considered on spaces of analytic
functions by several authors, see [1]-[6], [10],[11],[16],[20]-[22] and
references therein. Motivated by results in [8] and [17], in this paper, we
characterize nuclear Volterra composition operators between Bloch and weighted
type spaces of analytic functions, when the weights are normal. Recall that a
radial weight $\nu$ is a non-increasing, non-negative continuous function
defined on $\mathbb{D}$ such that $\nu(z)=\nu(|z|)$ for all $z\in\mathbb{D}.$
A radial weight $\nu$ is normal in the sense of Shields and Williams if it
satisfies following two conditions:

  1. ($\mathbf{I}$).

There exists $\beta>0$ such that the function
$r\rightarrow\frac{\nu(r)}{(1-r^{2})^{\beta}}$ is almost increasing, or
equivalently, $\inf_{n}\frac{\nu(1-2^{-(n+1)})}{\nu(1-2^{-n})}>0$.

  2. ($\mathbf{II}$).

There exists $\gamma>0$ such that the function
$r\rightarrow\frac{\nu(r)}{(1-r^{2})^{\gamma}}$ is almost decreasing, or
equivalently, these is some $k\in\mathbb{N}$ such that
$\limsup_{n}\frac{\nu(1-2^{-(n+k)})}{\nu(1-2^{-n})}<1$.

It is clear that the standard weights
$\nu_{\alpha}(z)=(1-|z|^{2})^{\alpha},(\alpha>0)$ are normal weights.
Throughout this paper, we denote the set of all normal weights on $\mathbb{D}$
by $N_{W}(\mathbb{D}).$ For more about normal weights, we refer to [29],[6]
and [15]. The weighted Banach spaces $H^{\infty}_{\nu}$ and $H_{\nu}^{0}$ are
defined as follows:

$\mathcal{H}_{\nu}^{\infty}=\bigg{\\{}{f\in{H}(\mathbb{D}):\|f\|_{\mathcal{H}_{\nu}^{\infty}}=\sup_{z\in\mathbb{D}}\nu(z)|f(z)|<\infty\bigg{\\}}}$

and

$H_{\nu}^{0}=\bigg{\\{}{f\in{H}(\mathbb{D}):\lim_{|z|\rightarrow
1^{-}}\nu(z)|f(z)|=0\bigg{\\}}}.$

The weighted Bloch space $\mathcal{B}_{\nu}$ is the space of all functions $f$
in ${H}(\mathbb{D})$ such that
$\|f\|_{\nu}=\sup_{z\in\mathbb{D}}\nu(z)|f^{\prime}(z)|<{\infty}.$ Note that
$\|\cdot\|_{\nu}$ is a semi-norm on $\mathcal{B}_{\nu}$ and a norm can be
defined on $\mathcal{B}_{\nu}$ as

$\|f\|_{\mathcal{B}_{\nu}}=|f(0)|+\sup_{z\in\mathbb{D}}\nu(z)|f^{\prime}(z)|.$

Endowed with the norm $\|\cdot\|_{\mathcal{B}_{\nu}}$, $\mathcal{B}_{\nu}$
becomes a Banach space. Similarly, the little weighted Bloch space
$\mathcal{B}^{0}_{\nu}$ is defined as $\lim_{|z|\rightarrow
1^{-}}\nu(z)|f^{\prime}(z)|=0.$ If $\lim_{|z|\rightarrow 1^{-}}\nu(z)=0$, then
the weight $\nu$ is called typical and for typical weights $H_{\nu}^{\infty}$
is the bidual space of $H_{\nu}^{0}$. If the weight $\nu$ satisfies the
condition ($\mathbf{I}$), then $\nu$ satisfies condition ($\mathbf{II}$) if
and only if $H_{\nu}^{0}=\mathcal{B}^{0}_{\nu(z)(1-|z|^{2})}$, see [25]. In
particular, if $\nu\in N_{W}(\mathbb{D})$, then
$H_{\nu}^{0}=\mathcal{B}_{\nu(z)(1-|z|^{2})}^{0}$ and by duality,
$H_{\nu}^{\infty}=\mathcal{B}_{\nu(z)(1-|z|^{2})}$, see [23]. Moreover, if
$\nu\in N_{W}(\mathbb{D})$, then the topological isomorphisms
$H_{\nu}^{0}\simeq c_{0}$ and $H_{\nu}^{\infty}\simeq l_{\infty}$ hold, see
[8],[30] and [23]-[26]. Next, let us recall a duality pairing between
$\mathcal{H}^{\infty}_{\nu}$ and a weighted Bergman space, which was
established by Shields and Williams in [29]. Following [29], let $\nu$ be a
typical weight and $\omega$ a radial, positive and continuous function which
satisfies $\int_{0}^{1}\omega(r)dr<\infty.$ Let $A^{1}_{\omega}$ be the
subspace of $H(\mathbb{D})$ consists of functions $f$ such that
$\|f\|_{A^{1}_{\omega}}=\int_{\mathbb{D}}|f(z)|\omega(z)dA(z)<\infty,$ where
$dA$ is the normalized Lebesgue measure on $\mathbb{D}.$ For a normal weight
$\nu$ and $\omega$ a function defined as above, the pair $\\{\nu,\omega\\}$ is
called a normal pair if

$\nu(r)\omega(r)=(1-r^{2})^{\alpha},\;\;\;0\leq r<1$ (1.1)

for some $\alpha>\beta-1$, where $\beta$ is a positive real number defined as
in $(\mathbf{I}).$ For a normal pair $\\{\nu,\omega\\}$, the following pairing
between $\mathcal{H}^{\infty}_{\nu}$ and $A^{1}_{\omega}$ is defined in [29],
see p. 292.

$\displaystyle(f,g)$
$\displaystyle=\int_{\mathbb{D}}f(z)g(\bar{z})(1-|z|^{2})^{\alpha}dA(z)=\int_{\mathbb{D}}f(\bar{z})g({z})(1-|z|^{2})^{\alpha}dA(z).$
(1.2)

The pairing $(\cdot,\cdot)$ define a duality relation
$(\mathcal{H}^{0}_{\nu})^{\prime}=A^{1}_{\omega},$ see [29], Theorem 2, p.
296. Thus we also have
$(\mathcal{H}^{0}_{\nu})^{\prime\prime}=\mathcal{H}^{\infty}_{\nu}.$ Moreover,
if $K_{\zeta}(z)=(\alpha+1)/(1-{\zeta}z)^{\alpha+2},$ $\alpha>-1$ is the
kernel function and $\\{\nu,\omega\\}$ a normal pair, then
$K_{\zeta}\in\mathcal{H}^{0}_{\nu}\cap A^{1}_{\omega},$
$g(\zeta)=(K_{\zeta},g)$ for all $g\in A^{1}_{\omega}$ and
$f(\zeta)=(f,K_{\zeta})$ for all $f\in\mathcal{H}^{\infty}_{\nu},$ see Lemma
10 in [29].

Recall that a linear operator $T$ between two Banach space $X$ and $Y$ is

  1. (a)

Nuclear if there is a sequence $(x^{\prime}_{k})\subset X^{\prime}$ and a
sequence $(y_{k})\subset Y$ such that

$\sum_{k}\|x^{\prime}_{k}\|\|y_{k}\|<{\infty}\mbox{ and
}T=\sum_{k=1}^{\infty}x^{\prime}_{k}\otimes y_{k},$

where $x^{\prime}_{k}\otimes y_{k}:X\rightarrow Y$ is the mapping defined as
$x\rightarrow x^{\prime}_{k}(x)y_{k}$.

  2. (b)

Absolutely summing if there is $C>0$ such that for all finite sequences
$(x_{i})_{i=1}^{n}\subset X$ we have

$\sum_{i=1}^{n}\|Tx_{i}\|\leq C\sup_{\|x^{\prime}\|\leq
1}\sum_{i=1}^{n}|x^{\prime}(x_{i})|=C\sup_{|\eta_{i}|=1}\|\sum_{i=1}^{n}\eta_{i}x_{i}\|.$

Moreover, it is well known that any nuclear operator is compact and absolutely
summing. Also if $X=c_{0}$ or $l_{\infty}$, then a linear operator
$T:X\rightarrow Y$ is absolutely summing if and only if it is nuclear, see
[17].

Throughout this paper, $C$ denotes a positive constant, the exact value of
which may be not be same at each occurrence. The expression $A\lesssim B$
means that there is a positive constant $C$ such that $A\leq CB$ and the
expression $A\asymp B$ means that there are positive constants $C$ and $D$
such that $CB\leq A\leq DB$.

##  2\. Nuclear Volterra composition operators

In this section, we characterize nuclear Volterra composition operator between
Bloch and weighted spaces of analytic functions.
Next we present characterizations of bounded Volterra composition operators
between weighted spaces of analytic functions. The arguments are standard and
may possibly appear in literature in a more general sense, so we give an
outline proof for completeness.

######  Theorem 1.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $g\in H(\mathbb{D})$ and $\phi$ a self-map
of $\mathbb{D}$. Then

  1. ($i$)

The following are equivalent.

    1. ($a$)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.

    2. ($b$)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.

    3. ($c$)

$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:

$M_{1}=\sup_{z\in\mathbb{D}}(1-|z|^{2})\frac{\mu(z)}{\nu(\phi(z))}|g^{\prime}(z)|<\infty.$
(2.1)

Moreover,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.1)
holds.

  2. ($ii$)

The following are equivalent.

    1. ($d$)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.

    2. ($e$)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.

    3. ($f$)

$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:

$\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})}{(1-|\phi(z)|^{2})}\frac{\mu(z)}{\nu(\phi(z))}|g(z)|<\infty.$
(2.2)

Moreover,
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.2)
holds.

###### Proof.

$(i)$ Since $\mu$ is a normal weight, so
$\mathcal{H}_{\mu(z)}=\mathcal{B}_{(1-|z|^{2})\mu(z)}.$ Thus boundedness of
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to the boundedness of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}.$
By Proposition 2 in [15], every normal weight is essential, that is there is a
constant $C>0$ such that $\nu(z)\leq\tilde{\nu}(z)\leq C\nu(z)$ for each
$z\in\mathbb{D}$.
$(a)\Leftrightarrow(c)$ Then by Proposition 3.1 in [9], we have that
$(a)\Leftrightarrow(c).$
It is obvious that $(a)\Rightarrow(b).$ Thus to complete the proof, we need to
show that $(b)\Rightarrow(c).$
$(b)\Rightarrow(c).$ Suppose that
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$
is bounded. For a fixed $\zeta\in\mathbb{D},$ consider the function
$f_{\zeta}$ defined on $\mathbb{D}$ as

$f_{\zeta}(z)=\frac{(1-|\phi(\zeta)|^{2})^{\beta+1}}{\nu(\phi(\zeta))(1-\overline{\phi(\zeta)}z)^{\beta+1}},$

where $\beta>0$ is as in condition $(\mathbf{I})$ of normal weight. Then it is
easy to see that $f_{\zeta}\in\mathcal{H}^{0}_{\nu}$ and
$\sup_{\zeta\in\mathbb{D}}\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$ Moreover, $f_{\zeta}(\phi(\zeta))=1/\nu(\phi(\zeta)).$ Thus by the
boundedness of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$,
we have that

$(1-|\zeta|^{2})\mu(\zeta)|g^{\prime}(\zeta)||f_{\zeta}(\phi(\zeta))|\leq\|M_{g^{\prime}}C_{\phi}f_{\zeta}\|_{\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}}\lesssim\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$

Taking supermum over $\zeta\in\mathbb{D},$ we get (2.1).
By Proposition 3.2 in [9], we have that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.1)
holds. This completes the proof of $(i)$.
$(ii)$ Thus boundedness of
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to the boundedness of
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)},$
where $D$ is the differentiation operator. We write an outline proof of
$(d)\Leftrightarrow(f)$ and $(e)\Leftrightarrow(f)$ The details of all other
cases are omitted.
$(d)\Leftrightarrow(f)$ By Theorem 7 in [27], we have that
$(d)\Leftrightarrow(f).$
$(e)\Rightarrow(f).$ Suppose that
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$
is bounded. For a fixed $\zeta\in\mathbb{D},$ consider the function
$f_{\zeta}$ defined as in case $(i).$ Then proceeding as in case $(i)$, we
have that
$f^{\prime}_{\zeta}(\phi(\zeta))=\overline{\phi(\zeta)}/(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta)).$
Thus by the boundedness of
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$,
we have that

$(1-|\zeta|^{2})\mu(\zeta)|g(\zeta)||f^{\prime}_{\zeta}(\phi(\zeta))|\leq\|M_{g}C_{\phi}Df_{\zeta}\|_{\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}}\lesssim\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$

Taking supermum over $\zeta\in\mathbb{D},$ we get

$\sup_{z\in\mathbb{D}}\frac{(1-|\zeta|^{2})\mu(\zeta)}{(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta))}|\phi(\zeta)||g(\zeta)|<\infty.$

Thus

$\sup_{|\phi(\zeta)|>1/2}\frac{(1-|\zeta|^{2})\mu(\zeta)}{(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta))}|g(\zeta)|<\infty.$
(2.3)

The above condition easily yields (2.2) by bifurcating the supermum over
$z\in\mathbb{D}$ in (2.2) as sum of supermum over $|\phi(\zeta)|\leq 1/2$ and
supermum over $|\phi(\zeta)|>1/2$. Thus the proof can be finished, using the
fact that the first term in this sum is definitely finite and the second term
in this sum is finite by (2.3).∎

######  Theorem 2.

Let $\nu,\mu\in N_{W}(\mathbb{D}).$ If $g\in H(\mathbb{D})$ and $\phi$ a self-
map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then

  1. ($i$)

The following are equivalent.

    1. ($a$)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is compact.

    2. ($b$)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.

    3. ($c$)

The bi-transpose $(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$.

  2. ($ii$)

If $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$, then
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded and the bi-transpose $(T^{\phi}_{g})^{\prime\prime}$ of $T^{\phi}_{g}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$. Moreover,
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
compact if and only if
$T^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$

###### Proof.

$(i)$ $(a)\Leftrightarrow(b)$ Since
$\mathcal{H}^{\infty}_{\mu}=\mathcal{B}_{(1-|z|^{2})\mu(z)}$, so
$T_{g}^{\phi}f\in H^{\infty}_{\mu}$ if and only if $M_{g^{\prime}}C_{\phi}f\in
H^{\infty}_{(1-|z|^{2})\mu}.$ Moreover, $T_{g}^{\phi}f$ and
$M_{g^{\prime}}C_{\phi}f$ have comparable norm. Thus compactness (and/or weak
compactness) of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}.$
By Theorem 1 in [7], $M_{g^{\prime}}C_{\phi}f$ is compact if and only if it is
weakly compact. Therefore, it follows that $(a)$ and $(b)$ are equivalent.
$(a)\Rightarrow(c)$ Assume that $(a)$ holds. Then by Gantmacher-Nakamura’s
theorem, see Theorem 5.5 in [12], the bi-transpose
$(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
acts continuously from $\mathcal{H}^{\infty}_{\nu}$ with $w^{*}$-topology to
$\mathcal{H}^{\infty}_{\mu}$ with $w^{*}$-topology. Since pointwise topology
$\tau_{p}$ is weaker than $w^{*}$-topology on $\mathcal{H}^{\infty}_{\nu}$ and
$\mathcal{H}^{\infty}_{\mu}$, so $(T_{g}^{\phi})^{\prime\prime}$ and
$T_{g}^{\phi}$ are both $w^{*}$ and $\tau_{p}$ continuous and they agree in
$\mathcal{H}^{0}_{\nu}$ which is $w^{*}$ dense in
$\mathcal{H}^{\infty}_{\nu}.$ Therefore, $(c)$ holds.
$(c)\Rightarrow(b)$ Assume that the bi-transpose
$(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$. Then it holds that
$(T^{\phi}_{g})^{\prime\prime}(\mathcal{H}^{\infty}_{\nu})\subset\mathcal{H}^{\infty}_{\mu}.$
Using the fact that a bounded linear operator $T$ acting from a Banach space
$X$ to a Banach space $Y$ is weakly compact if and only if
$T^{\prime\prime}(X^{\prime\prime})\subset Y,$ see p. 482 in [14], we have
that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.
$(ii)$ If $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$, then by Theorem 1,
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded. Moreover, by replacing $T_{g}$ in Lemma 1 in [6] by $T^{\phi}_{g}$,
we can easily prove that $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$.
Finally, by Gantmacher-Nakamura’s theorem it holds that the compactness of
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
equivalent to $T^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$
This completes the proof. ∎

######  Theorem 3.

Let $\nu,\mu\in N_{W}(\mathbb{D}).$ If $g\in H(\mathbb{D})$ and $\phi$ a self-
map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then

  1. ($i$)

The following are equivalent.

    1. ($a$)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is compact.

    2. ($b$)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.

    3. ($c$)

The bi-transpose $(S_{g}^{\phi})^{\prime\prime}$ of
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$.

  2. ($ii$)

If $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$, then
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded and the bi-transpose $(S^{\phi}_{g})^{\prime\prime}$ of $S^{\phi}_{g}$
satisfies $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$. Moreover,
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
compact if and only if
$S^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$

###### Proof.

(i) Proceeding as in Theorem 2, $S_{g}^{\phi}f\in H^{\infty}_{\mu}$ if and
only if $M_{g^{\prime}}C_{\phi}Df\in H^{\infty}_{(1-|z|^{2})\mu(z)}$ with
comparable norm. Thus compactness (and/or weak compactness) of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
which is further equivalent to is compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}\longrightarrow\mathcal{H}^{\infty}_{\mu}.$
Thus the first part of theorem can now be settles as the proof of first part
of Theorem 2.
(ii) If $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$, then by Theorem 1,
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded. Moreover, by replacing $S_{g}$ in Lemma 2.12 in [21], we can easily
prove that $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$. The last part can be
settle by Gantmacher-Nakamura’s theorem as in the second part of Theorem 2.∎

Finally, we recall Piestsch’s theorem which plays an important role in the
proof of main results of this paper. For more about the Piestsch’s theorem, we
refer the readers to Theorem 2.12 of [13].

######  Theorem 4.

(Pietsch’s Theorem) Let $E$ and $F$ be Banach spaces and $1\leq p<+\infty$. A
bounded linear operator $T:E\longrightarrow F$ is absolutely p-summing if and
only if there is a constant $C$ and a regular Borel probability measure
$\varrho$ on closed unit ball $B_{E^{*}}$ of $E^{*}$ with the weak∗-topology
$\sigma(E^{*},E)$ such that for every $x\in E$,

$\|T(x)\|\leq
C\left(\int_{B_{E^{*}}}|\phi(x)|^{p}d\varrho(\phi)\right)^{1/p}.$

######  Theorem 5.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ be such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  2. (b)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  3. (c)

$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:

$M_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

###### Proof.

$(c)\Rightarrow(a)$ Since $\mathcal{H}^{0}_{\nu}\simeq c_{0}$, so
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear if and only if
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. Thus to complete the proof of the implication
$(c)\Rightarrow(a)$, we need to prove that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. First note that if $\mu$ is normal,
$\mathcal{H}^{\infty}_{\mu(z)}=\mathcal{B}_{\mu(z)(1-|z|^{2})}$,
$\|T_{g}^{\phi}(f)\|_{\mathcal{H}^{\infty}_{\mu}}\asymp\|T_{g}^{\phi}(f)\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}$
for any $f\in H^{\infty}_{\nu}$ and $T_{g}^{\phi}(f)$ vanishes at origin for
any $f\in H(\mathbb{D})$. Now polynomials are dense in
$\mathcal{H}^{0}_{\mu}$, so it is sufficient to consider polynomials
$f_{1},f_{2},\cdots,f_{N}$ in the definition of absolutely summing. Now for
any $C>1,$ we can select $z_{i}$, $i=1,2,\cdots N$ such that

$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{\mathcal{H}^{\infty}_{\mu}}$
$\displaystyle\asymp\displaystyle\sum_{i=1}^{N}\left(\big{|}(T_{g}^{\phi}f_{i})(0)\big{|}+\sup_{z\in\mathbb{D}}(1-|z|^{2})\mu(z)\big{|}(T_{g}^{\phi}f_{i})^{\prime}(z)\big{|}\right)$
$\displaystyle=\sum_{i=1}^{N}\displaystyle\sup_{z\in\mathbb{D}}(1-|z|^{2})\mu(z)\big{|}f_{i}(\phi(z))g^{\prime}(z)\big{|}$
$\displaystyle\leq
C\displaystyle\sum_{i=1}^{N}(1-|z_{i}|^{2})\mu(z_{i})\big{|}f_{i}(\phi(z_{i}))g^{\prime}(z_{i})\big{|}$
(2.4)

Using the duality pairing in (1.2), we have that

$\displaystyle\big{|}f_{i}(\phi(z_{i}))\big{|}=\big{|}(f_{i},K_{\phi(z_{i})})\big{|}=\bigg{|}(\alpha+1)\int_{\mathbb{D}}f_{i}(\zeta)\frac{(1-|\zeta|^{2})^{\alpha}}{|1-\bar{\zeta}\phi(z_{i})|^{\alpha+2}}dA(\zeta)\bigg{|}.$

Thus from (2.4), we have that

$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}^{\infty}_{\mu}}}$
$\displaystyle\leq
C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}$
$\displaystyle\quad\times\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)\big{|}g^{\prime}(z)\big{|}}{\big{|}1-\bar{\zeta}\phi(z)\big{|}^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle=C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}M_{\alpha}$
(2.5)

Using the equality

$\sup_{w\in\mathbb{D}}\sum_{i=1}^{N}|f_{i}(w)|\nu(w)=\sup_{|\eta_{i}|=1}\sup_{w\in\mathbb{D}}\bigg{|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}(w)\bigg{|}\nu(w)=\sup_{|\eta_{i}|=1}\bigg{\|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}\bigg{\|}_{\mathcal{H}^{\infty}_{\mu}}$

in (2.5), we have that

$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}^{\infty}_{\mu}}}$
$\displaystyle\leq
C\sup_{|\eta_{i}|=1}\bigg{\|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}\bigg{\|}_{\mathcal{H}^{\infty}_{\mu}}.$

Since $C>0$ is arbitrary, so
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing.

$(a)\Rightarrow(c)$ Since every nuclear operator is absolutely summing,
therefore,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. By Theorem 4, there is a probability Borel measure
$\varrho$ on $\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$-compact unit ball
$\mathbb{B}_{1}$ of $A^{1}_{\omega}$, where
$\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$ is the weak∗-topology on
$(\mathcal{H}^{0}_{\mu})^{\prime}=A^{1}_{\omega},$ and some $\xi$ in
$A^{1}_{\omega}$ such that

$\|T_{g}^{\phi}f\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\int_{\mathbb{B}_{1}}|\xi(f)|d\varrho(\xi)$ (2.6)

for every $f\in\mathcal{H}^{0}_{\mu}$ and some constant $C>0$ independent of
$f$. For every $\zeta\in\mathbb{D}$, we have that
$K_{{\zeta}}\in\mathcal{H}^{0}_{\mu}$. Therefore,

$\displaystyle(\alpha+1)\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-{\zeta}\phi(z)|^{\alpha+2}}\leq\|T_{g}^{\phi}K_{{\zeta}}\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}\leq
C\|T_{g}^{\phi}K_{{\zeta}}\|_{\mathcal{H}^{\infty}_{\mu(z)}}.$ (2.7)

Replacing $f$ in (2.6) by $K_{{\zeta}}$, and then using (2.7), we have that

$\displaystyle\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-{\zeta}\phi(z)|^{\alpha+2}}\leq\frac{C}{\alpha+1}\int_{\mathbb{B}_{1}}|\xi(K_{w})|d\varrho(\xi).$

Integrating over $\mathbb{D}$ with respect to $\omega dA$, using (1.1) and
then applying Fubini’s theorem, we get

$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}$
$\displaystyle\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad\quad\leq\frac{C}{\alpha+1}\int_{\mathbb{B}_{1}}\int_{\mathbb{D}}|\xi(K_{\zeta})|\omega(z)dA(\zeta)d\varrho(\xi).$

Now $K_{\zeta}\in\mathcal{H}^{0}_{\mu}$ and $\xi$ in $A^{1}_{\omega}$, so
$\xi(K_{\zeta})=\langle K_{\zeta},\xi\rangle=\xi(w).$ Thus using the fact that
$\varrho$ is a probability Borel measure, we get

$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}$
$\displaystyle\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad\quad\leq\frac{C}{\alpha+1}\sup_{\xi\in\mathbb{B}_{1}}\int_{\mathbb{D}}|\xi(\zeta)|\omega(\zeta)dA(\zeta)$
$\displaystyle\quad\quad=\frac{C}{\alpha+1}\sup_{\xi\in\mathbb{B}_{1}}\|\xi\|_{A^{1}_{\omega}}=\frac{C}{\alpha+1}.$

This completes the proof of the implication $(a)\Rightarrow(c)$.
$(a)\Rightarrow(b)$ Since transpose of a nuclear operator is nuclear, see
3.1.8 in [15], so by an application of Theorem 2(1), it follows that
$(a)\Rightarrow(b)$.
$(b)\Rightarrow(a)$ Recall that class of nuclear operators forms an operator
ideal, that is, nuclear operators are a class of operators which do not fixes
a copy of $l^{\infty},$ (see [7] and [8]), and $\mathcal{H}^{0}_{\nu}$ is a
closed subspace of $\mathcal{H}^{\infty}_{\mu}$, so it is obvious that
$(b)\Rightarrow(a)$. This completes the proof. ∎

######  Theorem 6.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and $\phi$ a self-map of
$\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  2. (b)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  3. (c)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.

  4. (d)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$
is nuclear.

  5. (e)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$M_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

###### Proof.

By Theorem 4, we have that $(a)\Leftrightarrow(b)\Leftrightarrow(e).$ To
complete the proof it is sufficient to prove that
$(a)\Rightarrow(c)\Rightarrow(d)\Rightarrow(a).$
$(a)\Rightarrow(c)$ Suppose that $(a)$ holds. Then by Theorem 2(2)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded. Moreover,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing if and only if
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}.$ Thus
the implication $(a)\Rightarrow(d)$ follows.
$(c)\Rightarrow(d)$ Using Theorem 2(ii) and the fact that transpose of a
nuclear operator is nuclear the implication follows.
$(d)\Rightarrow(a).$ The proof follows as the proof of implication
$(b)\Rightarrow(a)$ in Theorem 4. This completes the proof.∎

######  Corollary 1.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded,
then the following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  2. (b)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 2.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu}$ and $\phi$ a self-map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  2. (b)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  4. (d)

$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$
is nuclear.

  5. (e)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

We can easily obtain the following corollaries.

######  Corollary 3.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ is a
self-map of $\mathbb{D}$ such that
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.

  1. (a)

$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 4.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$\phi\in\mathcal{B}^{0}_{\mu(z)}$ and
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.

  1. (a)

$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  4. (d)

$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  5. (e)

$\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 5.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in{H}(\mathbb{D})$ be such that
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then the
following statements are equivalent.

  1. (a)

$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  3. (c)

For $g$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 6.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ be such that
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is bounded,
then the following statements are equivalent.

  1. (a)

$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  3. (c)

$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is nuclear.

  4. (d)

$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  5. (e)

For $g$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Theorem 7.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.

  1. (a)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.

  2. (b)

$S_{g}^{\phi}:\mathcal{H}_{\nu}\longrightarrow\mathcal{H}_{\mu}$ is nuclear.

  3. (c)

$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:

$N_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

###### Proof.

That $(a)\Leftrightarrow(b)$ can be proved, proceeding as in the proof of
Theorem 5. Thus to complete the proof, we need to prove that
$(a)\Leftrightarrow(c)$. $(c)\Rightarrow(a)$ Proceeding as in the proof of
$(c)\Rightarrow(a)$ of Theorem 5, for any $C>1,$ we can select $z_{i}$,
$i=1,2,\cdots N$ such that

$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\displaystyle\sum_{i=1}^{N}(1-|z_{i}|^{2})\mu(z_{i})\big{|}f_{i}(\phi(z_{i}))g(z_{i})\big{|}$
(2.8)

and

$\displaystyle f_{i}(w)=\langle
f_{i},K_{w}\rangle=(\alpha+1)\int_{\mathbb{D}}f_{i}(\zeta)\frac{(1-|\zeta|^{2})^{\alpha}}{(1-\bar{\zeta}w)^{\alpha+2}}dA(\zeta).$

Differentiating with respect to $w,$ we have that

$\displaystyle
f^{\prime}_{i}(w)=(\alpha+1)(\alpha+2)\int_{\mathbb{D}}f_{i}(\zeta)\frac{\bar{\zeta}(1-|\zeta|^{2})^{\alpha}}{(1-\bar{\zeta}w)^{\alpha+3}}dA(\zeta).$

Thus

$\displaystyle\big{|}f^{\prime}_{i}(\phi(z_{i}))\big{|}\leq(\alpha+1)(\alpha+2)\int_{\mathbb{D}}|f_{i}(\zeta)|\frac{\bar{\zeta}(1-|\zeta|^{2})^{\alpha}}{|1-\bar{\zeta}\phi(z_{i})|^{\alpha+3}}dA(\zeta).$
(2.9)

Using (2.8) and (2.9 ), we have that

$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}_{\mu}^{\infty}}}$
$\displaystyle\leq
C(\alpha+1)(\alpha+2)\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}$
$\displaystyle\quad\times\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)\big{|}g^{\prime}(z)\big{|}}{\big{|}1-\bar{\zeta}\phi(z)\big{|}^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle=C(\alpha+1)(\alpha+2)\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}N_{\alpha}$
(2.10)

The proof can now be completed proceeding as in the Theorem 5. We omit the
details.
$(a)\Rightarrow(c)$ Once again, proceeding as in the proof of Theorem 5, there
exists a probability Borel measure $\varrho$ on
$\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$-compact unit ball
$\mathbb{B}_{1}$ of $A^{1}_{\omega}$ and $\xi$ in $A^{1}_{\omega}$ such that

$\|S_{g}^{\phi}f\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\int_{\mathbb{B}_{1}}|\xi(f)|d\varrho(\xi)$ (2.11)

for every $f\in\mathcal{H}^{0}_{\mu}$ and some constant $C>0$ independent of
$f$. For every $\zeta\in\mathbb{D}$, we have that
$K_{\zeta}\in\mathcal{H}^{0}_{\mu}$.

$\displaystyle(\alpha+1)(\alpha+2)\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\leq\|S_{g}^{\phi}K_{\zeta}\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}\leq
C\|S_{g}^{\phi}K_{\zeta}\|_{\mathcal{H}_{\mu(z)}^{\infty}}.$ (2.12)

Replacing $f$ in (2.11) by $K_{\zeta}$, and then using (2.12), we have that

$\displaystyle\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\leq\frac{C}{(\alpha+1)(\alpha+2)}\int_{\mathbb{B}_{1}}|\xi(K_{\zeta})|d\varrho(\xi).$
(2.13)

Integrating (2.13) over $\mathbb{D}$ with respect to $\omega dA$ and
proceeding as in the Theorem 5, we get

$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{C}{(\alpha+1)(\alpha+2)}.$
(2.14)

Now if $|\zeta|>1/2,$ then by (2.14), we have that

$\displaystyle\int_{|\zeta|>1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{2C}{(\alpha+1)(\alpha+2)}.$
(2.15)

Further, if $|\zeta|\leq 1/2,$ then we have that

$\displaystyle\int_{|\zeta|\leq
1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{2^{\alpha+1}\|g\|_{\mathcal{H}_{\mu}^{\infty}}}{\nu(1/2)}.$
(2.16)

Thus

$\displaystyle N_{\alpha}$ $\displaystyle\leq\int_{|\zeta|\leq
1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad+\int_{|\zeta|>1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\leq\frac{2^{\alpha+1}\|g\|_{\mathcal{H}_{\mu}^{\infty}}}{\nu(1/2)}+\frac{2C}{(\alpha+1)(\alpha+2)}.$

This completes the proof. ∎

######  Corollary 7.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$ and $\phi$ an analytic self-map of
$\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.

  1. (a)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  2. (b)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.

  3. (c)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.

  4. (d)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$
is nuclear.

  5. (e)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$N_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 8.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in{H}(\mathbb{D})and$ $\phi$ an analytic self-map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded,
then the following statements are equivalent.

  1. (a)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  2. (b)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 9.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ and $\phi$ an analytic self-map of $\mathbb{D}$
such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.

  1. (a)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  2. (b)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  4. (d)

$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$
is nuclear.

  5. (e)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

We can also obtain the following corollaries.

######  Corollary 10.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ is an
analytic self-map of $\mathbb{D}$ such that
$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then
the following statements are equivalent.

  1. (a)

$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 11.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ an
analytic self-map of $\mathbb{D}$ such that $\phi\in\mathcal{H}^{0}_{\mu(z)}$
and $C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.

  1. (a)

$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.

  3. (c)

$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  4. (d)

$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  5. (e)

$\phi$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 12.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If Let
$g\in{H}(\mathbb{D})$ be such that
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then the
following statements are equivalent.

  1. (a)

$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  3. (c)

For $g$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

######  Corollary 13.

Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ be such that
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is bounded,
then the following statements are equivalent.

  1. (a)

$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  2. (b)

$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.

  3. (c)

$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is nuclear.

  4. (d)

$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.

  5. (e)

$g$ and $\nu$ satisfy the following condition:

$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$

##  3\. Nuclear $T_{g}^{\phi}$ and $S_{g}^{\phi}$ between Bloch spaces of
order $\beta$

Finally, we apply our results to $T_{g}^{\phi}$ and $S_{g}^{\phi}$ acting
between weighted Banach spaces and Bloch spaces of order $\beta,$ where
$\beta>0.$ Recall that if $\nu_{\beta}$ is a classical weight, that is
$\nu_{\beta}(z)=(1-|z|^{2})^{\beta}$, then the spaces
$\mathcal{H}^{\infty}_{\nu}$, $\mathcal{H}^{0}_{\nu}$, $\mathcal{B}_{\nu}$ and
$\mathcal{B}^{0}_{\mu}$ reduces respectively to weighted and Bloch spaces of
order $\beta$, denoted by $\mathcal{H}^{\infty}_{\beta}$,
$\mathcal{H}^{0}_{\beta}$, $\mathcal{B}_{\beta}$ and
$\mathcal{B}^{0}_{\beta}$. Moreover, for $\beta>1$, we have that
$\mathcal{H}^{0}_{\beta-1}=\mathcal{B}^{0}_{\beta}$ and
$\mathcal{H}^{\infty}_{\beta-1}=\mathcal{B}^{\infty}_{\beta}$ with equivalent
norm. Thus we have the following corollary.

######  Corollary 14.

Let $\beta>1$, $\gamma>0$, $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-
map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
bounded. Then the following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{B}^{0}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.

  2. (b)

$T_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.

  3. (c)

$g$, $\beta,$ $\gamma$ and $\phi$ satisfy the following condition:

$P_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})^{\gamma}|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}{(1-|\zeta|^{2})^{\alpha-\beta+1}}dA(\zeta)<\infty.$

The case $\beta=1$ is settled in the next theorem.

######  Theorem 8.

Let $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-map of $\mathbb{D}$ such
that $T_{g}^{\phi}:\mathcal{B}\longrightarrow\mathcal{B}$ is bounded. Then the
following statements are equivalent.

  1. (a)

$T_{g}^{\phi}:\mathcal{B}_{0}\longrightarrow\mathcal{B}$ is nuclear.

  2. (b)

$g$ and $\phi$ satisfy the following condition:

$M=\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|\phi^{\prime}(z)|}{|1-\bar{w}\phi(z)|^{3}}|g(z)|dA(w)<\infty.$

###### Proof.

$(b)\Rightarrow(a)$ Since $c_{0}\simeq\mathcal{B}_{0}$ and polynomials are
dense in $\mathcal{B}_{0}$. Moreover, by Corollary $1.5$ in [18], we have that
$f\in\mathcal{B},$ then we have that

$f^{\prime}(z)=2\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-z\bar{w})^{3}}f^{\prime}(w)dA(w).$
(3.1)

Consider polynomials $p_{1},p_{2},\cdots,p_{N}$ in the definition of
absolutely summing and then using (3.1), we have

$\sum_{i=1}^{N}\|T_{g}^{\phi}p_{i}\|_{\mathcal{B}}\leq
C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}p^{\prime}_{i}(w)\big{|}(1-|w|^{2})\bigg{)}\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\big{|}\phi^{\prime}(z)\big{|}}{\big{|}1-\bar{w}\phi(z)\big{|}^{3}}\big{|}g(z)\big{|}dA(w).$

The proof can be settled proceeding as in the proof of Theorem 5.
$(a)\Rightarrow(b)$ By Theorem 1, there exists a probability Borel measure
$\varrho$ on $\sigma((\mathcal{B}_{0})^{*},\mathcal{B})$-compact unit ball
$B^{\prime}$ of $(\mathcal{B}_{0})^{*}=A^{1}$ such that

$\|T_{g}^{\phi}f\|_{\mathcal{B}}\leq
C\int_{B^{{}^{\prime}}}|\xi(f)|d\varrho(\xi)$

for every $f\in\mathcal{B}_{0}$ and some constant $C>0$ independent of $f$.
Next for every $w\in\mathbb{D}$, we consider $f_{w}(z)={1}/{(1-\bar{w}z)^{2}}$
which lies in $\mathcal{B}_{0}\cap H^{\infty}$. By duality and reproducing
property of $f_{w}$ in the Bergman space $(\mathcal{B}_{0})^{*}=A^{1}$, we
have

$\xi(f_{w})=\langle
f_{w},h\rangle=\int_{\mathbb{D}}h(z)\overline{f_{w}(z)}dA(z)=\int_{\mathbb{D}}\dfrac{h(z)}{(1-\bar{z}w)^{2}}dA(z)=h(w).$

Also,

$\sup_{z\in\mathbb{D}}\dfrac{2|w|(1-|z|^{2})|\phi^{\prime}(z)||g(z)|}{|1-\bar{w}\phi(z)|^{3}}=\|T_{g}^{\phi}f_{w}\|_{1}\leq\|T_{g}^{\phi}f_{w}\|_{\mathcal{B}}$

Now proceeding as in the proof of Theorem 7, we can easily see that $(b)$
holds. This completes the proof.∎

Similarly, we have the following corollary.

######  Corollary 15.

Let $\beta>1$, $\gamma>0$, $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-
map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
bounded. Then the following statements are equivalent.

  1. (a)

$S_{g}^{\phi}:\mathcal{B}^{0}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.

  2. (b)

$S_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.

  3. (c)

For $g$, $\phi$ and $\nu$ satisfy the following condition:

$Q_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})^{\gamma}|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}{(1-|\zeta|^{2})^{\alpha-\beta+1}}dA(\zeta)<\infty.$

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# Giant near-field radiative energy transfer due to residual charge modes

Minggang Luo  School of Energy Science and Engineering, Harbin Institute of
Technology, 92 West Street, Harbin 150001, China  Laboratoire Charles Coulomb
(L2C) UMR 5221 CNRS-Université de Montpellier, F- 34095 Montpellier, France
Jiaqi Zhu  Center for Composite Materials and Structures, Harbin Institute of
Technology, Harbin 150080, China  S.-A. Biehs<EMAIL_ADDRESS>Institut für Physik,Carl von Ossietzky Universität, D-26111 Oldenburg, Germany
Junming Zhao<EMAIL_ADDRESS>School of Energy Science and Engineering,
Harbin Institute of Technology, 92 West Street, Harbin 150001, China  Key
Laboratory of Aerospace Thermophysics, Ministry of Industry and Information
Technology, Harbin 150001, China  Linhua Liu  School of Energy and Power
Engineering, Shandong University, Qingdao 266237, China

###### Abstract

We study the modifications of near-field radiative energy transfer caused by
residual charges, which are common in MEMS/NEMS, colloid solutions and lunar
dusts etc. Besides the well-known resonant modes (surface polariton and
localized surface polariton), the residual charges give rise to an additional
temperature-dependent mode providing a new heat flux channel. For low
temperatures we find a giant charge-induced enhancement of the NFRET, whereas
for temperatures larger than 200K the low-frequency charge-induced mode
results in a weakening of the NFRET.

The energy transfer between two objects with a separation distance comparable
to or less than the thermal wavelength $\lambda_{T}=\hbar c/k_{B}T$ via
thermal photons can exceed by several orders of magnitude the blackbody limit,
which has been investigated widely from both theoretical and experimental side
for many different geometrical configurations (e.g., planar structures Rytov
_et al._ (1989); Polder and Van Hove (1971); Loomis and Maris (1994);
Carminati and Greffet (1999); Volokitin and Persson (2001); Ottens  _et al._
(2011); Watjen  _et al._ (2016); Ghashami  _et al._ (2018); DeSutter  _et al._
(2019) and nanoparticles-involved structures Chapuis  _et al._ (2008a);
Narayanaswamy and Chen (2008); Manjavacas and García de Abajo (2012); Chapuis
_et al._ (2008b); Messina  _et al._ (2018); Dong  _et al._ (2018); Shen  _et
al._ (2009); Rousseau  _et al._ (2009); Song  _et al._ (2015)) as reviewed in
Ref. Biehs  _et al._ (2021), for instance. In the aforementioned works, only
electrically neutral objects are involved. However, electrically neutral
objects can easily be charged by exchanging electrons or ions with their
environment Bohren and Hunt (1977); Spitzer (1998); Vladimirov  _et al._
(2005); Kocifaj  _et al._ (2020). The presence of residual charges
significantly affects the optical properties of nanostructures, e.g. the
absorption and scattering cross sections of nanoparticles Klačka and Kocifaj
(2007); Klačka  _et al._ (2015); Kocifaj  _et al._ (2020); Zhang  _et al._
(2020a), the reflection and transmission coefficients of planar structures
Zhang  _et al._ (2020b) in the infrared region. Since the optical properties
of nanostructures in the infrared determine their ability to exchange heat
with their environment by thermal radiation, it can be expected that thermal
radiation of charged objects in general and near-field radiative energy
transfer (NFRET) in particular will be affected by the presence of residual
charges. Due to the lack of studies on this topic, it is largely unknown how
the presence of residual charges will affect NFRET in different geometries.

In this letter, the NFRET transfer between charged objects, e.g. charged
nanoparticles and charged planar slabs [see Fig. 1 (a) and (b)], is analyzed
within the framework of fluctuational electrodynamics Rytov  _et al._ (1989);
Polder and Van Hove (1971); Ben-Abdallah  _et al._ (2011), by introducing the
residual charge modification. We will show that the presence of residual
charges provides a new heat flux channel via an additional charge-induced mode
which will typically enhance the NFRET in the temperature-dependent Planck
window. This effect is particularly large for low temperatures around 10K, as
considered for the planar slabs, because in this case the Planck window
matches well with the charge-induced mode, which results in a giant charge-
induced effect on NFRET. However, for room or higher temperatures as
considered for the particle case, the Planck window is far away from the low-
frequency charge-induced mode, so that NFRET is only weakly affected even for
very high surface charge densities corresponding to high bias voltages between
the particles.

Figure 1: Sketch of the considered NFRET geometries: (a) two charged
nanoparticles and (b) two charged planar slabs.

The residual charges can be assumed to be localized at the surfaces of the
nanostructures Bohren and Hunt (1977); Kocifaj  _et al._ (2020); Zhang  _et
al._ (2020b). The surface conductivity due to the residual charges is given by
Klačka and Kocifaj (2007).

$\sigma_{s}=\frac{\eta e/m_{e}}{k_{\rm{B}}T/\hbar-i\omega},$ (1)

where $\eta$ is the surface charge density, $e$ is the electron charge,
$m_{e}$ is the electron mass, $k_{\rm{B}}$ is the Boltzmann constant, and
$\hbar$ is the reduced Planck constant. For the charged slabs, the surface
charge density is $\eta=eN$ where $N$ [m-2] is the surface charge number
density Zhang  _et al._ (2020b) while for charged nanoparticles of the radius
$a$ we have $\eta=\epsilon_{0}\phi/a$ where $\phi$ [V] is the surface
potential Klačka and Kocifaj (2007); Zhang  _et al._ (2020a) and
$\epsilon_{0}$ is the vacuum permittivity. From Eq. (1) it can be seen that
the surface conductivity induced by the residual charges is temperature-
dependent. In particular, the surface conductivity decreases when increasing
the temperature, because charge motion and collisions become more significant
in that case.

Let’s start by considering NFRET between two identical charged nanoparticles.
The expression for the exchanged power $P$ due to NFRET between a particle $1$
and particle $2$ within the framework of fluctuational electrodynamics for
small particles within the dipole approximation can be written in a Landauer
form as Ben-Abdallah  _et al._ (2011); Dong  _et al._ (2017); Luo  _et al._
(2019)

$P=3\int_{0}^{+\infty}\frac{\mathrm{d}\omega}{2\pi}\left[\Theta(\omega,T_{1})-\Theta(\omega,T_{2})\right]\mathcal{T}_{12}(\omega),$
(2)

where $\omega$ is the angular frequency,
$\Theta(\omega,T)=\hbar\omega/(\exp(\hbar\omega/k_{\rm B}T)-1)$ is the mean
energy of the harmonic oscillator at temperature $T$ and
$\mathcal{T}_{12}(\omega)$ is the transmission coefficient between both
particles Ben-Abdallah  _et al._ (2011); Dong  _et al._ (2017); Luo  _et al._
(2019). The transmission coefficient $\mathcal{T}_{12}(\omega)$ depends on the
optical properties of the two particles, i.e. their electric and magnetic
dipolar polarizabilities ${{\alpha}_{E}}=\frac{i6\pi}{{{k}^{3}_{0}}}{{a}_{1}}$
and ${{\alpha}_{H}}=\frac{i6\pi}{{{k}^{3}_{0}}}{{b}_{1}}$, where
$k_{0}=\omega/c$ with the vacuum light velocity $c$, $a_{1}$ and $b_{1}$ are
the dipolar scattering coefficients. Due to the presence of residual charges
these scattering coefficients derived from Mie theory are modified such that
Zhang  _et al._ (2020a); Klačka and Kocifaj (2007)

$\displaystyle a_{1}$
$\displaystyle=\frac{\sqrt{\varepsilon}\Psi_{1}(y)\Psi_{1}^{\prime}(x)-\Psi_{1}(x)\Psi_{1}^{\prime}(y)+\frac{i\mu_{0}\omega{\color[rgb]{1,0,0}\sigma_{s}}}{k_{0}}\Psi_{1}^{\prime}(x)\Psi_{1}^{\prime}(y)}{\sqrt{\varepsilon}\Psi_{1}(y)\xi_{1}^{\prime}(x)-\xi_{1}(x)\Psi_{1}^{\prime}(y)+\frac{i\mu_{0}\omega{\color[rgb]{1,0,0}\sigma_{s}}}{k_{0}}\xi_{1}^{\prime}(x)\Psi_{1}^{\prime}(y)},$
(3) $\displaystyle b_{1}$
$\displaystyle=\frac{\Psi_{1}(y)\Psi_{1}^{\prime}(x)-\sqrt{\varepsilon}\Psi_{1}(x)\Psi_{1}^{\prime}(y)+\frac{i\mu_{0}\omega{\color[rgb]{1,0,0}\sigma_{s}}}{k_{0}}\Psi_{1}(x)\Psi_{1}(y)}{\Psi_{1}(y)\xi_{1}^{\prime}(x)-\sqrt{\varepsilon}\xi_{1}(x)\Psi^{\prime}(y)+\frac{i\mu_{0}\omega{\color[rgb]{1,0,0}\sigma_{s}}}{k_{0}}\xi_{1}(x)\Psi_{1}(y)},$
(4)

where $x=k_{0}a$, $y=\sqrt{\varepsilon}k_{0}a$, $\varepsilon$ is the relative
dielectric permittivity, $\mu_{0}$ is the vacuum permeability,
$\Psi_{1}(x)=xj_{1}(x)$, $\xi_{1}(x)=xh_{1}^{(1)}(x)$ are the Riccatti bessel
functions and ${{j}_{\text{1}}}(x)$, $h_{\text{1}}^{(1)}(x)$ are the spherical
Bessel and Hankel functions. For the neutral particles ($\sigma_{s}=0$), the
coefficients $a_{1}$ and $b_{1}$ defined by the Eqs. (3) and (4) reduce to the
well-known Mie scattering coefficients Bohren and Huffman (1983).

Similarly, for two identical charged planar slabs with the surface
conductivity $\sigma_{s}$ fluctuational electrodynamics allows to determine
the flux $\Phi$ [W/m2] between those slabs yielding again a Landauer form
Rytov  _et al._ (1989); Polder and Van Hove (1971); Biehs  _et al._ (2010)

${\Phi}=\int_{0}^{\infty}\\!\\!\frac{\text{d}\omega}{2\pi}\,\left[\Theta(\omega,{{T}_{1}})-\Theta(\omega,{{T}_{2}})\right]\int_{0}^{\infty}\\!\\!\frac{\text{d}\beta}{2\pi}\,\beta\sum_{i={\rm
s},{\rm p}}\mathcal{T}_{i}(\omega,\beta),$ (5)

where $\beta$ is the magnitude of the in-plane wave vector and the
$\mathcal{T}_{i}(\omega,\beta)$ ($i={\rm s},{\rm p}$) is the energy
transmission coefficient for the s- and p-polarized waves, which again
strongly dependends on the optical properties of the two slabs via their
reflection coefficients $r_{i}$ ($i=s,p$). Hence, the impact of residual
charges modifies the heat flux by modifying the reflection as follows Shi  _et
al._ (2017); Zhao  _et al._ (2017); Zhang  _et al._ (2020b)

$\displaystyle r_{\rm s}$
$\displaystyle=\frac{k_{z}^{(0)}-k_{z}-{\color[rgb]{1,0,0}\sigma_{s}}\mu_{0}\omega}{k_{z}^{(0)}+k_{z}+{\color[rgb]{1,0,0}\sigma_{s}}\mu_{0}\omega},$
(6) $\displaystyle r_{\rm p}$ $\displaystyle=\frac{k_{z}^{(0)}\varepsilon-
k_{z}\varepsilon^{(0)}+{\color[rgb]{1,0,0}\sigma_{s}}k_{z}^{(0)}k_{z}\varepsilon_{0}^{-1}\omega^{-1}}{k_{z}^{(0)}\varepsilon^{(1)}+k_{z}\varepsilon^{(0)}+{\color[rgb]{1,0,0}\sigma_{s}}k_{z}^{(0)}k_{z}\varepsilon_{0}^{-1}\omega^{-1}},$
(7)

where $k_{z}^{(0)}=\sqrt{\varepsilon_{0}k_{0}^{2}-\beta^{2}}$ and
$k_{z}=\sqrt{\varepsilon k_{0}^{2}-\beta^{2}}$. For the neutral slabs
($\sigma_{0}=0$), the above reflection coefficients Eqs. (6) and (7) reduce to
the well-known Fresnel reflection coefficients Volokitin and Persson (2001);
Shi  _et al._ (2017).

In order to study the impact of residual charges on the exchanged heat between
two nanoparticles or two planar slabs, we choose for the numerical calculation
the typical dielectric material SiC for both nanoparticles and slabs. The
dielectric functions is described by the Lorentz-Drude model
$\varepsilon(\omega)=\varepsilon_{\infty}(\omega^{2}-\omega_{l}^{2}+i\gamma\omega)/(\omega^{2}-\omega_{t}^{2}+i\gamma\omega)$
with $\varepsilon_{\infty}$ = 6.7, $\omega_{l}$ = 1.827 $\times$ 1014
rad$\cdot$s-1, $\omega_{t}$ = 1.495 $\times$ 1014 rad$\cdot$s-1, and $\gamma$
= 0.9 $\times$ 1012 rad$\cdot$s-1 Palik (1998). The particle radius will be
set to $a=50$ nm. The separation edge to edge between two nanoparticles or
slabs is labeled by $d$. The minimum $d_{\rm{min}}$ between the nanoparticles
is 2$a$ guaranteeing the validity of the dipole approximation Dong  _et al._
(2017); Biehs  _et al._ (2021). Below, we will seperately discuss the residual
charge effect on NFRET for small particles and slabs.

To study the impact of residual charges we first define the relative deviation
$\Delta=(P-P_{0})/P_{0}$ of the exchanged power $P$ with residual charges with
respect to the exchanged power $P_{0}$ without residual charges. The
dependence of $\Delta$ on the separation $d$ between the two charged
nanoparticles is shown in Figs. 2 (a) and (b). In Figs. 2 (a), the two
particles are fixed at 200 K and 0 K and the voltage (or amount of residual
charges) is varied. The surface potentials used in this work are comparable to
that of the Refs. Klačka and Kocifaj (2007); Zhang  _et al._ (2020a). It can
be seen that in the near-field regime with separations $d<4~{}\mu$m there is a
clear enhancement of the exchanged power when increasing the amount of
residual charges up to about $2.5P_{0}$. That there is only a strong residual
charge effect for such small distances is due to fact that the residual
charges are strongly localized at the particle surfaces having a strongly
decaying electric field. In Figs. 2 (b), again one particle (receiver) is
fixed at 0 K and temperature of the other particle (emitter) is varied from
200 K to 600 K choosing $\varphi=50\,{\rm V}$. It can be observed that for low
temperatures the enhancement due to the residual charges is strong. When
increasing the temperature $\Delta$ gets smaller. For temperatures larger than
200K there is also a sign change at a given distance. When increasing the
temperature this distance moves to smaller values such that at 400K we find
negative $\Delta$ for all shown distance. Hence, for relatively large
temperature the exchanged power is diminished by the presence of the residual
charges.

Figure 2: Dependence of the $(P-P_{0})/P_{0}$ on the separation $d$ between
two nanoparticles. $P$ is radiative heat flux with residual charges, $P_{0}$
is the radiative heat flux with no charge. (a) The two particles are fixed at
200 K and 0 K. $\phi=$25 V, 50 V, 75 V and 100 V, respectively. The considered
surface potentials are comparable to these used in the Refs. Klačka and
Kocifaj (2007); Zhang  _et al._ (2020a). (b) $\phi=50$ V. One particle is
fixed at 0 K. The temperature of the other particle takes the following
values, $T=$200 K, 300 K, 400 K, 500 K and 600 K.

Since the NFRET between the nanoparticles strongly depends on their
absorptivities which are proportional to ${\rm Im}(\alpha_{\rm E})$ Bohren and
Huffman (1983), we can get some insight in the underlying mechanism behind the
observed charge-induced effects by focusing on ${\rm Im}(\alpha_{\rm E})$ of
the SiC nanoparticles as shown in Fig. 3. Besides the well-know Frölich
resonance at $\omega_{\rm r}=1.756\times 10^{14}\,{\rm rad/s}$ from the
condition ${\rm Re}[\varepsilon(\omega_{\rm r})]=-2$, a new charge-induced
resonance can be observed for $\omega<\omega_{\rm r}$. This resonance is
consistent with the reported charge-induced resonance around wavelength of 1
$\mu$m of SiO2 and some other particles Kocifaj  _et al._ (2015). Due to the
fact, that this resonance has such a small frequency, it is clear that it has
a particularly large impact at low temperatures. Furthermore, the resonance
peak monotonically increases when increasing the potential so that $\Delta$
will inevitably show a monotonically increasing behavior when increasing the
potential due to this resonance as seen in Fig. 2(a). To understand now, why
for large temperatures $\Delta$ can change its sign as observed in Fig. 2(b),
we show in Fig. 3 a close-up of the Fröhlich resonance peak. Obviously, the
resonance peak is blue shifted due the presence of the residual charges and by
increasing the temperature it is becoming weaker. Hence, for larger
temperatures where the Fröhlich resonance dominates the heat transfer rather
than the charge-induced resonance, we find a smaller NFRET $P$ when the
particles are charged compared to the neutral case $P_{0}$. By increasing the
temperature $P$ becomes smaller and smaller compared to $P_{0}$ as observed in
Fig. 2(b).

Figure 3: Charged SiC nanoparticle “absorptivity” ${\rm Im}(\alpha_{\rm E})$:
(a) for different potential (i.e., 25 V, 50 V, 75 V and 100 V) at $T=200$ K
and (b) at different temperatures (i.e., $T=200$ K, 300 K, 400 K, 500 K and
600 K) and fixed potential $\phi=50$ V. The polarizability of the neutral SiC
nanoparticle ($\phi=0$) is shown as reference. The region around the Frölich
resonance peak at $\omega_{\rm r}=1.756\times 10^{14}\,{\rm rad/s}$ is
enlarged.

.

Residual charge effect on NFRET of slabs.– From our analysis of the NFRET
between nanoparticles we can expect that there is also an enhancement of NFRET
for charged slabs which might be particularly significant at low temperatures.
Therefore, we will now focus on the NFRET between two charged slabs at 10K so
that the Planck window matches well with the spectral position of the charge-
induced resonance Kocifaj  _et al._ (2015). We note, that the NFRET between
two neutral slabs has already been experimentally studied at the such low
temperatures Kralik  _et al._ (2012, 2021), recently. Instead of studying the
heat flux as epressed in Eq. (5) we focus on the heat transfer coefficient
(HTC, $h$ [W/(m${}^{2}\cdot$K)] defined by

$h=\lim\limits_{T_{1}\rightarrow
T_{2}}\frac{\Phi}{T_{1}-T_{2}}=\int_{0}^{\infty}h_{\omega}{\rm d}\omega,$ (8)

where $h_{\omega}$ [W/(m${}^{2}\cdot$K$\cdot$rad$\cdot$s-1)] is the spectral
HTC.

In Fig. 4, we show the full HTC as function of separation distance $d$ for
different surface number densities $N$. It can be seen that due to the
presence of surface charges there is a strong near-field enhancement of about
two to three orders of magnitude for $d=100\,{\rm nm}$ and the chosen values
of $N$. This observation is radically different from the negligible
electrostatic effect for the metal planar structures at room temperature
reported recently in Ref. Guo  _et al._ (2022). Note that the charges induce a
heat flux proportional to $1/d^{3}$, so that at distances below 100nm the
enhancement would be even much larger. However, it has to be kept in mind that
the charges also induce a force on the plates and in addition if the gap
between the slabs is not filled with vacuum but a gas, for instance, there can
be a discharge when the voltage reaches the breakdown voltage of the gas. For
the largest values of $N=10^{15}\,{\rm m}^{-2}$ and $3\times 10^{14}\,{\rm
m}^{-2}$ we find a presure of $F/A=1.4\,{\rm kPa}$ and $130\,{\rm Pa}$ which
is about 100 or 10 times the Casimir force at 100nm. For the voltage we find
$1.8\,{\rm V}$ or $0.5\,{\rm V}$ at $100\,{\rm nm}$ so that the value
corresponds to 6 or 1.7 times the breakdown voltage of air. Hence, values of
$N>5\times 10^{13}\,{\rm m}^{-2}$ can only be achieved under vacuum conditions
as typically used in near-field experiments Kralik  _et al._ (2012, 2021).

To understand the mechanism behind the enhancement of HTC, we show the
corresponding spectral HTCs in Fig. 5. Two kind of peaks can be observed: (1)
a broad peak due to the Planck spectrum (basically shonw by the curve for
$N=0$) which is independent of the charge density; (2) a low frequency peak
corresponding to the charge-induced resonance. Obviously, when increasing the
charge number density this peak enhances the HTC by shifting towards the
Planck window. This can be explained by increased charge collisions resulting
in a blueshift of the charge induced resonance. Similarly to the case of two
nanoparticles, when the separation $d$ increases the charge-induced
enhancement effect decreases dramatically, which can again be attributed to
the fact that the resonant mode due to residual charges is highly localized on
the surfaces of the slabs. Note, that the surface phonon polariton resonance
of SiC is at $\omega=1.787\times 10^{14}\,{\rm rad/s}$ and therefore cannot be
thermally excited at 10K. This also explains, why there the HTC does not show
any $1/d^{2}$ dependence associated with the surface phonon polaritons in Fig.
4.

Figure 4: Distance dependence of the HTC $h$ between two charged SiC slabs at
10 K considering the number densities $N=0,~{}5\times
10^{13}$m${}^{-2},~{}3\times 10^{14}$m-2 and $1\times 10^{15}$m-2. The line of
$1/d^{3}$ is added for reference.

Figure 5: Spectrum of the HTC $h_{\omega}$ at $T=10$ K for $N=0,~{}5\times
10^{13}$m${}^{-2},~{}3\times 10^{14}$m-2 and $1\times 10^{15}$m-2 chosing the
separation distance $d=100\,{\rm nm}$.

Finally, we show the transmission coefficient $\mathcal{T}_{\rm
p}(\omega,\beta)$ of the p-polarized waves for $d=100\,{\rm nm}$ in Fig. 6 for
$N=$ 3.0$\times 10^{14}$m-2 and 1.0$\times 10^{15}$m-2. The enhanced
transmission due to the charge-induced mode can be nicely seen, whereas the
surface phonon polariton is outside the plotted frequency range. When
increasing the charge number density [from Fig. 6 (a) to Fig. 6 (b)], the
charge-induced mode blueshifts and approaches the Planck window at 10K. Hence,
more and more energy can be transferred between the two slabs when increasing
the charge density, which accounts for the increasing enhancement of HTC
observed in Fig. 4. In Fig. 6, the dispersion relation of the charge-induced
resonance is shown as dashed-symbol line. It is determined by the poles of the
reflection coefficient $r_{\rm p}$ in Eq. (7). For the strong evanescent limit
$\beta\gg k_{0}$ it is given by

$\beta=\frac{(\varepsilon+1)\varepsilon_{0}\omega
m_{e}(ik_{B}T/\hbar+\omega)}{Ne^{2}}.$ (9)

From this relation and Fig. 6 we can easily understand the blueshift of the
charge-induced resonance. Because the evanescent waves with $\beta\approx 1/d$
dominate the heat flux between the slabs at a given distance $d$ Biehs  _et
al._ (2010) we could chose a fixed value of $\beta=1/d$ in the dispersion
relation in Eq. (9). It is then clear that when increasing $N$ the numerator
has to increase as well to obtain $1/d$ which can be achieved by increasing
$\omega$ (assuming that dispersion can be neglected in that frequency range)
explaining the observed blueshift in the spectral HTC in Fig. 5. Likewise, one
can think of a vertical line in Fig. 6 at $\beta=1/d$ which is about 30
$\beta_{0}$ for $d=100\,{\rm nm}$. The intersection point of that line with
the dispersion relation marks the spectral position of the resonance. Then it
is obvious from Fig. 6 that the resonances are blue-shifted when increasing
$N$.

Figure 6: The transmission coefficient $\mathcal{T}_{\rm p}(\omega,\beta)$ for
$p-$polarized waves for two SiC slabs at separation $d=100$ nm with a charge
number density $N$ given by (a) 3.0$\times 10^{14}$m-2 and (b) 1.0$\times
10^{15}$m-2. The frequencies and wave vectors are normalized to
$\omega_{0}=$1.0$\times 10^{14}$rad$\cdot$s-1 and $\beta_{0}=\omega_{0}/c$.

In summary, we have studied the effect of residual charges on the NFRET
between two nanoparticles and two planar slabs made of SiC. The residual
charges on the nanoparticles and slabs give rise to an additional charge-
induced resonant mode in the low-frequency infrared range. This mode provides
a new heat flux channel enhancing the NFRET. For temperatures as low as 10 K
as considered for the two slabs the Planck window matches well with the
charge-induced mode resonance resulting in a giant charge-induced enhancement
of three orders of magnitude at $d=100\,{\rm nm}$. Consequently, for
temperatures around room temperature as considered for the nanoparticles, the
Planck window is far away from the low-frequency charge-induced mode, which
accounts for a much weaker residual charge effect on NFRET with an enhancement
factor of 2.5. Furthermore, for temperatures larger than 200K we find that the
NFRET is rather inhibited than enhanced due to the surface-charge effect.
These observations pave the way for understanding NFRET between nanostructures
under extreme electromagnetic conditions like in colloid solutions and lunar
dusts or in nanostructures like MEMS/NEMS where residual charges are
unavoidable.

###### Acknowledgements.

The authors thank Prof. Achim Kittel and Dr. Shangyu Zhang for the fruitful
discussions. The support of this work by the National Natural Science
Foundation of China (No. 51976045; No. 52206081) is gratefully acknowledged.
M.G.L. also thanks for support from the China Postdoctoral Science Foundation
(2021M700991). S.-A. B. acknowledges support from Heisenberg Programme of the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the
project No. 404073166.

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