#  2-complex symmetric composition operators on $H^{2}$

Lian Hu, Songxiao Li∗ and Rong Yang  Lian Hu
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, 610054, Chengdu, Sichuan, P.R. China.
<EMAIL_ADDRESS>Songxiao Li
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, 610054, Chengdu, Sichuan, P.R. China.
<EMAIL_ADDRESS>Rong Yang
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, 610054, Chengdu, Sichuan, P.R. China.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study 2-complex symmetric composition operators with the
conjugation $J$ on the Hardy space $H^{2}$. More precisely, we obtain the
necessary and sufficient condition for the composition operator $C_{\phi}$ to
be 2-complex symmetric when the symbols $\phi$ is an automorphism of
${\mathbb{D}}$. We also characterize the 2-complex symmetric composition
operator $C_{\phi}$ on the Hardy space $H^{2}$ when $\phi$ is a linear
fractional self-map of ${\mathbb{D}}$. Keywords: Composition operator;
$m$-complex symmetric; normal

###### 2010 Mathematics Subject Classification:

32A36, 47B38

$\ast$ Corresponding author.

##  1\. Introduction

Let ${\mathbb{D}}$ and $\partial{\mathbb{D}}$ be the open unit disk
$\mathbb{C}$ and the unit circle in the complex plane, respectively. Let
$H({\mathbb{D}})$ be the set of all analytic functions on ${\mathbb{D}}$. The
Hardy space $H^{2}({\mathbb{D}})$ is the space of all $f\in H({\mathbb{D}})$
such that

$\|f\|^{2}=\sup_{0<r<1}{1\over
2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}d\theta<\infty.$

The space $H^{2}({\mathbb{D}})$ is a reproducing kernel Hilbert space, that
is, for each $w\in{\mathbb{D}}$ and $f\in H^{2}({\mathbb{D}})$, there is a
unique function $K_{w}\in H^{2}({\mathbb{D}})$ such that

$\langle f,K_{w}\rangle=f(w),$

where $K_{w}={1\over 1-\bar{w}z}$ is said to be the reproducing kernel at $w$.
For $g\in L^{\infty}(\partial{\mathbb{D}})$, the Toeplitz operator $T_{g}$ is
defined as $T_{g}f=P(fg)$ for $f\in H^{2}({\mathbb{D}})$, where $P$ is the
orthogonal projection of $L^{2}$ onto $H^{2}({\mathbb{D}})$. Recall that

$T_{g}^{\ast}K_{w}=\overline{f(w)}K_{w}$

for each $w\in{\mathbb{D}}$ and $f\in H^{\infty}$.

Let $\phi$ be an analytic self-map of ${\mathbb{D}}$. Recall that the
composition operator $C_{\phi}$ is defined by

$C_{\phi}f(z)=f(\phi(z)),~{}~{}~{}~{}~{}~{}~{}f\in
H({\mathbb{D}}),~{}~{}~{}z\in{\mathbb{D}}.$

It is easy to see that $C_{\phi}^{\ast}K_{w}=K_{\phi(w)}$ foe each
$w\in{\mathbb{D}}$.

Let $H$ and $\mathcal{B}(H)$ stand for a separable complex Hilbert space and
the set of all continuous linear operators on $H$, respectively. An operator
$C:H\to H$ is said to be a conjugation on $H$ if it is

  1. (a)

anti-linear or conjugate-linear: $C(\alpha+\beta
y)=\bar{\alpha}C(x)+\bar{\beta}C(y)$, for all $\alpha,\beta\in\mathbb{C}$ and
$x,y\in H$,

  2. (b)

isometric: $\|Cx\|=\|x\|$, for all $x\in H$,

  3. (c)

involutive: $C^{2}=I_{d}$, where $I_{d}$ is an identity operator.

It is easy to check that $(Jf)(z)=\overline{f(\bar{z})}$ is a conjugation on
the space $H^{2}$.

An operator $T\in\mathcal{B}(H)$ is said to be complex symmetric if
$T=CT^{\ast}C$, where $C$ is a conjugation on $H$. We also say that $T$ is
complex symmetric with $C$ or $C$-symmetric operator. The class of complex
symmetric operators includes all normal operators, binormal operators, Hankel
operators, compressed Toeplitz operators and the Volterra integration
operator. The general study of complex symmetric operators was initiated by
Garcia, Putinar and Wogen in [8, 9, 10, 11]. See [13, 17, 19, 12, 6, 18, 20,
21, 7] for more results on complex symmetric operators.

Let $m$ be a positive integer and $T\in\mathcal{B}(H)$. In [1], the authors
defined $m$-complex symmetric operators as follows: $T$ is said to be an
$m$-complex symmetric operator if there exists some conjugation $C$ such that

$\sum_{j=0}^{m}(-1)^{m-j}\left(\begin{array}[]{c}m\\\
j\end{array}\right)T^{*j}CT^{m-j}C=0.$

In this case, $T$ is said to be $m$-complex symmetric with $C$. When $m=2$, we
get

$CT^{2}C-2T^{\ast}CTC+{T^{\ast}}^{2}=0,$

which is equivalent to

$CT^{2}-2T^{\ast}CT+{T^{\ast}}^{2}C=0.$

It is clear that $1$-complex symmetric operator is just the complex symmetric
operator. By a simple calculation, we see that all complex symmetric operators
are 2-complex symmetric operators. In [5], the authors studied $m$-complex
symmetric weighted shifts on $\mathbb{C}^{n}$. We refer the reader to [1, 2,
5, 14] for more details about $m$-complex symmetric operators.

In this paper, we study 2-complex symmetric composition operators with $J$ on
the Hardy space $H^{2}$. When the symbols $\phi$ is an automorphism of
${\mathbb{D}}$, we see that the composition operator $C_{\phi}$ is 2-complex
symmetric with $J$ if and only if $C_{\phi}$ is normal. Furthermore, we also
characterize the 2-complex symmetric composition operators on $H^{2}$ when the
induced map is a linear fractional self-map of ${\mathbb{D}}$.

##  2\. Main results

We begin this section with Cowen’s formula for the adjoint of a linear
fractional self-map. For a linear fractional self-map $\phi(z)={az+b\over
cz+d}$ of ${\mathbb{D}}$, where $a,b,c,d\in\mathbb{C}$ and $ad-bc\neq 0$,
Cowen in [3] obtained the following important formula

$C_{\phi}^{\ast}=T_{g}C_{\sigma}T_{h}^{\ast},$

where

$\sigma(z)={\bar{a}z-\bar{c}\over-\bar{b}z+\bar{d}},~{}~{}g(z)={1\over-\bar{b}z+\bar{d}},~{}~{}h(z)=cz+d.$

######  Lemma 1.

[4] Let $\phi$ be an analytic self-map of ${\mathbb{D}}$. Then $C_{\phi}$ is
normal if and only if $\phi(z)=az$ with $|a|\leq 1$.

######  Lemma 2.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional self-map of
${\mathbb{D}}$. If $w\in{\mathbb{D}}$ satisfies $\bar{a}w\neq\bar{c}$, then
the following statements hold:

  1. (a)

$JC_{\phi}^{2}K_{w}(z)=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})$,

  2. (b)

$C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=-\bar{c}{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z),$

  3. (c)

${C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z),$

  4. (d)

$JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)=K_{\phi(\bar{w})}(\overline{\phi(\bar{z})})$

where $\sigma(z)={\bar{a}z-\bar{c}\over-\bar{b}z+\bar{d}}$,
$g(z)={1\over-\bar{b}z+\bar{d}}$, and $h(z)=cz+d$.

###### Proof.

Since $Jf(z)=\overline{f(\bar{z})}$ and $f(w)=\langle f,K_{w}\rangle$ for any
$w\in{\mathbb{D}}$ and $f\in H^{2}$, then $(a)$ and $(c)$ hold obviously. Now
we only verify $(b)$ and $(d)$. For any $w\in{\mathbb{D}}$ with
$\bar{a}w\neq\bar{c}$, we obtain

$\displaystyle C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=$ $\displaystyle
C_{\phi}^{\ast}K_{\bar{w}}(\overline{\phi(\bar{z})})$ $\displaystyle=$
$\displaystyle
C_{\phi}^{\ast}{\bar{c}z+\bar{d}\over(\bar{c}-\bar{a}w)z+(\bar{d}-\bar{b}w)}$
$\displaystyle=$ $\displaystyle
C_{\phi}^{\ast}\left({\bar{c}\over\bar{c}-\bar{a}w}+\left(\bar{d}-{\bar{c}(\bar{d}-\bar{b}w)\over\bar{c}-\bar{a}w}\right){1\over\bar{d}-\bar{b}w}{1\over
1-{\bar{a}w-\bar{c}\over\bar{d}-\bar{b}w}z}\right)$ $\displaystyle=$
$\displaystyle{\bar{c}\over\bar{c}-\bar{a}w}C_{\phi}^{\ast}K_{0}(z)+\left(\bar{d}-{\bar{c}(\bar{d}-\bar{b}w)\over\bar{c}-\bar{a}w}\right){1\over\bar{d}-\bar{b}w}C_{\phi}^{\ast}K_{\overline{\sigma(w)}}(z)$
$\displaystyle=$
$\displaystyle-\bar{c}{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z).$

Clearly,

$\displaystyle JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)$
$\displaystyle=JC_{\phi}JC_{\phi}^{\ast}K_{\bar{w}}(z)=JC_{\phi}JK_{\phi(\bar{w})}(z)$
$\displaystyle=JC_{\phi}K_{\overline{\phi(\bar{w})}}(z)=JK_{\overline{\phi(\bar{w})}}(\phi(z))$
$\displaystyle=K_{\phi(\bar{w})}(\overline{\phi(\bar{z})}).$

The proof is complete. ∎

######  Lemma 3.

Let $\phi$ be an analytic self-map of ${\mathbb{D}}$. If $C_{\phi}$ is normal,
then $C_{\phi}^{\ast}$ and $C_{\phi}$ is 2-complex symmetric with $J$.

###### Proof.

Assume that $C_{\phi}$ is normal. From Corollary 3.10 in [16] we see that
$C_{\phi}$ is complex symmetric with $J$. Hence, $C_{\phi}$ is 2-complex
symmetric with $J$. Write $\phi(z)=az$ with $|a|\leq 1$ from Lemma 1. We
notice that $T^{*}$ is $2$-complex symmetric with $J$ if and only if

${T^{\ast}}^{2}J-2JTJT^{\ast}J+JT^{2}=0.$

In view of Lemma 2, set $\sigma(z)=\bar{a}z$, $g(z)=h(z)=1$, we have

${C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over
1-\bar{a}^{2}wz},$

$\displaystyle JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)$
$\displaystyle=K_{\phi(\bar{w})}(\overline{\phi(\bar{z})})={1\over
1-\bar{a}^{2}wz}$

and

$JC_{\phi}^{2}K_{w}(z)=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})={1\over
1-w\overline{\phi_{2}(\bar{z})}}={1\over 1-\bar{a}^{2}wz}.$

Thus, it is obvious that
${C^{\ast}_{\phi}}^{2}J-2JC_{\phi}JC^{\ast}_{\phi}J+JC^{2}_{\phi}=0$, which
means that $C_{\phi}^{\ast}$ is 2-complex symmetric with $J$. ∎

######  Lemma 4.

Let $\phi$ be an analytic self-map of ${\mathbb{D}}$. If $C_{\phi}$ is
2-complex symmetric with $J$, then

$2\phi_{2}(0)-4\phi(0)+\phi(0)\phi_{2}(0)=0.$

###### Proof.

Since $f(w)=\langle f,K_{w}\rangle$ for any $w\in{\mathbb{D}}$ and $f\in
H^{2}$, we have

$\langle JC_{\phi}^{2}K_{0},K_{1\over 2}\rangle=\langle JK_{0},K_{1\over
2}\rangle=1,$ $\langle C_{\phi}^{\ast}JC_{\phi}K_{0},K_{1\over
2}\rangle=\langle C_{\phi}^{\ast}K_{0},K_{1\over 2}\rangle=\langle
K_{\phi(0)},K_{1\over 2}\rangle={2\over 2-\overline{\phi(0)}}$

and

$\langle{C_{\phi}^{\ast}}^{2}JK_{0},K_{1\over
2}\rangle=\langle{C_{\phi}^{\ast}}^{2}K_{0},K_{1\over 2}\rangle=\langle
K_{\phi_{2}(0)},K_{1\over 2}\rangle={2\over 2-\overline{\phi_{2}(0)}}.$

By the assumption that $C_{\phi}$ is 2-complex symmetric with $J$, we get that

$1-{4\over 2-\overline{\phi(0)}}+{2\over 2-\overline{\phi_{2}(0)}}=0.$

By a simple calculation, we get the desired result. ∎

######  Theorem 1.

Let $\phi$ be an automorphism of ${\mathbb{D}}$. Then $C_{\phi}$ is 2-complex
symmetric with $J$ if and only if $C_{\phi}$ is normal.

###### Proof.

Assume that $C_{\phi}$ is normal. From Lemma 3 we see that $C_{\phi}$ is
2-complex symmetric with $J$.

Now suppose that $\phi(z)=\lambda{a-z\over 1-\bar{a}z}$ with $|\lambda|=1$,
$a\in{\mathbb{D}}$ and $C_{\phi}$ is 2-complex symmetric with $J$. Let
$\sigma(z)={-\bar{\lambda}z+a\over-\overline{\lambda a}z+1}$,
$g(z)={1\over-\overline{\lambda a}z+1}$ and $h(z)=-\bar{a}z+1$. Noting that

$\phi_{2}(z)={\lambda(\lambda z-\lambda a)-\lambda
a(\bar{a}z-1)\over\bar{a}(\lambda z-\lambda a)-(\bar{a}z-1)},$

then for $\bar{\lambda}w\neq a$, Lemma 2 gives that

$\displaystyle
JC_{\phi}^{2}K_{w}(z)=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})={a(\bar{\lambda}z-\overline{\lambda
a})-(az-1)\over a(\bar{\lambda}z-\overline{\lambda
a})-(az-1)-w[\bar{\lambda}(\bar{\lambda}z-\overline{\lambda
a})-\overline{\lambda a}(az-1)]},$ (1) $\displaystyle
C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=$ $\displaystyle
a{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z)$
(2) $\displaystyle=$
$\displaystyle{a\over(-\bar{\lambda}w+a)(1-\overline{\lambda
a}z)}+{\bar{\lambda}w(|a|^{2}-1)\over(-\bar{\lambda}w+a)(-\overline{\lambda
a}w+1)}$ $\displaystyle\cdot{-\overline{\lambda
a}w+1+a\bar{\lambda}w-{a}^{2}\over-\overline{\lambda
a}w+1+a\bar{\lambda}w-{a}^{2}-(\bar{\lambda}^{2}{w}-\bar{\lambda}^{2}\bar{a}^{2}{w}-\bar{\lambda}a+\overline{\lambda
a})z}$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={a(\bar{\lambda}w-\overline{\lambda
a})-(aw-1)\over a(\bar{\lambda}w-\overline{\lambda
a})-(aw-1)-[\bar{\lambda}(\bar{\lambda}w-\overline{\lambda
a})-\overline{\lambda a}(aw-1)]z}$ (3)

for any $w,z\in{\mathbb{D}}$. Taking $w=0$ in (1), (2) and (3), we have

$JC_{\phi}^{2}K_{0}(z)=1,\,\,\,\,\,\,C_{\phi}^{\ast}JC_{\phi}K_{0}(z)={1\over
1-\overline{\lambda a}z},$

and

${C_{\phi}^{\ast}}^{2}JK_{0}(z)={-\bar{\lambda}|a|^{2}+1\over-\bar{\lambda}|a|^{2}+1+(\bar{\lambda}^{2}\bar{a}-\overline{\lambda
a})z}$

for any $z\in{\mathbb{D}}$. Since $C_{\phi}$ is 2-complex symmetric with $J$,
we get

$\displaystyle{1+\overline{\lambda a}z\over 1-\overline{\lambda
a}z}={1-\bar{\lambda}|a|^{2}\over-\bar{\lambda}|a|^{2}+1+(\bar{\lambda}^{2}\bar{a}-\overline{\lambda
a})z}.$ (4)

Calculating and noting that the coefficients of $z^{2}$ must be $0$, we obtain
that $\lambda^{2}a^{2}(\lambda-1)=0$. Since $\lambda$ and $a$ are nonzero,
then $\lambda=1$. Hence, (4) becomes

${1+\bar{a}z\over 1-\bar{a}z}=1,$

which implies that $\bar{a}z=0$ for all $z\in{\mathbb{D}}$. Thus $a=0$. Hence
$\phi(z)=-\lambda z$ with $|\lambda|=1$. Lemma 1 gives that $C_{\phi}$ is
normal. The proof is complete. ∎

##  3\. Linear fractional self-maps

In this section, we first consider 2-complex symmetric composition operators
with $J$ which are induced by linear fractional self-maps with $\phi(0)=0$.

######  Theorem 2.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$ and
$\phi(0)=0$. Then $C_{\phi}$ ($C_{\phi}^{\ast}$) is 2-complex symmetric with
$J$ if and only if $C_{\phi}$ is normal.

###### Proof.

Assume first that $C_{\phi}$ is normal. Lemma 3 gives that $C_{\phi}$
($C_{\phi}^{\ast}$) is 2-complex symmetric with $J$.

Conversely, suppose that $C_{\phi}$ is 2-complex symmetric with $J$. Since
$a\neq 0$ and $\phi(0)=0$, set $\phi(z)={z\over sz+t}$, where $s={c\over a}$
and $t={d\over a}$ satisfy $|t|\geq|s|+1$. If $s=0$, then $\phi(z)={1\over
t}z$. Therefore, Lemma 1 gives that $C_{\phi}$ is normal. Now, we suppose that
$s\neq 0$. Let $\sigma(z)={z-\bar{s}\over\bar{t}}$, $g(z)={1\over\bar{t}}$ and
$h(z)=sz+t$. Note that

$\phi_{2}(z)={z\over(s+st)z+t^{2}}$

and

$\phi(\overline{\sigma(z)})={\bar{z}-s\over s\bar{z}-s^{2}+t^{2}},\,\,\
h\left({1\over\overline{\sigma(z)}}\right)={st\over\bar{z}-s}+t.$

For any $w,z\in{\mathbb{D}}$ with $w\neq\bar{s}$, employing Lemma 2, we obtain
that

$\displaystyle
JC_{\phi}^{2}K_{w}(z)=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})={1\over
1-{zw\over\bar{s}z+\overline{st}z+\bar{t}^{2}}}={(\bar{s}+\overline{st})z+\bar{t}^{2}\over(\bar{s}+\overline{st})z+\bar{t}^{2}-wz},$
(5) $\displaystyle C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=$
$\displaystyle-\bar{s}{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z)$
(6) $\displaystyle=$ $\displaystyle{-\bar{s}\over
w-\bar{s}}+\left({\overline{st}\over{w}-\bar{s}}+\bar{t}\right)\cdot{1\over\bar{t}}{1\over
1-{w-\bar{s}\over\bar{s}w-\bar{s}^{2}+\bar{t}^{2}}z}$ $\displaystyle=$
$\displaystyle{\bar{s}\over\bar{s}-w}+{w\bar{t}\over(w-\bar{s})\bar{t}}{\bar{s}w-\bar{s}^{2}+\bar{t}^{2}\over\bar{s}w-\bar{s}^{2}+\bar{t}^{2}-(w-\bar{s})z},$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over
1-{zw\over\bar{s}w+\overline{st}w+\bar{t}^{2}}}={(\bar{s}+\overline{st})w+\bar{t}^{2}\over(\bar{s}+\overline{st})w+\bar{t}^{2}-wz}.$
(7)

Since $C_{\phi}$ is 2-complex symmetric with $J$, we obtain from (5), (6) and
(7) that

${(\bar{s}+\overline{st})z+\bar{t}^{2}\over(\bar{s}+\overline{st}-w)z+\bar{t}^{2}}+{(\bar{s}+\overline{st})w+\bar{t}^{2}\over(\bar{s}+\overline{st})w+\bar{t}^{2}-wz}$
$={2\bar{s}\over\bar{s}-w}+{2w\bar{t}\over(w-\bar{s})\bar{t}}{\bar{s}w-\bar{s}^{2}+\bar{t}^{2}\over\bar{s}w-\bar{s}^{2}+\bar{t}^{2}-(w-\bar{s})z}$

for any $w,z\in{\mathbb{D}}$ with $w\neq\bar{s}$, which gives that

$\displaystyle{[(\bar{s}+\overline{st})z+\bar{t}^{2}][(\bar{s}+\overline{st})w+\bar{t}^{2}-wz]+[(\bar{s}+\overline{st})w+\bar{t}^{2}][(\bar{s}+\overline{st}-w)z+\bar{t}^{2}]\over[(\bar{s}+\overline{st}-w)z+\bar{t}^{2}][(\bar{s}+\overline{st})w+\bar{t}^{2}-wz]}$
(8) $\displaystyle=$
$\displaystyle{2\overline{st}(w-\bar{s})[\bar{s}w-\bar{s}^{2}+\bar{t}^{2}-(w-\bar{s})z]+2w\bar{t}(\bar{s}-w)(\bar{s}w-\bar{s}^{2}+\bar{t}^{2})\over-(w-\bar{s})^{2}\bar{t}[\bar{s}w-\bar{s}^{2}+\bar{t}^{2}-(w-\bar{s})z]}$

for any $w,z\in{\mathbb{D}}$ with $w\neq\bar{s}$. Taking $w=0$ in (8), then

${\bar{t}^{2}[(\bar{s}-\overline{st})z+\bar{t}^{2}]+\bar{t}^{2}[(\bar{s}+\overline{st})z+\bar{t}^{2}]\over\bar{t}^{2}[(\bar{s}+\overline{st})z+\bar{t}^{2}]}={2\bar{s}^{2}\bar{t}(-\bar{s}^{2}+\bar{t}^{2}+\bar{s}z)\over\bar{s}^{2}\bar{t}(-\bar{s}^{2}+\bar{t}^{2}+\bar{s}z)}=2.$

Therefore,
$(\bar{s}+\overline{st})z+\bar{t}^{2}=(\bar{s}-\overline{st})z+\bar{t}^{2}$
for all $z\in{\mathbb{D}}$. $s\neq 0$ gives that $1+t=1-t$. Hence $t=0$, a
contradiction. So the hypothesis is not true, that is $s=0$.

Now, assume that $C_{\phi}^{\ast}$ is 2-complex symmetric with $J$. We also
assume that $s\neq 0$. After a calculation, we have

$\displaystyle JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)$
$\displaystyle=K_{\phi(\bar{w})}(\overline{\phi(\bar{z})})={1\over
1-{w\over\bar{s}w+\bar{t}}\cdot{z\over\bar{s}z+\bar{t}}}$ (9)
$\displaystyle={(\bar{s}w+\bar{t})(\bar{s}z+\bar{t})\over(\bar{s}w+\bar{t})(\bar{s}z+\bar{t})-wz}$

for any $w,z\in{\mathbb{D}}$. Since $C_{\phi}^{\ast}$ is 2-complex symmetric
with $J$, we obtain from (5), (7) and (9) that

$\displaystyle{[(\bar{s}+\overline{st})z+\bar{t}^{2}][(\bar{s}+\overline{st})w+\bar{t}^{2}-wz]+[(\bar{s}+\overline{st})w+\bar{t}^{2}][(\bar{s}+\overline{st}-w)z+\bar{t}^{2}]\over[(\bar{s}+\overline{st}-w)z+\bar{t}^{2}][(\bar{s}+\overline{st})w+\bar{t}^{2}-wz]}$
(10) $\displaystyle=$
$\displaystyle{2(\bar{s}w+\bar{t})(\bar{s}z+\bar{t})\over(\bar{s}w+\bar{t})(\bar{s}z+\bar{t})-wz}$

for any $w,z\in{\mathbb{D}}$ with $w\neq\bar{s}$. Taking $w=0$ in (10), we
also have that

${\bar{t}^{2}[(\bar{s}-\overline{st})z+\bar{t}^{2}]+\bar{t}^{2}[(\bar{s}+\overline{st})z+\bar{t}^{2}]\over\bar{t}^{2}[(\bar{s}+\overline{st})z+\bar{t}^{2}]}=2.$

The other arguments are similar to the case of $C_{\phi}$. Then we obtain the
desired result. The proof is complete. ∎

We now consider 2-complex symmetric composition operators with $J$ which are
induced by linear fractional self-maps with $c=0$.

######  Lemma 5.

[15] Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$
such that $c=0$. Then $C_{\phi}=C_{\xi}^{\ast}T_{\tau}^{\ast}$, where
$\xi(z)={\bar{a}z\over-\bar{b}z+\bar{d}}$ and
$\tau(z)={\bar{d}\over-\bar{b}z+\bar{d}}$.

######  Theorem 3.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$ such
that $c=0$. Then $C_{\phi}$ is 2-complex symmetric with $J$ if and only if
$C_{\phi}$ is normal.

###### Proof.

Assume first that $C_{\phi}$ is normal. Lemma 3 gives that $C_{\phi}$ is
2-complex symmetric with $J$.

Conversely, suppose that $C_{\phi}$ is 2-complex symmetric with $J$. Since
$c=0$, then $\phi(z)=mz+n$, where $m={a\over d}$ and $n={b\over d}$ satisfy
$|m|+|n|\leq 1$. Let $\xi(z)={\bar{m}z\over-\bar{n}z+1}$ and
$\tau(z)={1\over-\bar{n}z+1}$. Note that

$\phi_{2}(z)=m^{2}z+mn+n$

and

$\xi_{2}(z)={\bar{m}^{2}z\over-\overline{mn}z-\bar{n}z+1},\,\,\,\phi(\overline{\xi(z)})={m^{2}\bar{z}-n^{2}\bar{z}+n\over-n\bar{z}+1},\,\,\,\tau(\xi(z))={-\bar{n}z+1\over-\overline{mn}z-\bar{n}z+1}.$

Lemmas 2 and 5 give that

$\displaystyle JC_{\phi}^{2}K_{w}(z)$
$\displaystyle=JC_{\xi}^{\ast}T_{\tau}^{\ast}C_{\xi}^{\ast}T_{\tau}^{\ast}K_{w}(z)=J\overline{\tau(w)}C_{\xi}^{\ast}T_{\tau}^{\ast}K_{\xi(w)}(z)$
(11)
$\displaystyle=J\overline{\tau(w)\tau(\xi(w))}K_{\xi_{2}(w)}(z)=\tau(w)\tau(\xi(w))K_{\overline{\xi_{2}(w)}}(z)$
$\displaystyle={1\over-\bar{n}w+1}{-\bar{n}w+1\over-\overline{mn}w-\bar{n}w+1}\cdot{1\over
1-{\bar{m}^{2}w\over-\overline{mn}w-\bar{n}w+1}z}$
$\displaystyle={1\over-\overline{mn}w-\bar{n}w+1-\bar{m}^{2}wz},$
$\displaystyle C_{\phi}^{\ast}JC_{\phi}K_{w}(z)$
$\displaystyle=C_{\phi}^{\ast}JC_{\xi}^{\ast}T_{\tau}^{\ast}K_{w}(z)=C_{\phi}^{\ast}J\overline{\tau(w)}K_{\xi(w)}(z)=C_{\phi}^{\ast}\tau(w)\overline{K_{\xi(w)}(\bar{z})}$
(12)
$\displaystyle=C_{\phi}^{\ast}\tau(w)K_{\overline{\xi(w)}}(z)=\tau(w)K_{\phi(\overline{\xi(w)})}(z)$
$\displaystyle={1\over-\bar{n}w+1}\cdot{1\over
1-{\bar{m}^{2}w-\bar{n}^{2}w+\bar{n}\over-\bar{n}w+1}z}$
$\displaystyle={1\over-\bar{n}w+1-(\bar{m}^{2}w-\bar{n}^{2}w+\bar{n})z}$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over
1-(\bar{m}^{2}w+\overline{mn}+\bar{n})z}$ (13)

for any $w,z\in{\mathbb{D}}$. Since $C_{\phi}$ is 2-complex symmetric with
$J$, then we obtain from (11), (12) and (13) that

$\displaystyle{1\over-\overline{mn}w-\bar{n}w+1-\bar{m}^{2}wz}+{1\over
1-(\bar{m}^{2}w+\overline{mn}+\bar{n})z}$ (14)
$\displaystyle={2\over-\bar{n}w+1-(\bar{m}^{2}w-\bar{n}^{2}w+\bar{n})z}$

for any $w,z\in{\mathbb{D}}$. Taking $w=0$ in (14), we have that

$1+{1\over 1-(\overline{mn}+\bar{n})z}={2\over 1-\bar{n}z}$

for any $z\in{\mathbb{D}}$. Then

${2-(\overline{mn}+\bar{n})z\over 1-(\overline{mn}+\bar{n})z}={2\over
1-\bar{n}z}.$

Noting that the coefficient of $z^{2}$ must be $0$, then $n^{2}(m+1)=0$, which
means that $n=0$ or $m=-1$. Similarly, noting that the coefficient of $z$ must
be also $0$, then $n=mn$, which means that $n=0$ or $m=1$. Therefore, $n=0$.
This implies that $\phi(z)=mz$ with $|m|\leq 1$. Lemma 1 gives that $C_{\phi}$
is normal. ∎

######  Theorem 4.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$ such
that $c=0$. Then $C_{\phi}^{\ast}$ is 2-complex symmetric with $J$ if and only
if $C_{\phi}$ is normal.

###### Proof.

Since $c=0$, then set $\phi(z)=mz+n$ where $m={a\over d}$ and $n={b\over d}$
satisfy $|m|+|n|\leq 1$. Let $\xi(z)={\bar{m}z\over-\bar{n}z+1}$ and
$\tau(z)={1\over-\bar{n}z+1}$. Noting that

$\phi_{2}(z)=m^{2}z+mn+n,$

and

$\tau(\overline{\phi(\bar{z})})={1\over-\bar{n}(\bar{m}z+\bar{n})+1},\,\,\,\,\xi(\overline{\phi(\bar{z})})={\bar{m}(\bar{m}z+\bar{n})\over-\bar{n}(\bar{m}z+\bar{n})+1}.$

Using Lemma 5 and Theorem 3, we obtain that

$\displaystyle
JC_{\phi}^{2}K_{w}(z)=JC_{\xi}^{\ast}T_{\tau}^{\ast}C_{\xi}^{\ast}T_{\tau}^{\ast}K_{w}(z)$
$\displaystyle={1\over-\overline{mn}w-\bar{n}w+1-\bar{m}^{2}wz},$ (15)
$\displaystyle JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)$
$\displaystyle=JC_{\xi}^{\ast}T_{\tau}^{\ast}K_{\overline{\phi(\bar{w})}}(z)$
(16)
$\displaystyle=J\overline{\tau(\overline{\phi(\bar{w})})}K_{\xi(\overline{\phi(\bar{w})})}(z)=\tau(\overline{\phi(\bar{w})})K_{\overline{\xi(\overline{\phi(\bar{w})})}}(z)$
$\displaystyle={1\over-\bar{n}(\bar{m}w+\bar{n})+1}\cdot{1\over
1-{\bar{m}(\bar{m}w+\bar{n})z\over-\bar{n}(\bar{m}w+\bar{n})+1}}$
$\displaystyle={1\over 1-(\bar{m}w+\bar{n})(\bar{m}z+\bar{n})}$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over
1-(\bar{m}^{2}w+\overline{mn}+\bar{n})z}$ (17)

for any $w,z\in{\mathbb{D}}$. Since $C_{\phi}^{\ast}$ is 2-complex symmetric
with $J$, then we obtain from (15), (16) and (17) that

$\displaystyle{1\over-\overline{mn}w-\bar{n}w+1-\bar{m}^{2}wz}+{1\over
1-(\bar{m}^{2}w+\overline{mn}+\bar{n})z}={2\over
1-(\bar{m}w+\bar{n})(\bar{m}z+\bar{n})}$ (18)

for any $w,z\in{\mathbb{D}}$. Taking $w=0$, we get

${2-(\overline{mn}+\bar{n})z\over 1-(\overline{mn}+\bar{n})z}={2\over
1-\bar{n}(\bar{m}z+\bar{n})}.$

Noting that the coefficient of constant term must be $0$, then
$\bar{n}^{2}=0$, which means that $n=0$. Thus, $\phi(z)=mz$ with $|m|\leq 1$.
Lemma 1 deduces that $C_{\phi}$ is normal.

Conversely, assume that $C_{\phi}$ is normal. Lemma 3 gives that
$C_{\phi}^{\ast}$ is 2-complex symmetric with $J$. ∎

Furthermore, we prove that there is no 2-complex symmetric composition
operators with $J$ which are induced by linear fractional self-maps with $a=0$
and $\phi(0)\neq 0$.

######  Lemma 6.

Let $\phi:{\mathbb{D}}\to{\mathbb{D}}$ be a constant function. Then $C_{\phi}$
$(C_{\phi}^{\ast})$ is 2-complex symmetric with $J$ if and only if
$\phi(z)\equiv 0$.

###### Proof.

The sufficiency is obvious.

Now we assume that $\phi(z)\equiv c$ for some $c\in{\mathbb{D}}$ and
$C_{\phi}$ is 2-complex symmetric with $J$. Then we obtain that

$JC_{\phi}^{2}K_{w}(z)=JK_{w}(\phi_{2}(z))=J{1\over 1-\bar{w}c}={1\over
1-w\bar{c}},$
$C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=C_{\phi}^{\ast}JK_{w}(\phi(z))={1\over
1-w\bar{c}}C_{\phi}^{\ast}K_{0}(z)={1\over(1-w\bar{c})(1-\bar{c}z)},$
${C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over 1-\bar{c}z}$

and

$JC_{\phi}JC_{\phi}^{\ast}JK_{w}(z)=K_{\phi(\bar{w})}(\overline{\phi(\bar{z})})={1\over
1-\bar{c}^{2}}$

for any $w,z\in{\mathbb{D}}$. Since $C_{\phi}$ is 2-complex symmetric with
$J$, we get

$\displaystyle{1\over 1-w\bar{c}}-{2\over(1-w\bar{c})(1-\bar{c}z)}+{1\over
1-\bar{c}z}=0$ (19)

for any $w,z\in{\mathbb{D}}$. By a simple calculation, we see that $c=0$.

Assume that $C_{\phi}^{\ast}$ is 2-complex symmetric with $J$. Similarly, we
have

$\displaystyle{1\over 1-w\bar{c}}-{2\over 1-\bar{c}^{2}}+{1\over
1-\bar{c}z}=0$ (20)

for any $w,z\in{\mathbb{D}}$. Therefore, $c=0$. The proof is complete. ∎

######  Theorem 5.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$ such
that $a=0$ and $\phi(0)\neq 0$. Then $C_{\phi}$ is not 2-complex symmetric
with $J$.

###### Proof.

Since $a=0$ and $\phi(0)\neq 0$, we set $\phi(z)={1\over mz+n}$, where
$m={c\over b}$ and $n={d\over b}$. When $m=0$, $\phi(z)={1\over n}\neq 0$,
then Lemma 6 gives that $C_{\phi}$ is not 2-complex symmetric with $J$. Now,
we assume that $m\neq 0$, $n\neq 0$ and $C_{\phi}$ is 2-complex symmetric with
$J$. Let $\sigma(z)={-\bar{m}\over-z+\bar{n}}$, $g(z)={1\over-z+\bar{n}}$ and
$h(z)=mz+n$. Note that

$\phi_{2}(z)={mz+n\over
m+mnz+n^{2}},\,\,h\left({1\over\overline{\sigma(z)}}\right)=\bar{z},\,\,\phi(\overline{\sigma(z)})={-\bar{z}+n\over-m^{2}-n\bar{z}+n^{2}}.$

Lemma 2 gives that

$\displaystyle
JC_{\phi}^{2}K_{w}(z)=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})={1\over
1-w{\bar{m}z+\bar{n}\over\bar{m}+\overline{mn}z+\bar{n}^{2}}}={\bar{m}+\overline{mn}z+\bar{n}^{2}\over\bar{m}+\overline{mn}z+\bar{n}^{2}-(\bar{m}z+\bar{n})w},$
(21) $\displaystyle C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=$
$\displaystyle-\bar{m}{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z)$
(22) $\displaystyle=$ $\displaystyle
K_{\phi(0)}(z)+{w\over-w+\bar{n}}K_{\phi(\overline{\sigma(w)})}(z)$
$\displaystyle=$ $\displaystyle{1\over
1-{z\over\bar{n}}}+{w\over-w+\bar{n}}\cdot{1\over
1-{-w+\bar{n}\over-\bar{m}^{2}-\bar{n}w+\bar{n}^{2}}z}$ $\displaystyle=$
$\displaystyle{\bar{n}\over\bar{n}-z}+{w\over-w+\bar{n}}\cdot{-\bar{m}^{2}-\bar{n}w+\bar{n}^{2}\over-\bar{m}^{2}-\bar{n}w+\bar{n}^{2}-(-w+\bar{n})z}$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)=K_{\phi_{2}(\bar{w})}(z)={1\over
1-z{\bar{m}w+n\over\bar{m}+\overline{mn}w+\bar{n}^{2}}}={\bar{m}+\overline{mn}w+\bar{n}^{2}\over\bar{m}+\overline{mn}w+\bar{n}^{2}-(\bar{m}w+\bar{n})z}$
(23)

for any $w,z\in{\mathbb{D}}$. Since $C_{\phi}$ is 2-complex symmetric with
$J$, we obtain from (21), (22) and (23) that

$\displaystyle{\bar{m}+\overline{mn}z+\bar{n}^{2}\over\bar{m}+\overline{mn}z+\bar{n}^{2}-(\bar{m}z+\bar{n})w}+{\bar{m}+\overline{mn}w+\bar{n}^{2}\over\bar{m}+\overline{mn}w+\bar{n}^{2}-(\bar{m}w+n)z}$
(24) $\displaystyle=$ $\displaystyle
2\left({\bar{n}\over\bar{n}-z}+{w\over-w+\bar{n}}\cdot{-\bar{m}^{2}-\bar{n}w+\bar{n}^{2}\over-\bar{m}^{2}-\bar{n}w+\bar{n}^{2}-(-w+\bar{n})z}\right)$

for any $w,z\in{\mathbb{D}}$. Taking $w=0$ in (24), we have that

$\displaystyle{(-\bar{n}z+\bar{m}+\bar{n}^{2})+(\bar{m}+\bar{n}^{2})\over-\bar{n}z+\bar{m}+\bar{n}^{2}}={2\bar{n}\
\over\bar{n}-kz}$ (25)

for any $z\in{\mathbb{D}}$. Thus the coefficient of $z^{2}$ must be $0$. This
implies that $n=0$, which is a contradiction. The proof is complete. ∎

In the remainder of this paper, we consider 2-complex symmetric composition
operators with $J$ which are induced by linear fractional self-maps with
$a\neq 0$, $c\neq 0$ and $\phi(0)\neq 0$.

######  Theorem 6.

Let $\phi(z)={az+b\over cz+d}$ be a linear fractional of ${\mathbb{D}}$ such
that $a\neq 0$, $c\neq 0$ and $\phi(0)\neq 0$. Then $C_{\phi}$ is not
2-complex symmetric with $J$.

###### Proof.

We prove it by contradiction. Assume that $C_{\phi}$ is 2-complex symmetric
with $J$. Since $a\neq 0$, $c\neq 0$ and $\phi(0)\neq 0$, set
$\phi(z)={mz+n\over sz+1}$, where $m={a\over d}$, $n={b\over d}$ and
$s={c\over d}$. Let $\sigma(z)={\bar{m}z-\bar{s}\over-\bar{n}z+1}$,
$g(z)={1\over-\bar{n}z+1}$ and $h(z)=sz+1$. Note that

$\phi_{2}(z)={(m^{2}+ns)z+mn+n\over(ms+s)z+ns+1},\,~{}~{}~{}~{}~{}~{}~{}\,h\left({1\over\overline{\sigma(z)}}\right)={(m-ns)\bar{z}\over
m\bar{z}-s}$

and

$\phi(\overline{\sigma(z)})={(m^{2}-n^{2})\bar{z}+n-ms\over(ms-n)\bar{z}+1-s^{2}}.$

Lemma 2 gives that

$\displaystyle JC_{\phi}^{2}K_{w}(z)$
$\displaystyle=K_{\bar{w}}(\overline{\phi_{2}(\bar{z})})={1\over
1-w{(\bar{m}^{2}+\overline{ns})z+\overline{mn}+\bar{n}\over(\overline{ms}+\bar{s})z+\overline{ns}+1}}$
(26)
$\displaystyle={(\overline{ms}+\bar{s})z+\overline{ns}+1\over(\overline{ms}+\bar{s})z+\overline{ns}+1-[(\bar{m}^{2}+\overline{ns})z+\overline{mn}+\bar{n}]w},$
$\displaystyle C_{\phi}^{\ast}JC_{\phi}K_{w}(z)=$
$\displaystyle-\bar{s}{g(w)\over\sigma(w)}K_{\phi(0)}(z)+\overline{h({1\over\overline{\sigma(w)}})}g(w)K_{\phi(\overline{\sigma(w)})}(z)$
(27) $\displaystyle=$ $\displaystyle{-\bar{s}\over\bar{m}w-\bar{s}}{1\over
1-\bar{n}z}+{(\bar{m}-\overline{ns})w\over\bar{m}w-\bar{s}}{1\over-\bar{n}w+1}\cdot{1\over
1-{(\bar{m}^{2}-\bar{n}^{2})w-\overline{ms}+\bar{n}\over(\overline{ms}-\bar{n})w+1-\bar{s}^{2}}z}$
$\displaystyle=$
$\displaystyle{-\bar{s}\over(\bar{m}w-\bar{s})(1-\bar{n}z)}+{(\bar{m}-\overline{ns})w\over(\bar{m}w-\bar{s})(-\bar{n}w+1)}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdot{(\overline{ms}-\bar{n})w+1-\bar{s}^{2}\over(\overline{ms}-\bar{n})w+1-\bar{s}^{2}-[(\bar{m}^{2}-\bar{n}^{2})w-\overline{ms}+\bar{n}]z}$

and

$\displaystyle{C_{\phi}^{\ast}}^{2}JK_{w}(z)$
$\displaystyle=K_{\phi_{2}(\bar{w})}(z)={1\over
1-z{(\bar{m}^{2}+\overline{ns})w+\overline{mn}+\bar{n}\over(\overline{ms}+\bar{s})w+\overline{ns}+1}}$
(28)
$\displaystyle={(\overline{ms}+\bar{s})w+\overline{ns}+1\over(\overline{ms}+\bar{s})w+\overline{ns}+1-[(\bar{m}^{2}+\overline{ns})w+\overline{mn}+\bar{n}]z}$

for any $w,z\in{\mathbb{D}}$. Taking $w=0$, then we obtain from (26), (27) and
(28) that

$JC_{\phi}^{2}K_{0}(z)=1,\,\,\,\,C_{\phi}^{\ast}JC_{\phi}K_{0}(z)={1\over
1-\bar{n}z}$

and

${C_{\phi}^{\ast}}^{2}JK_{0}(z)={\overline{ns}+1\over\overline{ns}+1-(\overline{mn}+\bar{n})z}$

for any $z\in{\mathbb{D}}$. Since $C_{\phi}$ is 2-complex symmetric with $J$,
we obtain that

$\displaystyle
1+{\overline{ns}+1\over\overline{ns}+1-(\overline{mn}+\bar{n})z}={2\over
1-\bar{n}z}$ (29)

for any $z\in{\mathbb{D}}$, which implies that

$\displaystyle{2\overline{ns}+2-(\overline{mn}+\bar{n})z\over\overline{ns}+1-(\overline{mn}+\bar{n})z}={2\over
1-\bar{n}z}$ (30)

for any $z\in{\mathbb{D}}$. Noting that the coefficients of $z^{2}$ must be
$0$, then we have that $\bar{n}^{2}(m+1)=0$. Since $\phi(0)\neq 0$, then
$m=-1$.

Noting that $\phi_{2}(z)={(m^{2}+ns)z+mn+n\over(ms+s)z+ns+1}$, then Lemma 4
gives that

$2\phi_{2}(0)-4\phi(0)+\phi(0)\phi_{2}(0)={2(mn+n)\over ns+1}-4n+{n(mn+n)\over
ns+1}=0.$

By a simple calculation, we see that $2m-4ns+mn+n=2$. Since $m=-1$, we get
that $ns=-1$. Therefore, $m-ns=0$, which means that $\phi$ is not a linear
fractional of ${\mathbb{D}}$, a contradiction. The proof is complete. ∎

## References

  * [1] M. Ch$\bar{o}$, E. Ko and J. Lee, On $m$-complex symmetric operators, Mediterr. J. Math.  13 (2016), 2025–2038.
  * [2] M. Ch$\bar{o}$, E. Ko and J. Lee, On $m$-complex symmetric operators, II, Mediterr. J. Math.  13 (2016), 3255–3264.
  * [3] C. Cowen, Composition operators on $H^{2}$, J. Operator Theory 9(1) (1983), 77–106.
  * [4] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. xii+388 pp. ISBN: 0-8493-8492-3
  * [5] G. Exner, J. Jin, I. Jung and J. Lee, On $m$-complex symmetric weighted shift operators on $\mathbb{C}^{n}$, Linear Algebra Appl.  603 (2020), 130–153.
  * [6] M. Fatehi, Complex symmetric weighted composition operators, Complex Var. Elliptic Equ. 64(4) (2019), 710–720.
  * [7] Y. Gao and Z. Zhou, Complex symmetric composition operators induced by linear fractional maps, Indiana Univ. Math. J. 69(2) (2020), 367–384.
  * [8] S. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358(3) (2006), 1285–1315.
  * [9] S. Garcia and M. Putinar, Complex symmetric operators and applications, II. Trans. Amer. Math. Soc. 359(8) (2007), 3913–3931.
  * [10] S. Garcia and W. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257(4) (2009), 1251–1260.
  * [11] S. Garcia and W. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362(11) (2010), 6065–6077.
  * [12] A. Gupta and A. Malhotra, Complex symmetric weighted composition operators on the space $H_{1}^{2}({\mathbb{D}})$, Complex Var. Elliptic Equ. 65(9) (2020), 1488–1500.
  * [13] K. Han and M. Wang, Weighted composition-differentiation operators on the Hardy space, Banach J. Math. Anal. 15(3) (2021), 1–18.
  * [14] N. Jeridi and R. Rabaoui, On $(A,m)$-symmetric operators in a Hilbert space, Results Math. 74(3) (2019), 1–33.
  * [15] S. Jung, Y. Kim and E. Ko, Composition operators for which $C_{\phi}^{\ast}C_{\phi}$ and $C_{\phi}+C_{\phi}^{\ast}$ commute, Complex Var. Elliptic Equ. 59(12) (2014), 1608–1625.
  * [16] S. Jung, Y. Kim and E. Ko and J. Lee, Complex symmetric weighted composition operators on $H^{2}(\mathbb{D})$, J. Funct. Anal. 267(4) (2014), 323–351.
  * [17] R. Lim and L. Khoi, Complex symmetric weighted composition operators on $H_{\gamma}({\mathbb{D}})$, J. Math. Anal. Appl. 464(1) (2018), 101–118.
  * [18] S. Narayan, D. Sievewright and D. Thompson, Complex symmetric composition operators on $H^{2}$, J. Math. Anal. Appl. 443(1) (2016), 625–630.
  * [19] S. Narayan, D. Sievewright and M. Tjani, Complex symmetric composition operators on weighted Hardy spaces, Proc. Amer. Math. Soc. 148(5) (2020), 2117–2127.
  * [20] D. Thompson, T. McClatchey and C. Holleman, Binormal, complex symmetric operators, Linear Mult. Algebra 69 (2021), 1705–1715.
  * [21] X. Yao, Complex symmetric composition operators on a Hilbert space of Dirichlet series, J. Math. Anal. Appl. 452(2) (2017), 1413–1419.


††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Stark Effect of Blue Quantum Emitters in Hexagonal Boron Nitride

Ivan Zhigulin  School of Mathematical and Physical Sciences, University of
Technology Sydney, Ultimo, New South Wales 2007, Australia  Jake Horder
School of Mathematical and Physical Sciences, University of Technology Sydney,
Ultimo, New South Wales 2007, Australia  Victor Ivády  Max-Planck-Institut für
Physik komplexer Systeme, Nöthnitzer, D-01187 Dresden, Germany  Department of
Physics, Chemistry and Biology, Linköping University, Linköping, Sweden  Simon
J. U. White  School of Mathematical and Physical Sciences, University of
Technology Sydney, Ultimo, New South Wales 2007, Australia  Angus Gale  School
of Mathematical and Physical Sciences, University of Technology Sydney,
Ultimo, New South Wales 2007, Australia  Chi Li  School of Mathematical and
Physical Sciences, University of Technology Sydney, Ultimo, New South Wales
2007, Australia  Charlene J. Lobo  School of Mathematical and Physical
Sciences, University of Technology Sydney, Ultimo, New South Wales 2007,
Australia  Milos Toth  School of Mathematical and Physical Sciences,
University of Technology Sydney, Ultimo, New South Wales 2007, Australia  ARC
Centre of Excellence for Transformative Meta-Optical Systems, University of
Technology Sydney, Ultimo, New South Wales 2007, Australia  Igor Aharonovich
<EMAIL_ADDRESS>School of Mathematical and Physical Sciences,
University of Technology Sydney, Ultimo, New South Wales 2007, Australia  ARC
Centre of Excellence for Transformative Meta-Optical Systems, University of
Technology Sydney, Ultimo, New South Wales 2007, Australia  Mehran Kianinia
<EMAIL_ADDRESS>School of Mathematical and Physical Sciences,
University of Technology Sydney, Ultimo, New South Wales 2007, Australia

###### Abstract

Inhomogeneous broadening is a major limitation for the application of quantum
emitters in hBN to integrated quantum photonics. Here we demonstrate that so-
called ‘blue emitters’ with an emission wavelength of
$436\text{\,}\mathrm{n}\mathrm{m}$ are less sensitive to electric fields than
other quantum emitters in hBN. Our measurements reveal a weak, predominantly
quadratic Stark effect that indicates a negligible transition dipole moment of
the defect. Using these results, we discuss implications for the symmetry of
the defect and use density functional theory simulations to identify a likely
atomic structure of blue emitters in hBN.

Quantum emitters in hexagonal boron nitride (hBN) are recognized for their
brightness and robustness against temperature and harsh environments [1, 2, 3,
4, 5, 6, 7]. Most recently, optical access to spin properties for some of
these defects has been demonstrated at room temperature [8, 9, 10, 11, 12].
Moreover, the van der Waals structure of the host crystals provides compelling
opportunities for the development of quantum technologies based on quantum
emitters in hBN [13, 14]. However, broad adoption of hBN quantum emitters in
integrated quantum photonics has been limited by inhomogeneous broadening of
the emission lines. The dominant cause of inhomogeneous broadening are
fluctuating electric fields resulting from charge transitions in the vicinity
of quantum emitters. The fields interact with electric dipoles of the defects
and give rise to spectral diffusion. The effect has been studied extensively
in various solid state systems such as NV centers in diamond [15], point
defects in silicon carbide, and rare earth doped crystals [16]. In the case of
hBN, this interaction is very strong for some emitters, and has been utilized
to realize massive Stark shifts using applied electric fields [17, 18, 19].
Whilst this is useful for spectral tuning of quantum emitters, the strong
interaction gives rise to linewidth broadening and spectral diffusion that
limit the usefulness of these emitters.

To this end, there has been work focused on eliminating spectral diffusion by
surface passivation, and optical re-pumping [20, 21]. Fourier-limited
linewidth has been achieved for a defect in hBN by charge
manipulation/depletion using heterostructure devices [22]. An alternative
approach to overcome spectral diffusion is to identify point defects with
small or ideally zero permanent electric dipole moments, such as the SiV
center in silicon carbide or centrosymmetric defects such as group IV defects
in diamond [23, 24]. The absence of a permanent transition dipole in these
defects results in reduced sensitivity to electric fields and small
inhomogeneous linewidth broadening.

Recently, a group of emitters in hBN known as ‘blue emitters’ has been
identified as being spectrally stable with a narrow emission at
$436\text{\,}\mathrm{n}\mathrm{m}$ [25, 26]. These defects can be created
deterministically using electron beam irradiation and have been associated
with the presence of a UV emission at $4.1\text{\,}\mathrm{e}\mathrm{V}$ [27].
Interestingly, the blue emitters exhibit stable single photon emission at
cryogenic temperatures with spectral diffusion that is negligible compared to
other quantum emitters studied to date in hBN [28]. Sub-GHz linewidths have
been measured in this group of quantum emitters in the absence of applied
electric fields or post-processing treatments.

Here we investigate the Stark shift of blue quantum emitters in hBN. We
performed coherent excitation to measure the Stark shift upon applying in-
plane and out-of-plane electric fields to characterize the electric dipole
interaction for blue emitters. Building on these results, we discuss
implications for defect symmetry and use density functional theory (DFT) to
identify the likely defect structure. Our results constitute an important step
towards identifying spectrally-stable quantum emitters in hBN and are
particularly valuable for creating site-specific quantum emitters that are
resonant with optical cavities or with other quantum emitters.

Figure 1: Blue single photon emitters in hBN. (a) Atomic defects in hBN as a
source of quantum emission. (b) Room temperature autocorrelation measurement
of a single defect in hBN emitting at $436\text{\,}\mathrm{n}\mathrm{m}$. The
solid line is a fit to the data. (c) Photoluminescence spectra of a blue
emitter obtained at cryogenic and room temperatures. (d) Resonant excitation
of a blue emitter fit with a Gaussian function that has a line width of
$1.9\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$.

In brief, the blue emitters were created by transferring an exfoliated hBN
flake onto a silicon substrate with a $285\text{\,}\mathrm{n}\mathrm{m}$
thermal oxide, then exposing the flake to a focused electron beam using a
commercial scanning electron microscope. Further details on the generation of
these emitters in hBN can be found in the Methods section of the Supplementary
Information. Figure 1(a) shows a schematic illustration of the hBN lattice
hosting a point defect, and an autocorrelation measurement confirming the
quantum nature of a blue emitter that was fabricated by an electron beam (Fig.
1(b)). Whilst the emission energy of single photon emitters in hBN spans a
broad spectral range, from UV up to NIR, the zero phonon line (ZPL) of blue
emitters is centered consistently on $436\text{\,}\mathrm{n}\mathrm{m}$ (Fig.
1(c)) and varies by less than $1\text{\,}\mathrm{n}\mathrm{m}$ [26, 28]. To
measure the linewidth of these emitters, the sample was loaded into a closed-
loop cryostat and cooled to $5\text{\,}\mathrm{K}$. When excited with a
$405\text{\,}\mathrm{n}\mathrm{m}$ laser, the linewidth of these emitters at
cryogenic temperature is beyond the
$130\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ resolution of our spectrometer.
With a lifetime of approximately $3\text{\,}\mathrm{n}\mathrm{s}$, a natural
linewidth of $53\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ is expected from
these emitters. By scanning a narrow bandwidth laser across the ZPL, the full
width at half maximum of this particular emission is revealed to be
$1.9\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The resonant photoluminescence
excitation (PLE) scan shown in Fig. 1(d) is the average of ten individual
scans over the ZPL to account for temporal fluctuations of the emission. The
best peak fit is Gaussian, indicative of inhomogeneous broadening caused by
spectral diffusion. We note that the range of spectral diffusion observed in
PLE measurements varied between emitters, but was insufficient to be observed
in photoluminescence (PL) spectra which acquired with a spectral resolution of
$130\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. This degree of inhomogeneous
broadening is small compared to that of other emitters studied previously in
hBN, indicating an inherent insensitivity of the blue emitters to
environmental effects such as electric field fluctuations in the hBN lattice.

Figure 2: Stark effect in a vertical structure used to generate an out-of-
plane electric field. (a) Schematic illustration of a vertical structure with
hBN located between the bottom gold electrode and a top multilayer graphene
(MLG) electrode. (b) Microscope image of the lateral device (the scale bar is
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$). Colored outlines show the
edges of each layer: the hBN layer hosting blue emitters is covered with MGL
and a protective hBN capping layer. (c) Emission linewidth and ZPL shift
versus applied electric field determined by Equ. 3. A linear fit yields a
gradient of
$0.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{M}\mathrm{V}\mathrm{m}^{-1}$.

To characterize the interaction of blue emitters with an external electric
field, the Stark effect of these emitters was measured. The transition energy
of a color center coupled to an external DC electric field shifts as

$\Delta E=-\Delta\mu F$ (1)

where $\Delta E$ is the Stark shift of the ZPL, $\Delta\mu$ is the difference
between the dipole moment of the ground and excited states, and $F$ is the
electric field. A nonlinear Stark effect originates from the polarization and
hyperpolarization of the defect states that make $\Delta\mu$ dependent on the
electric field. In this case, the energy shift $\Delta E$ of the transition
line can be expanded as a power series of the electric field [23, 29]

$\Delta E=-\Delta\mu(F)F=-\Delta\mu_{perm}F-\frac{1}{2}\Delta\alpha F^{2}$ (2)

where $\Delta\mu_{perm}$ is the difference between the permanent dipole of the
ground and excited states, and $\Delta\alpha$ is the difference between the
polarizability of the ground and excited states. Here the higher order Stark
shifts related to hyperpolarizabilities are not considered as their
contribution is negligible. Nonlinear electric field dependence is observed
when the linear term vanishes, either due to the vanishing permanent dipole
moment in both the ground and excited states, or due to the perfect
cancellation of the ground and excited state permanent dipoles. The latter is
less frequent and appears only in special cases [30].

We fabricated two devices used to apply in-plane and out-of-plane fields to
blue emitters. We first built a vertical device (which produces an out-of-
plane electric field) by transferring a $36\text{\,}\mathrm{n}\mathrm{m}$ hBN
flake containing blue emitters over gold electrodes, as shown in Fig. 2(a).
The electrodes were fabricated using photolithography, followed by vacuum
thermal deposition of $5\text{\,}\mathrm{n}\mathrm{m}$ Cr then
$100\text{\,}\mathrm{n}\mathrm{m}$ Au. Multilayer graphene (MLG) was placed on
top of the hBN to act as a top electrode, such that we could image the
underlying hBN in a confocal microscope. An additional hBN flake was used as a
capping layer to cover the MLG and protect it from oxidation during the
experiments. The materials were stacked in the manner described in the Methods
section of the Supplementary Information, in order to maintain a clean
interface. The thickness of each layer was confirmed by AFM. The vertical
structure was then irradiated by an electron beam in order to generate blue
emitters in the bottom hBN layer. To ensure that emitters were created only in
the bottom hBN layer, both hBN flakes were pre-screened prior to stacking for
the presence of a signature UV emission, as is detailed in [27]. An optical
image of the fabricated vertical device is shown in Fig. 2(b). The hBN flake
containing blue emitters is highlighted with a blue line around its periphery.
The MLG and capping hBN layers are indicated by black and yellow lines,
respectively. The device was loaded into a cryostat, and the measured leakage
current was limited to few tens of nanoamperes at biases of $\pm
15\text{\,}\mathrm{V}$ applied to the electrodes.

For the investigated emitter with a linewidth of
$1.9\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, a small linear Stark shift was
observed upon application of an out-of-plane electric field by the device. The
strength of the applied electric field in the region of the emitter was
estimated using the Lorentz local field approximation

$F=(\varepsilon_{hBN}+2)\frac{V_{g}}{3d}$ (3)

where $\varepsilon_{hBN}=3.76$ is the out-of-plane dielectric constant [31],
$V_{g}$ is the applied potential field, and $d=36\textrm{ nm}$ is the
electrode gap, equal to the thickness of the hBN flake containing the blue
emitter. A relatively large electric field was required to produce the
observed ZPL shift. A linear fit yields a sensitivity of
$0.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{M}\mathrm{V}\mathrm{m}^{-1}$
or
$0.04\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{V}\mathrm{n}\mathrm{m}^{-1}$,
indicating that the transition dipole moment of the emitter is either very
small or is oriented prediminantly within the plane of the hBN lattice, and
hence perpendicular to the electric field. Similarly, the lack of an obvious
quadratic dependence suggests that the system has a low transition
polarizability in the out-of-plane direction. The emission linewidth was
approximately constant, and did not change systematically with applied
electric field, confirming that heating caused by the leakage current was
negligible. We note that for this particular blue emitter, the emission
intensity gradually decreased with increasing negative bias, until the PLE
emission was no longer observable which occurred below
$-200\text{\,}\mathrm{M}\mathrm{V}\mathrm{m}^{-1}$. Similar behavior has been
observed in other quantum emitters in hBN and was attributed to charge
transfer caused by tuning of the Fermi level [32, 33].

Figure 3: Stark effect in a lateral structure used to generate an in-plane
electric field. (a) Schematic illustration of a lateral structure with hBN
between two gold electrodes. (b) Microscope image of the lateral device (the
scale bar is $50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$). A hBN flake
is positioned on top of interdigitated gold electrodes, with gap separations
of $2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. (c) Linewidth and ZPL
shifts versus applied electric field, as determined by Equ. 3. The ZPL shift
data are fit to Equ. 2, yielding the values $\Delta\mu=0.10\textrm{ D}$ and
$\Delta\alpha=1078\textrm{ \AA${}^{3}$}$.

We next built a lateral device used to generate an in-plane electric field by
transferring a $100\text{\,}\mathrm{n}\mathrm{m}$ hBN flake over gold
electrodes, as shown in Fig. 3(a),(b). The electrode spacing is
$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, which is not large enough
for the hBN flake to conform fully to the silicon oxide surface – the flake
remains mostly suspended over the gaps. Nevertheless, the local electric field
is assumed to be predominantly in-plane for emitters positioned equidistant
between two neighboring electrodes, so that the field strength at the emitter
is taken to be equal to the maximum field strength given by the electrode
geometry. As with the vertical device, here the local applied electric field
is approximated by Equ. 3, although it is now appropriate to use the in-plane
dielectric constant of $\varepsilon_{hBN}=6.93$ [31], whilst
$d=\textrm{$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$}$ is given by the
separation of the electrodes. Figure 3(b) shows an optical image of the hBN
flake overlaying the gold electrodes. The hBN flake was then irradiated by an
electron beam to generate blue emitters in the gap between the electrodes. PL
characterization of the emitter was performed after confirming that the
lateral device did not show any detectable leakage current.

In contrast to the out-of-plane electric field, the lateral device produced a
clear quadratic shift of the emitter ZPL, as shown in Fig. 3(c). The shift
magnitude per unit electric field strength is over 200 times greater than the
corresponding shift in the vertical, in-plane electric field device. A
measurement of the emission polarization showed that, for this particular
emitter, the light was linearly polarized at an angle of
$18\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the local applied
electric field direction. Assuming that the transition dipole moment is
aligned parallel to the polarization of the emitted light [23], the projection
of the transition dipole moment onto the applied electric field can be
determined. Using Equ. 2 to fit the ZPL data in Fig. 3(c), we get the value of
$\Delta\mu=0.10\textrm{ D}$ for the permanent transition dipole moment, and
$\Delta\alpha=1078\textrm{ \AA}^{3}$ for the transition polarizability along
the local field direction. A positive transition polarizability indicates that
the excited state has greater polarizability than the ground state [29]. These
results suggest that the blue emitter has a relatively small permanent
transition dipole moment, and a relatively large transition polarizability,
relative to visible emitters studied previously in hBN [17, 19, 18].

In the same lateral structure, a nearby emitter was found to also display a
small quadratic ZPL wavelength shift in response to the applied electric
field. This further suggests that the total transition dipole moment of the
blue emitter has a predominant contribution from the transition
polarizability, with minimal contribution from the permanent transition dipole
moment. Variation in polarizability between emitters has been attributed to
both strain [23] and the dipole orientation within the electric field [29,
34]. Each of these factors can produce a lifting of the degeneracy of the
excited state in proportion to its magnitude, leading to a modulated quadratic
shift. Emission polarization measurements indicate that these two blue
emitters are oriented at $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ with
respect to one another. This orientation variance, in addition to inherent
local strain gradients, could be the cause of the differing quadratic shift.

Figure 4: Representation of (a) Nitrogen split interstitial defect. (b)
$C^{2}_{N}$ defect in the hBN lattice. (c) The electric field dependence of
the ZPL energy of the nitrogen split interstitial defect as obtained from
first principles calculations.

The absence of a permanent dipole can be a consequence of defect symmetry.
Indeed, group IV-vacancy complexes in diamond may possess $D_{3d}$ symmetry
where the inversion symmetry element of the group ensures a vanishing
permanent dipole for these defects. The nonlinear Stark effect has been
recently reported for several such complexes in diamond [23]. The $D_{3d}$
point group symmetry of hBN does not contain inversion symmetry, however, the
combination of the 3-fold rotation symmetry and in-plane reflection symmetry
may also forbid the presence of a permanent dipole. Using symmetry arguments,
the defect that exhibits a nonlinear Stark effect due to the absence of a
permanent dipole has to have D3h point group symmetry in both the ground and
the excited states.

The observation of a non-linear Stark shift helps us to narrow down the
possible point defect configurations that may give rise to the blue PL
emission in hBN. The simplest configurations that have $D_{3h}$ symmetry are
in-plane single site defects, such as anti-sites, substitutional impurities,
and vacancies. Boron and nitrogen vacancies are well known in the literature
[35, 36], but according to the current understanding of these defects, they
are not related to the blue emitter in hBN. Furthermore, we exclude both anti-
site and carbon interstitial defects. These defects form strong $sp^{2}$
hydride bonding states with the neighboring atoms, within the valence band of
hBN. The defect states that appear in the band gap for these defects are
related to the $p_{z}$ state of the anti-site or the substitutional carbon.
Due to the lack of other defect states in the band gap, the possible optical
transitions involve excitations from the valence band to the conduction band
of the host.

For more complex defects the symmetry requirements are fairly restrictive. In-
plane two site defects can be ruled out due to the lack of 3-fold rotation
symmetry. Consequently, the $C_{B}C_{N}$ defect (associated with the
$4.1\text{\,}\mathrm{e}\mathrm{V}$ emission) and related defects can be ruled
out too.

Next, we consider interstitial defects. The most favorable configuration of
the nitrogen interstitial is a split interstitial configuration [36], where
two out-of-plane nitrogen atoms sit at a nitrogen site, see Fig. 4(a). The
negative charge state of the defect retains its $D_{3h}$ symmetry in the first
optical excited state and possesses no permanent dipole in the ground and
excited states. As depicted in Fig. 4(c), the ZPL energy calculated on DFT-PBE
level of theory depends nonlinearly on the electric field, which is in
qualitative agreement with our measurements. However, we note, however, that
the blue emitters are generated by electron irradiation of hBN flakes
characterized by a $4.1\text{\,}\mathrm{e}\mathrm{V}$ UV emission that is
associated with carbon [27]. We therefore consider a related atomic structure
obtained by replacing the two nitrogen atoms with two carbon atoms as shown in
Fig. 4(b). We dub this configuration $C_{N}^{2}$. Interestingly, this
configuration adopts $D_{3h}$ symmetry, exhibits similar properties of
nitrogen split interstitial defect, and can be obtained from $C_{B}C_{N}$ by
the addition of a boron interstitial as

$C_{B}C_{N}+B_{i}\rightarrow C^{2}_{N}+2.0\ eV$ (4)

Alternatively, dissociation of the $4.1\text{\,}\mathrm{e}\mathrm{V}$ emitter
due to the electron irradiation as suggested before [27, 28] may lead to
interstitial carbon atoms which may recombine with carbon substitutional
defect and form the $C_{N}^{2}$ complex as

$C_{N}+C_{i}\rightarrow C^{2}_{N}+2.8\ eV$ (5)

The positive energy gain of the reconstruction of the defects ensures the
stability of $C_{N}^{2}$ in both processes. Furthermore, since both
interstitial boron and carbon atoms are mobile at room temperature [36], the
formation of the $C_{N}^{2}$ defect may happen without any annealing step
solely due to electron irradiation.

In conclusion, the emission tunability of blue quantum emitters in hBN has
been investigated through the use of two separate devices used to generate in-
plane and out-of-plane electric fields. The transition energy is influenced by
out-of-plane fields, but the magnitude of the observed linear Stark shift is
small relative to the linewidth. In contrast, in-plane electric fields
produced significantly greater ZPL modulation, yielding a clear quadratic
Stark shift response. Using these results we provide insight into the possible
atomic structure of defects responsible for the reported blue emitters in hBN.
We suggest the transition dipole moment resides mainly within the plane of the
hBN lattice, which may result in decreased sensitivity to local field
fluctuations that cause spectral diffusion.

###### Acknowledgements.

The authors acknowledge financial support from the Australian Research Council
(CE200100010) and the Office of Naval Research Global (N62909-22-1-2028) for
financial support. The authors thank the UTS node of Optofab ANFF for the
assistance with nanofabrication. V.I. acknowledges the support from the Knut
and Alice Wallenberg Foundation through WBSQD2 project (Grant No.2018.0071).
The calculations were performed on resources provided by the Swedish National
Infrastructure for Computing (SNIC) at the National Supercomputer Center
(NSC).

## References

  * Tran  _et al._ [2016] T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, Quantum emission from hexagonal boron nitride monolayers, Nature Nanotechnology 11, 37 (2016).
  * Li  _et al._ [2021] C. Li, J. E. FrÃ¶ch, M. Nonahal, T. N. Tran, M. Toth, S. Kim, and I. Aharonovich, Integration of hBN Quantum Emitters in Monolithically Fabricated Waveguides, ACS Photonics 8, 2966 (2021).
  * Vogl  _et al._ [2019] T. Vogl, K. Sripathy, A. Sharma, P. Reddy, J. Sullivan, J. R. Machacek, L. Zhang, F. Karouta, B. C. Buchler, M. W. Doherty, Y. Lu, and P. K. Lam, Radiation tolerance of two-dimensional material-based devices for space applications, Nature Communications 10, 1202 (2019).
  * Kianinia  _et al._ [2022] M. Kianinia, Z. Q. Xu, M. Toth, and I. Aharonovich, Quantum emitters in 2d materials: Emitter engineering, photophysics, and integration in photonic nanostructures, Applied Physics Reviews 9, 011306 (2022).
  * Malein  _et al._ [2021] R. N. E. Malein, P. Khatri, A. J. Ramsay, and I. J. Luxmoore, Stimulated emission depletion spectroscopy of color centers in hexagonal boron nitride, ACS Photonics 8, 1007â€“1012 (2021).
  * Xu  _et al._ [2021] X. Xu, Z. O. Martin, D. Sychev, A. S. Lagutchev, Y. P. Chen, T. Taniguchi, K. Watanabe, V. M. Shalaev, and A. Boltasseva, Creating quantum emitters in hexagonal boron nitride deterministically on chip-compatible substrates, Nano Letters 21, 8182â€“8189 (2021).
  * Stewart  _et al._ [2021] J. C. Stewart, Y. Fan, J. S. H. Danial, A. Goetz, A. S. Prasad, O. J. Burton, J. A. Alexander-Webber, S. F. Lee, S. M. Skoff, V. Babenko, and S. Hofmann, Quantum emitter localization in layer-engineered hexagonal boron nitride, ACS Nano 15, 13591â€“13603 (2021).
  * AtatÃ¼re  _et al._ [2018] M. AtatÃ¼re, D. Englund, N. Vamivakas, S.-Y. Lee, and J. Wrachtrup, Material platforms for spin-based photonic quantum technologies, Nature Reviews Materials 3, 38 (2018).
  * Gottscholl  _et al._ [2020] A. Gottscholl, M. Kianinia, V. Soltamov, S. Orlinskii, G. Mamin, C. Bradac, C. Kasper, K. Krambrock, A. Sperlich, M. Toth, I. Aharonovich, and V. Dyakonov, Initialization and read-out of intrinsic spin defects in a van der Waals crystal at room temperature, Nature Materials 19, 540 (2020).
  * Gao  _et al._ [2021] X. Gao, S. Pandey, M. Kianinia, J. Ahn, P. Ju, I. Aharonovich, N. Shivaram, and T. Li, Femtosecond laser writing of spin defects in hexagonal boron nitride, ACS Photonics 8, 994â€“1000 (2021).
  * Chejanovsky  _et al._ [2021] N. Chejanovsky, A. Mukherjee, J. Geng, Y.-C. Chen, Y. Kim, A. Denisenko, A. Finkler, T. Taniguchi, K. Watanabe, D. B. R. Dasari, P. Auburger, A. Gali, J. H. Smet, and J. Wrachtrup, Single-spin resonance in a van der waals embedded paramagnetic defect, Nature Materials 20, 1079â€“1084 (2021).
  * Stern  _et al._ [2022] H. L. Stern, Q. Gu, J. Jarman, S. Eizagirre Barker, N. Mendelson, D. Chugh, S. Schott, H. H. Tan, H. Sirringhaus, I. Aharonovich, and M. AtatÃ¼re, Room-temperature optically detected magnetic resonance of single defects in hexagonal boron nitride, Nature Communications 13, 618 (2022).
  * Proscia  _et al._ [2020] N. V. Proscia, H. Jayakumar, X. Ge, G. Lopez-Morales, Z. Shotan, W. Zhou, C. A. Meriles, and V. M. Menon, Microcavity-coupled emitters in hexagonal boron nitride, Nanophotonics 9, 2937 (2020).
  * FrÃ¶ch  _et al._ [2020] J. E. FrÃ¶ch, S. Kim, N. Mendelson, M. Kianinia, M. Toth, and I. Aharonovich, Coupling Hexagonal Boron Nitride Quantum Emitters to Photonic Crystal Cavities, ACS Nano 14, 7085 (2020).
  * Ruf  _et al._ [2019] M. Ruf, M. IJspeert, S. van Dam, N. de Jong, H. van den Berg, G. Evers, and R. Hanson, Optically Coherent Nitrogen-Vacancy Centers in Micrometer-Thin Etched Diamond Membranes, Nano Letters 19, 3987 (2019).
  * Zhong  _et al._ [2015] T. Zhong, J. M. Kindem, E. Miyazono, and A. Faraon, Nanophotonic coherent lightâ€“matter interfaces based on rare-earth-doped crystals, Nature Communications 6, 8206 (2015).
  * Xia  _et al._ [2019] Y. Xia, Q. Li, J. Kim, W. Bao, C. Gong, S. Yang, Y. Wang, and X. Zhang, Room-Temperature Giant Stark Effect of Single Photon Emitter in van der Waals Material, Nano Letters 19, 7100 (2019).
  * Noh  _et al._ [2018] G. Noh, D. Choi, J.-H. Kim, D.-G. Im, Y.-H. Kim, H. Seo, and J. Lee, Stark Tuning of Single-Photon Emitters in Hexagonal Boron Nitride, Nano Letters 18, 4710 (2018).
  * Nikolay  _et al._ [2019] N. Nikolay, N. Mendelson, N. Sadzak, F. BÃ¶hm, T. T. Tran, B. Sontheimer, I. Aharonovich, and O. Benson, Very Large and Reversible Stark-Shift Tuning of Single Emitters in Layered Hexagonal Boron Nitride, Physical Review Applied 11, 041001 (2019).
  * White  _et al._ [2020] S. J. White, N. M. H. Duong, A. S. Solntsev, J.-H. Kim, M. Kianinia, and I. Aharonovich, Optical Repumping of Resonantly Excited Quantum Emitters in Hexagonal Boron Nitride, Physical Review Applied 14, 044017 (2020).
  * Li  _et al._ [2017] X. Li, G. D. Shepard, A. Cupo, N. Camporeale, K. Shayan, Y. Luo, V. Meunier, and S. Strauf, Nonmagnetic Quantum Emitters in Boron Nitride with Ultranarrow and Sideband-Free Emission Spectra, ACS Nano 11, 6652 (2017).
  * Akbari  _et al._ [2022] H. Akbari, S. Biswas, P. K. Jha, J. Wong, and B. Vest, Lifetime Limited and Tunable Quantum Light Emission in h-BN via Electric Field Modulation, ArXiv Preprint arXiv:2206.10725 (2022).
  * De Santis  _et al._ [2021] L. De Santis, M. E. Trusheim, K. C. Chen, and D. R. Englund, Investigation of the Stark Effect on a Centrosymmetric Quantum Emitter in Diamond, Physical Review Letters 127, 147402 (2021).
  * RÃ¼hl  _et al._ [2020] M. RÃ¼hl, L. Bergmann, M. Krieger, and H. B. Weber, Stark Tuning of the Silicon Vacancy in Silicon Carbide, Nano Letters 20, 658 (2020).
  * Shevitski  _et al._ [2019] B. Shevitski, S. M. Gilbert, C. T. Chen, C. Kastl, E. S. Barnard, E. Wong, D. F. Ogletree, K. Watanabe, T. Taniguchi, A. Zettl, and S. Aloni, Blue-light-emitting color centers in high-quality hexagonal boron nitride, Physical Review B 100, 155419 (2019).
  * Fournier  _et al._ [2021] C. Fournier, A. Plaud, S. Roux, A. Pierret, M. Rosticher, K. Watanabe, T. Taniguchi, S. Buil, X. QuÃ©lin, J. Barjon, J.-P. Hermier, and A. Delteil, Position-controlled quantum emitters with reproducible emission wavelength in hexagonal boron nitride, Nature Communications 12, 3779 (2021).
  * Gale  _et al._ [2022] A. Gale, C. Li, Y. Chen, K. Watanabe, T. Taniguchi, I. Aharonovich, and M. Toth, Site-Specific Fabrication of Blue Quantum Emitters in Hexagonal Boron Nitride, ACS Photonics 9, 2170 (2022).
  * Horder  _et al._ [2022] J. Horder, S. J. U. White, A. Gale, C. Li, K. Watanabe, M. Kianinia, I. Aharonovich, and M. Toth, Coherence properties of electron beam activated emitters in hexagonal boron nitride under resonant excitation, ArXiv Preprint arXiv:2205.08810 (2022).
  * Tamarat  _et al._ [2006] P. Tamarat, T. Gaebel, J. R. Rabeau, M. Khan, A. D. Greentree, H. Wilson, L. C. L. Hollenberg, S. Prawer, P. Hemmer, F. Jelezko, and J. Wrachtrup, Stark Shift Control of Single Optical Centers in Diamond, Physical Review Letters 97, 083002 (2006).
  * Udvarhelyi  _et al._ [2019] P. Udvarhelyi, R. Nagy, F. Kaiser, S.-Y. Lee, J. Wrachtrup, and A. Gali, Spectrally Stable Defect Qubits with no Inversion Symmetry for Robust Spin-To-Photon Interface, Physical Review Applied 11, 044022 (2019).
  * Laturia  _et al._ [2018] A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk, npj 2D Materials and Applications 2, 6 (2018).
  * White  _et al._ [2022] S. J. U. White, T. Yang, N. Dontschuk, C. Li, Z.-Q. Xu, M. Kianinia, A. Stacey, M. Toth, and I. Aharonovich, Electrical control of quantum emitters in a Van der Waals heterostructure, Light, Science & Applications 11, 186 (2022).
  * Yu  _et al._ [2022] M. Yu, D. Yim, H. Seo, and J. Lee, Electrical charge control of h-BN single photon sources, 2D Materials 9, 035020 (2022).
  * Maze  _et al._ [2011] J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, Properties of nitrogen-vacancy centers in diamond: the group theoretic approach, New Journal of Physics 13, 025025 (2011).
  * IvÃ¡dy  _et al._ [2020] V. IvÃ¡dy, G. Barcza, G. Thiering, S. Li, H. Hamdi, J.-P. Chou, Ã. Legeza, and A. Gali, Ab initio theory of the negatively charged boron vacancy qubit in hexagonal boron nitride, npj Computational Materials 6, 1 (2020).
  * Weston  _et al._ [2018] L. Weston, D. Wickramaratne, M. Mackoit, A. Alkauskas, and C. G. Van de Walle, Native point defects and impurities in hexagonal boron nitride, Physical Review B 97, 214104 (2018).


* [63] Y. Freund and R. E. Schapire, _A decision-theoretic generalization of on-line learning and an application to boosting_ , J. Comput. Syst. Sci. 55 (1997) 119
  * [64] N. L. Johnson, _Systems of frequency curves generated by methods of translation_ , Biometrika 36 (1949) 149
  * [65] L. Anderlini et al., _The PIDCalib package_ , LHCb-PUB-2016-021, 2016
  * [66] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [67] Y. Xie, _sFit: a method for background subtraction in maximum likelihood fit_ , arXiv:0905.0724
  * [68] LHCb collaboration, R. Aaij et al., _Opposite-side flavour tagging of $B$ mesons at the LHCb experiment_, Eur. Phys. J. C72 (2012) 2022, arXiv:1202.4979
  * [69] LHCb collaboration, R. Aaij et al., _$B$ flavour tagging using charm decays at the LHCb experiment_, JINST 10 (2015) P10005, arXiv:1507.07892
  * [70] LHCb collaboration, R. Aaij et al., _A new algorithm for identifying the flavour of ${B}^{0}_{s}$ mesons at LHCb_, JINST 11 (2016) P05010, arXiv:1602.07252
  * [71] LHCb collaboration, R. Aaij et al., _Precision measurement of the ${B}^{0}_{s}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ oscillation frequency in the decay ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$_, New J. Phys. 15 (2013) 053021, arXiv:1304.4741
  * [72] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ violation in ${{B}^{0}}\\!\rightarrow D^{\pm}{{\pi}^{\mp}}$ decays_, JHEP 06 (2018) 084, arXiv:1805.03448
  * [73] G. Punzi, _Comments on likelihood fits with variable resolution_ , eConf C030908 (2003) WELT002, arXiv:physics/0401045
  * [74] Geometric Modeling with Splines: An Introduction, A. K. Peters, Ltd., Natick, MA, USA, 2001
  * [75] P. L. Butzer, M. Schmidt, and E. L. Stark, _Observations on the history of central b-splines_ , Archive for History of Exact Sciences 39 (1988) 137
  * [76] T. M. Karbach, G. Raven, and M. Schiller, _Decay time integrals in neutral meson mixing and their efficient evaluation_ , arXiv:1407.0748
  * [77] D. Fazzini, _Flavour Tagging in the LHCb experiment_ , in Proceedings, 6th Large Hadron Collider Physics Conference (LHCP 2018): Bologna, Italy, June 4-9, 2018, LHCP2018 230, 2018
  * [78] LHCb collaboration, R. Aaij et al., _Measurement of ${B}^{0}$, ${B}^{0}_{s}$, ${{B}^{+}}$ and ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries in 7 and 8 TeV proton-proton collisions_, Phys. Lett. B774 (2017) 139, arXiv:1703.08464
  * [79] A. Davis et al., _Measurement of the instrumental asymmetry for $K^{-}\pi^{+}$-pairs at LHCb in Run 2_, LHCb-PUB-2018-004, 2018
  * [80] MARK-III collaboration, D. Coffman et al., _Resonant substructure in ${{\kern 1.79993pt\overline{\kern-1.79993ptK}}}\pi\pi\pi$ decays of $D$ mesons_, Phys. Rev. D45 (1992) 2196
  * [81] FOCUS collaboration, J. M. Link et al., _Study of the $D^{0}\rightarrow\pi^{-}\pi^{+}\pi^{-}\pi^{+}$ decay_, Phys. Rev. D75 (2007) 052003, arXiv:hep-ex/0701001
  * [82] CLEO collaboration, M. Artuso et al., _Amplitude analysis of $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$_, Phys. Rev. D85 (2012) 122002, arXiv:1201.5716
  * [83] R. Tibshirani, _Regression shrinkage and selection via the Lasso_ , Journal of the Royal Statistical Society, Series B 58 (1994) 267
  * [84] B. Guegan, J. Hardin, J. Stevens, and M. Williams, _Model selection for amplitude analysis_ , JINST 10 (2015) P09002, arXiv:1505.05133
  * [85] G. Schwarz, _Estimating the dimension of a model_ , Ann. Statist. 6 (1978) 461
  * [86] M. Kenzie et al., _GammaCombo: A statistical analysis framework for combining measurements, fitting datasets and producing confidence intervals_ ,  doi: 10.5281/zenodo.3371421
  * [87] LHCb collaboration, R. Aaij et al., _Measurement of the CKM angle $\gamma$ from a combination of LHCb results_, JHEP 12 (2016) 087, arXiv:1611.03076
  * [88] D. King, A. Lenz, and T. Rauh, _B s mixing observables and —Vtd/Vts— from sum rules_, JHEP 05 (2019) 034, arXiv:1904.00940
  * [89] LHCb collaboration, R. Aaij et al., _Studies of the resonance structure in ${{D}^{0}}\\!\rightarrow K^{\mp}\pi^{\pm}{{\pi}^{+}}{{\pi}^{-}}$ decays_, Eur. Phys. J. C78 (2018) 443, arXiv:1712.08609

LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala84, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, Z. Aliouche61,
G. Alkhazov37, P. Alvarez Cartelle47, S. Amato2, Y. Amhis11, L. An21, L.
Anderlini21, A. Andreianov37, M. Andreotti20, F. Archilli16, A. Artamonov43,
M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B. Audurier11, S.
Bachmann16, M. Bachmayer48, J.J. Back55, S. Baker60, P. Baladron Rodriguez45,
V. Balagura11, W. Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11,
W. Barter60, M. Bartolini23,i, F. Baryshnikov80, J.M. Basels13, G. Bassi28, B.
Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A.
Beiter67, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43, I. Belov39, I.
Belyaev38, G. Bencivenni22, E. Ben-Haim12, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, M. Birch60,
F.C.R. Bishop54, A. Bizzeti21,s, M. Bjørn62, M.P. Blago47, T. Blake55, F.
Blanc48, S. Blusk67, D. Bobulska58, J.A. Boelhauve14, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev81, A. Bondar42, N. Bondar37, S. Borghi61, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48, T.J.V. Bowcock59, A.
Boyer47, C. Bozzi20, M.J. Bradley60, S. Braun65, A. Brea Rodriguez45, M.
Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D. Brundu26, A. Buonaura49, C.
Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L. Calefice14,12, L.
Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,j, M. Calvo Gomez83, P.
Camargo Magalhaes53, A. Camboni44, P. Campana22, D.H. Campora Perez47, A.F.
Campoverde Quezada5, S. Capelli24,j, L. Capriotti19,e, A. Carbone19,e, G.
Carboni29, R. Cardinale23,i, A. Cardini26, I. Carli6, P. Carniti24,j, L.
Carus13, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M. Cattaneo47, G.
Cavallero47, S. Celani48, J. Cerasoli10, A.J. Chadwick59, M.G. Chapman53, M.
Charles12, Ph. Charpentier47, G. Chatzikonstantinidis52, C.A. Chavez
Barajas59, M. Chefdeville8, C. Chen3, S. Chen26, A. Chernov33, S.-G. Chitic47,
V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37, V. Chulikov37, P.
Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, P.E.L. Clarke57, M.
Clemencic47, H.V. Cliff54, J. Closier47, J.L. Cobbledick61, V. Coco47, J.A.B.
Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu36, P. Collins47, T.
Colombo47, L. Congedo18,d, A. Contu26, N. Cooke52, G. Coombs58, G. Corti47,
C.M. Costa Sobral55, B. Couturier47, D.C. Craik63, J. Crkovská66, M. Cruz
Torres1, R. Currie57, C.L. Da Silva66, E. Dall’Occo14, J. Dalseno45, C.
D’Ambrosio47, A. Danilina38, P. d’Argent47, A. Davis61, O. De Aguiar
Francisco61, K. De Bruyn77, S. De Capua61, M. De Cian48, J.M. De Miranda1, L.
De Paula2, M. De Serio18,d, D. De Simone49, P. De Simone22, J.A. de Vries78,
C.T. Dean66, W. Dean84, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P.
Dembinski14, A. Dendek34, V. Denysenko49, D. Derkach81, O. Deschamps9, F.
Desse11, F. Dettori26,f, B. Dey72, P. Di Nezza22, S. Didenko80, L. Dieste
Maronas45, H. Dijkstra47, V. Dobishuk51, A.M. Donohoe17, F. Dordei26, A.C. dos
Reis1, L. Douglas58, A. Dovbnya50, A.G. Downes8, K. Dreimanis59, M.W. Dudek33,
L. Dufour47, V. Duk76, P. Durante47, J.M. Durham66, D. Dutta61, M.
Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede68, V.
Egorychev38, S. Eidelman42,v, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S.
Ely67, A. Ene36, E. Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47,
A. Falabella19, J. Fan3, Y. Fan5, B. Fang72, N. Farley52, S. Farry59, D.
Fazzini24,j, P. Fedin38, M. Féo47, P. Fernandez Declara47, A. Fernandez
Prieto45, J.M. Fernandez-tenllado Arribas44, F. Ferrari19,e, L. Ferreira
Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49, M. Ferro-
Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g, M. Firlej34, K.M.
Fischer62, C. Fitzpatrick61, T. Fiutowski34, F. Fleuret11,b, M. Fontana12, F.
Fontanelli23,i, R. Forty47, V. Franco Lima59, M. Franco Sevilla65, M. Frank47,
E. Franzoso20, G. Frau16, C. Frei47, D.A. Friday58, J. Fu25, Q. Fuehring14, W.
Funk47, E. Gabriel31, T. Gaintseva41, A. Gallas Torreira45, D. Galli19,e, S.
Gambetta57,47, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4, M. Garau26, L.M.
Garcia Martin55, P. Garcia Moreno44, J. García Pardiñas49, B. Garcia Plana45,
F.A. Garcia Rosales11, L. Garrido44, C. Gaspar47, R.E. Geertsema31, D.
Gerick16, L.L. Gerken14, E. Gersabeck61, M. Gersabeck61, T. Gershon55, D.
Gerstel10, Ph. Ghez8, V. Gibson54, M. Giovannetti22,k, A. Gioventù45, P.
Gironella Gironell44, L. Giubega36, C. Giugliano20,47,g, K. Gizdov57, E.L.
Gkougkousis47, V.V. Gligorov12, C. Göbel69, E. Golobardes83, D. Golubkov38, A.
Golutvin60,80, A. Gomes1,a, S. Gomez Fernandez44, F. Goncalves Abrantes69, M.
Goncerz33, G. Gong3, P. Gorbounov38, I.V. Gorelov39, C. Gotti24,j, E.
Govorkova47, J.P. Grabowski16, R. Graciani Diaz44, T. Grammatico12, L.A.
Granado Cardoso47, E. Graugés44, E. Graverini48, G. Graziani21, A. Grecu36,
L.M. Greeven31, P. Griffith20, L. Grillo61, S. Gromov80, B.R. Gruberg Cazon62,
C. Gu3, M. Guarise20, P. A. Günther16, E. Gushchin40, A. Guth13, Y. Guz43,47,
T. Gys47, T. Hadavizadeh68, G. Haefeli48, C. Haen47, J. Haimberger47, S.C.
Haines54, T. Halewood-leagas59, P.M. Hamilton65, Q. Han7, X. Han16, T.H.
Hancock62, S. Hansmann-Menzemer16, N. Harnew62, T. Harrison59, C. Hasse47, M.
Hatch47, J. He5, M. Hecker60, K. Heijhoff31, K. Heinicke14, A.M. Hennequin47,
K. Hennessy59, L. Henry25,46, J. Heuel13, A. Hicheur2, D. Hill62, M. Hilton61,
S.E. Hollitt14, J. Hu16, J. Hu71, W. Hu7, W. Huang5, X. Huang72, W.
Hulsbergen31, R.J. Hunter55, M. Hushchyn81, D. Hutchcroft59, D. Hynds31, P.
Ibis14, M. Idzik34, D. Ilin37, P. Ilten64, A. Inglessi37, A. Ishteev80, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K. Jashal46, A.
Jawahery65, V. Jevtic14, M. Jezabek33, F. Jiang3, M. John62, D. Johnson47,
C.R. Jones54, T.P. Jones55, B. Jost47, N. Jurik47, S. Kandybei50, Y. Kang3, M.
Karacson47, M. Karpov81, N. Kazeev81, M. Kecke16, F. Keizer54,47, M. Kenzie55,
T. Ketel32, B. Khanji14, A. Kharisova82, S. Kholodenko43, K.E. Kim67, T.
Kirn13, V.S. Kirsebom48, O. Kitouni63, S. Klaver31, K. Klimaszewski35, S.
Koliiev51, A. Kondybayeva80, A. Konoplyannikov38, P. Kopciewicz34, R.
Kopecna16, P. Koppenburg31, M. Korolev39, I. Kostiuk31,51, O. Kot51, S.
Kotriakhova37,30, P. Kravchenko37, L. Kravchuk40, R.D. Krawczyk47, M. Kreps55,
F. Kress60, S. Kretzschmar13, P. Krokovny42,v, W. Krupa34, W. Krzemien35, W.
Kucewicz33,l, M. Kucharczyk33, V. Kudryavtsev42,v, H.S. Kuindersma31, G.J.
Kunde66, T. Kvaratskheliya38, D. Lacarrere47, G. Lafferty61, A. Lai26, A.
Lampis26, D. Lancierini49, J.J. Lane61, R. Lane53, G. Lanfranchi22, C.
Langenbruch13, J. Langer14, O. Lantwin49,80, T. Latham55, F. Lazzari28,t, R.
Le Gac10, S.H. Lee84, R. Lefèvre9, A. Leflat39, S. Legotin80, O. Leroy10, T.
Lesiak33, B. Leverington16, H. Li71, L. Li62, P. Li16, X. Li66, Y. Li6, Y.
Li6, Z. Li67, X. Liang67, T. Lin60, R. Lindner47, V. Lisovskyi14, R.
Litvinov26, G. Liu71, H. Liu5, S. Liu6, X. Liu3, A. Loi26, J. Lomba Castro45,
I. Longstaff58, J.H. Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D.
Lucchesi27,m, S. Luchuk40, M. Lucio Martinez31, V. Lukashenko31, Y. Luo3, A.
Lupato61, E. Luppi20,g, O. Lupton55, A. Lusiani28,r, X. Lyu5, L. Ma6, S.
Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48, P. Mackowiak14, S.
Maddrell-Mander53, O. Madejczyk34, L.R. Madhan Mohan53, O. Maev37, A.
Maevskiy81, D. Maisuzenko37, M.W. Majewski34, J.J. Malczewski33, S. Malde62,
B. Malecki47, A. Malinin79, T. Maltsev42,v, H. Malygina16, G. Manca26,f, G.
Mancinelli10, R. Manera Escalero44, D. Manuzzi19,e, D. Marangotto25,o, J.
Maratas9,u, J.F. Marchand8, U. Marconi19, S. Mariani21,47,h, C. Marin
Benito11, M. Marinangeli48, P. Marino48, J. Marks16, P.J. Marshall59, G.
Martellotti30, L. Martinazzoli47,j, M. Martinelli24,j, D. Martinez Santos45,
F. Martinez Vidal46, A. Massafferri1, M. Materok13, R. Matev47, A. Mathad49,
Z. Mathe47, V. Matiunin38, C. Matteuzzi24, K.R. Mattioli84, A. Mauri31, E.
Maurice11,b, J. Mauricio44, M. Mazurek35, M. McCann60, L. Mcconnell17, T.H.
Mcgrath61, A. McNab61, R. McNulty17, J.V. Mead59, B. Meadows64, C. Meaux10, G.
Meier14, N. Meinert75, D. Melnychuk35, S. Meloni24,j, M. Merk31,78, A.
Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A. Milanes73, E. Millard55, M.
Milovanovic47, M.-N. Minard8, L. Minzoni20,g, S.E. Mitchell57, B. Mitreska61,
D.S. Mitzel47, A. Mödden14, R.A. Mohammed62, R.D. Moise60, T. Mombächer14,
I.A. Monroy73, S. Monteil9, M. Morandin27, G. Morello22, M.J. Morello28,r, J.
Moron34, A.B. Morris74, A.G. Morris55, R. Mountain67, H. Mu3, F. Muheim57, M.
Mukherjee7, M. Mulder47, D. Müller47, K. Müller49, C.H. Murphy62, D. Murray61,
P. Muzzetto26,47, P. Naik53, T. Nakada48, R. Nandakumar56, T. Nanut48, I.
Nasteva2, M. Needham57, I. Neri20,g, N. Neri25,o, S. Neubert74, N. Neufeld47,
R. Newcombe60, T.D. Nguyen48, C. Nguyen-Mau48, E.M. Niel11, S. Nieswand13, N.
Nikitin39, N.S. Nolte47, C. Nunez84, A. Oblakowska-Mucha34, V. Obraztsov43,
D.P. O’Hanlon53, R. Oldeman26,f, M.E. Olivares67, C.J.G. Onderwater77, A.
Ossowska33, J.M. Otalora Goicochea2, T. Ovsiannikova38, P. Owen49, A.
Oyanguren46,47, B. Pagare55, P.R. Pais47, T. Pajero28,47,r, A. Palano18, M.
Palutan22, Y. Pan61, G. Panshin82, A. Papanestis56, M. Pappagallo18,d, L.L.
Pappalardo20,g, C. Pappenheimer64, W. Parker65, C. Parkes61, C.J. Parkinson45,
B. Passalacqua20, G. Passaleva21, A. Pastore18, M. Patel60, C. Patrignani19,e,
C.J. Pawley78, A. Pearce47, A. Pellegrino31, M. Pepe Altarelli47, S.
Perazzini19, D. Pereima38, P. Perret9, K. Petridis53, A. Petrolini23,i, A.
Petrov79, S. Petrucci57, M. Petruzzo25, T.T.H. Pham67, A. Philippov41, L.
Pica28, M. Piccini76, B. Pietrzyk8, G. Pietrzyk48, M. Pili62, D. Pinci30, F.
Pisani47, A. Piucci16, Resmi P.K10, V. Placinta36, J. Plews52, M. Plo
Casasus45, F. Polci12, M. Poli Lener22, M. Poliakova67, A. Poluektov10, N.
Polukhina80,c, I. Polyakov67, E. Polycarpo2, G.J. Pomery53, S. Ponce47, D.
Popov5,47, S. Popov41, S. Poslavskii43, K. Prasanth33, L. Promberger47, C.
Prouve45, V. Pugatch51, H. Pullen62, G. Punzi28,n, W. Qian5, J. Qin5, R.
Quagliani12, B. Quintana8, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34,
J.H. Rademacker53, M. Rama28, M. Ramos Pernas55, M.S. Rangel2, F.
Ratnikov41,81, G. Raven32, M. Reboud8, F. Redi48, F. Reiss12, C. Remon
Alepuz46, Z. Ren3, V. Renaudin62, R. Ribatti28, S. Ricciardi56, K. Rinnert59,
P. Robbe11, A. Robert12, G. Robertson57, A.B. Rodrigues48, E. Rodrigues59,
J.A. Rodriguez Lopez73, A. Rollings62, P. Roloff47, V. Romanovskiy43, M.
Romero Lamas45, A. Romero Vidal45, J.D. Roth84, M. Rotondo22, M.S. Rudolph67,
T. Ruf47, J. Ruiz Vidal46, A. Ryzhikov81, J. Ryzka34, J.J. Saborido Silva45,
N. Sagidova37, N. Sahoo55, B. Saitta26,f, D. Sanchez Gonzalo44, C. Sanchez
Gras31, R. Santacesaria30, C. Santamarina Rios45, M. Santimaria22, E.
Santovetti29,k, D. Saranin80, G. Sarpis58, M. Sarpis74, A. Sarti30, C.
Satriano30,q, A. Satta29, M. Saur5, D. Savrina38,39, H. Sazak9, L.G.
Scantlebury Smead62, S. Schael13, M. Schellenberg14, M. Schiller58, H.
Schindler47, M. Schmelling15, T. Schmelzer14, B. Schmidt47, O. Schneider48, A.
Schopper47, M. Schubiger31, S. Schulte48, M.H. Schune11, R. Schwemmer47, B.
Sciascia22, A. Sciubba30, S. Sellam45, A. Semennikov38, M. Senghi Soares32, A.
Sergi52,47, N. Serra49, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M.
Shangase84, M. Shapkin43, I. Shchemerov80, L. Shchutska48, T. Shears59, L.
Shekhtman42,v, Z. Shen4, V. Shevchenko79, E.B. Shields24,j, E. Shmanin80, J.D.
Shupperd67, B.G. Siddi20, R. Silva Coutinho49, G. Simi27, S. Simone18,d, I.
Skiba20,g, N. Skidmore74, T. Skwarnicki67, M.W. Slater52, J.C. Smallwood62,
J.G. Smeaton54, A. Smetkina38, E. Smith13, M. Smith60, A. Snoch31, M.
Soares19, L. Soares Lavra9, M.D. Sokoloff64, F.J.P. Soler58, A. Solovev37, I.
Solovyev37, F.L. Souza De Almeida2, B. Souza De Paula2, B. Spaan14, E. Spadaro
Norella25,o, P. Spradlin58, F. Stagni47, M. Stahl64, S. Stahl47, P. Stefko48,
O. Steinkamp49,80, S. Stemmle16, O. Stenyakin43, H. Stevens14, S. Stone67,
M.E. Stramaglia48, M. Straticiuc36, D. Strekalina80, S. Strokov82, F.
Suljik62, J. Sun26, L. Sun72, Y. Sun65, P. Svihra61, P.N. Swallow52, K.
Swientek34, A. Szabelski35, T. Szumlak34, M. Szymanski47, S. Taneja61, F.
Teubert47, E. Thomas47, K.A. Thomson59, M.J. Tilley60, V. Tisserand9, S.
T’Jampens8, M. Tobin6, S. Tolk47, L. Tomassetti20,g, D. Torres Machado1, D.Y.
Tou12, M. Traill58, M.T. Tran48, E. Trifonova80, C. Trippl48, G. Tuci28,n, A.
Tully48, N. Tuning31, A. Ukleja35, D.J. Unverzagt16, A. Usachov31, A.
Ustyuzhanin41,81, U. Uwer16, A. Vagner82, V. Vagnoni19, A. Valassi47, G.
Valenti19, N. Valls Canudas44, M. van Beuzekom31, M. Van Dijk48, H. Van
Hecke66, E. van Herwijnen80, C.B. Van Hulse17, M. van Veghel77, R. Vazquez
Gomez45, P. Vazquez Regueiro45, C. Vázquez Sierra31, S. Vecchi20, J.J.
Velthuis53, M. Veltri21,p, A. Venkateswaran67, M. Veronesi31, M. Vesterinen55,
D. Vieira64, M. Vieites Diaz48, H. Viemann75, X. Vilasis-Cardona83, E. Vilella
Figueras59, P. Vincent12, G. Vitali28, A. Vollhardt49, D. Vom Bruch12, A.
Vorobyev37, V. Vorobyev42,v, N. Voropaev37, R. Waldi75, J. Walsh28, C. Wang16,
J. Wang3, J. Wang72, J. Wang4, J. Wang6, M. Wang3, R. Wang53, Y. Wang7, Z.
Wang49, H.M. Wark59, N.K. Watson52, S.G. Weber12, D. Websdale60, C. Weisser63,
B.D.C. Westhenry53, D.J. White61, M. Whitehead53, D. Wiedner14, G.
Wilkinson62, M. Wilkinson67, I. Williams54, M. Williams63,68, M.R.J.
Williams57, F.F. Wilson56, W. Wislicki35, M. Witek33, L. Witola16, G.
Wormser11, S.A. Wotton54, H. Wu67, K. Wyllie47, Z. Xiang5, D. Xiao7, Y. Xie7,
A. Xu4, J. Xu5, L. Xu3, M. Xu7, Q. Xu5, Z. Xu5, Z. Xu4, D. Yang3, Y. Yang5, Z.
Yang3, Z. Yang65, Y. Yao67, L.E. Yeomans59, H. Yin7, J. Yu70, X. Yuan67, O.
Yushchenko43, E. Zaffaroni48, K.A. Zarebski52, M. Zavertyaev15,c, M. Zdybal33,
O. Zenaiev47, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, Y. Zhang4, Y.
Zhang62, A. Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V.
Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,e, D. Zuliani27, G. Zunica61.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
69Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
70Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 16
75Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
76INFN Sezione di Perugia, Perugia, Italy, associated to 20
77Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
78Universiteit Maastricht, Maastricht, Netherlands, associated to 31
79National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
80National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
81National Research University Higher School of Economics, Moscow, Russia,
associated to 41
82National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
83DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 44
84University of Michigan, Ann Arbor, United States, associated to 67

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Firenze, Firenze, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mUniversità di Padova, Padova, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità degli Studi di Milano, Milano, Italy
pUniversità di Urbino, Urbino, Italy
qUniversità della Basilicata, Potenza, Italy
rScuola Normale Superiore, Pisa, Italy
sUniversità di Modena e Reggio Emilia, Modena, Italy
tUniversità di Siena, Siena, Italy
uMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
vNovosibirsk State University, Novosibirsk, Russia