# Toeplitz operators and Hilbert modules on the symmetrized polydisc

Tirthankar Bhattacharyya  Department of Mathematics, Indian Institute of
Science, Bangalore 560 012, India<EMAIL_ADDRESS>,  B. Krishna Das
Department of Mathematics, Indian Institute of Technology Bombay, Powai,
Mumbai 400076, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Haripada
Sau  Department of Mathematics, Indian Institute of Science Education and
Research, Pashan, Pune 411008, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

When is the collection of $\mathsf{S}$-Toeplitz operators with respect to a
tuple of commuting bounded operators
$\mathsf{S}=(S_{1},S_{2},\ldots,S_{d-1},P)$, which has the symmetrized
polydisc as a spectral set, non-trivial? The answer is in terms of powers of
$P$ as well as in terms of a unitary extension. En route, Brown-Halmos
relations are investigated. A commutant lifting theorem is established.
Finally, we establish a general result connecting the $C^{*}$-algebra
generated by the commutant of $\mathsf{S}$ and the commutant of its unitary
extension $\mathsf{R}$.

###### Key words and phrases:

Symmetrized Polydisc, Polydisc, Toeplitz operator, contractive Hilbert
modules, contractive embeddings.

###### 2020 Mathematics Subject Classification:

47A13, 47A20, 47B35, 46L07

##  1\. Introduction

A famous result of Brown and Halmos shows that a bounded linear operator $A$
on the Hardy space $H^{2}$ is a Toeplitz operator if and only if
$M_{z}^{*}AM_{z}=A$, where $M_{z}$ denotes the unilateral shift. This led to
the study of operators $A$ on general Hilbert spaces satisfying an operator
equation of the form $T^{*}AT=A$ by many authors.

Recently, the study of Toeplitz operators has been extended to the Hardy space
of the symmetrized bidisc, see [2]. This motivates our study. The
characterizations obtained in this note are in terms of Hilbert modules which
is a natural settings for operator theory in several variables. Completely
positive maps were used by Prunaru in [15] because they were well-suited for
the Euclidean unit ball. We use both. It is this interplay of completely
positive maps with Hilbert modules which allows the results to be presented in
their most general and natural form.

Let $\mathbb{D}$ be the open unit disc while $\mathbb{D}^{d}$,
$\overline{\mathbb{D}}^{d}$ and $\mathbb{T}^{d}$ denote the open polydisc, the
closed polydisc, and the $d$-torus, respectively in $d$-dimensional complex
space for $d\geq 2$.

If $T_{1},T_{2},\ldots,T_{d}$ are commuting contractions on a Hilbert space
$\mathcal{H}$ and $P$ denotes the product $T_{1}T_{2}\ldots T_{d}$, then an
application of the arguments in [11] shows that there is a non-trivial bounded
operator $A$ on $\mathcal{H}$ satisfying $T_{i}^{*}AT_{i}=A$ for all
$i=1,2,\ldots,d$ if and only if $P^{n}$ does not converge to the zero operator
strongly. This may remind an astute reader of Corollary 2.2 in Prunaru ([15])
where he showed that a completely positive, completely contractive and
ultraweakly continuous linear map on $\mathcal{B}(\mathcal{H})$ has a non-zero
operator as a fixed point if and only if its powers at the identity operator
converge strongly to the zero operator. Prunaru’s results were attuned to the
Euclidean unit ball whereas the case of commuting contractions is attuned to
the polydisc. In this note, we discuss a third domain which has gained
prominence over the last two decades.

Consider the elementary symmetric polynomials $e_{k}$ for $k=1,2,\ldots,d$ in
$d$ variables:

$e_{k}(z_{1},z_{2},\ldots,z_{d})=\sum_{1\leq j_{1}<j_{2}<\cdots<j_{k}\leq
d}z_{j_{1}}z_{j_{2}}\ldots z_{j_{k}}.$

By convention, $e_{0}$ is the constant polynomial $1$. The closed symmetrized
polydisc is the polynomially convex set

$\Gamma_{d}=\\{(e_{1}(\bm{z}),e_{2}(\bm{z}),\ldots,e_{d}(\bm{z}):\bm{z}=(z_{1},z_{2},\ldots,z_{d})\in\overline{\mathbb{D}}^{d}\\}.$

A point of $\Gamma_{d}$ will usually be denoted by $(s_{1},s_{2},\ldots,p)$.
The number $d\geq 2$ will be fixed for this paper.

Let $\mathsf{S}=(S_{1},\dots,S_{d-1},P)$ be a commuting $d$-tuple of bounded
operators on a Hilbert space $\mathcal{H}$. Let
$\mathcal{A}=\mathbb{C}[z_{1},z_{2},\ldots,z_{d}]$ be the algebra of
polynomials in $d$ commuting variables. Consider the $\mathcal{A}$-module
structure induced on $\mathcal{H}$ as follows:

$f\cdot h=f(S_{1},\dots,S_{d-1},P)h\text{ for }f\in\mathcal{A}\text{ and
}h\in\mathcal{H}.$

Let $\|f\|$ be the supremum norm of $f$ over $\Gamma_{d}$. The
$\mathcal{A}$-module $\mathcal{H}$ is called a contractive Hilbert module if

$\|f(S_{1},\dots,S_{d-1},P)h\|\leq\|f\|\|h\|\text{ for }f\in\mathcal{A}\text{
and }h\in\mathcal{H}.$ (1.1)

The polynomial convexity of $\Gamma_{d}$ implies that (1.1) holds for every
function holomorphic in a neighbourhood of $\Gamma_{d}$. We could call such an
$\mathcal{H}$ a $\Gamma_{d}$-contractive Hilbert module because $\Gamma_{d}$
is a spectral set for $\mathsf{S}$. However, in this note, the set
$\Gamma_{d}$ is fixed. Hence, the brevity. The commuting tuple $\mathsf{S}$
satisfying (1.1) is also known as a $\Gamma_{d}$-contraction, see [4]. We
shall write a Hilbert module as above as $({\mathcal{H}},\mathsf{S})$ because
we shall often need the tuple $\mathsf{S}$. For an account of why the language
of Hilbert modules is a natural one for non-trivial extension of operator
theory results to several variables, see [8]. In the present context, see
[16].

If $(\mathcal{K},\mathsf{R})$ is a contractive module as above where
$\mathsf{R}=(R_{1},\ldots,R_{d-1},U)$ consists of normal operators and the
Taylor joint spectrum $\sigma(\mathsf{R})$ is contained in the distinguished
boundary $b\Gamma_{d}=\\{(s_{1},\ldots,s_{d-1},p):|p|=1\\}$ (see Theorem 2.4
(iii) in [4]), $(\mathcal{K},\mathsf{R})$ will be called a unitary Hilbert
module. Let $(\mathcal{M},\mathsf{T})$ with
$\mathsf{T}=(T_{1},\ldots,T_{d-1},V)$ be a submodule of a unitary Hilbert
module $(\mathcal{K},\mathsf{R})$. Then it is called an isometric Hilbert
module. The operator tuples $\mathsf{R}$ and $\mathsf{T}$ are called a
$\Gamma_{d}$-unitary and a $\Gamma_{d}$-isometry respectively.

If $\mathsf{S}^{*}\stackrel{{\scriptstyle\rm
def}}{{=}}(S_{1}^{*},\dots,S_{d-1}^{*},P^{*})$, then the adjoint module
$({\mathcal{H}},\mathsf{S}^{*})$ is a contractive module when
$({\mathcal{H}},\mathsf{S})$ is so. This is easy to see from the definition.
Clearly, the adjoint module of a unitary module is again a unitary module.
Obviously, the adjoint of an isometric module need not be an isometric module.

Given two Hilbert modules $({\mathcal{H}},\mathsf{S})$ and
$({\mathcal{K}},\mathsf{R})$, a Hilbert module homomorphism
$\mathfrak{J}:{\mathcal{H}}\to{\mathcal{K}}$ is a bounded operator
$\mathfrak{J}$ such that

$\mathfrak{J}(f\cdot h)=f\cdot\mathfrak{J}h\text{ for all
}f\in\mathcal{A}\text{ and }h\in\mathcal{H}.$

Such a homomorphism will often be called a module map. If, moreover,
$\mathfrak{J}$ is a contraction, we say that the module
$({\mathcal{H}},\mathsf{S})$ is contractively embedded as a submodule of
$({\mathcal{K}},\mathsf{R})$. A contractive module map
$\mathfrak{J}:{\mathcal{H}}\to{\mathcal{K}}$ is called a canonical module map
if $({\mathcal{K}},\mathsf{R})$ is minimal in the sense that there is no
submodule of $({\mathcal{K}},\mathsf{R})$ containing
$({\mathcal{H}},\mathsf{S})$ and reducing $\mathsf{R}$ and

$\displaystyle\mathfrak{J}^{*}\mathfrak{J}=\operatorname{SOT-}\lim
P^{*n}P^{n}.$ (1.2)

######  Definition 1.1.

Let $(\mathcal{H},\mathsf{S})$ be a contractive Hilbert module. A bounded
operator $A$ on $\mathcal{H}$ is said to be an $\mathsf{S}$-Toeplitz operator
if it satisfies the Brown-Halmos relations with respect to $\mathsf{S}$:

$\displaystyle S_{i}^{*}AP=AS_{d-i}\text{ for each $1\leq i\leq d-1$ and
}P^{*}AP=P.$ (1.3)

The $*$–closed and norm closed vector space of all $\mathsf{S}$-Toeplitz
operators is denoted by $\mathcal{T}(\mathsf{S})$.

See [2] and [6] for the motivation of the definition above.

######  Theorem 1.

Let $({\mathcal{H}},\mathsf{S})$ be a contractive Hilbert module. Then the
following are equivalent.

  1. (1)

$\mathcal{T}(\mathsf{S})$ is non-trivial.

  2. (2)

$Q=\operatorname{SOT-}\lim P^{*n}P^{n}\neq 0$.

  3. (3)

$(\mathcal{H},\mathsf{S})$ can be canonically embedded as a submodule of a
unitary module $(\mathcal{K},\mathsf{R})$, which is unique up to unitary
isomorphism.

  4. (4)

$({\mathcal{H}},\mathsf{S})$ can be contractively embedded as a submodule of
an isometric Hilbert module.

Moreover, in this case, for any unitary module
$({\mathcal{K}}^{\prime},\mathsf{R}^{\prime})$ and any contractive embedding
$\mathfrak{J}^{\prime}:{\mathcal{H}}\rightarrow{\mathcal{K}}^{\prime}$, the
following are true.

  1. (i)

$\mathfrak{J}^{\prime*}\mathfrak{J}^{\prime}\leq\operatorname{SOT-}\lim
P^{*n}P^{n}=\mathfrak{J}^{*}\mathfrak{J}$,

  2. (ii)

$({\mathcal{K}},\mathsf{R})$ is contractively embedded into
$({\mathcal{K}}^{\prime},\mathsf{R}^{\prime})$ through a contraction
$\mathfrak{T}:{\mathcal{K}}\to{\mathcal{K}}^{\prime}$ such that
$\mathfrak{T}\mathfrak{J}=\mathfrak{J}^{\prime}$.

This theorem is proved in section 2. In Theorem 1 above,
$\mathcal{T}(\mathsf{S})$ depends on the $S_{i}$ and $P$ whereas the condition
(2) is in terms of $P$ alone. That is the surprising strength.

The Toeplitz $C^{*}$-algebra, denoted by
$C^{*}(I_{{\mathcal{H}}},{\mathcal{T}}(\mathsf{S}))$, is the $C^{*}$-algebra
generated by $I_{{\mathcal{H}}}$ and the vector space
${\mathcal{T}}(\mathsf{S})$ of $\mathsf{S}$-Toeplitz operators.

######  Theorem 2.

Let $({\mathcal{H}},\mathsf{S})$ be a contractive Hilbert module satisfying

$Q=\operatorname{SOT-}\lim P^{*n}P^{n}\neq 0.$

Then $({\mathcal{H}},\mathsf{S})$ can be canonically embedded as a submodule
of a unitary module $({\mathcal{K}},\mathsf{R})$ by a module map
$\mathfrak{J}$ such that

  1. (1)

the map $\rho$ defined on the commutant algebra
$\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ by

$\rho(Y)=\mathfrak{J}^{*}Y\mathfrak{J},$

is a complete isometry onto $\mathcal{T}(\mathsf{S})$;

  2. (2)

there exists a surjective unital $*$-representation

$\pi:\mathcal{C}^{*}\\{I_{\mathcal{H}},\mathcal{T}(\mathsf{S})\\}\to\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$

such that $\pi\circ\rho=I;$

  3. (3)

there exists a completely contractive, unital and multiplicative mapping

$\Theta:\\{S_{1},\dots,S_{d-1},P\\}^{\prime}\to\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$

defined by $\Theta(X)=\pi(\mathfrak{J}^{*}\mathfrak{J}X)$ which satisfies

$\Theta(X)\mathfrak{J}=\mathfrak{J}X.$

When the contractive module above is an isometric one, the following stronger
version holds.

######  Theorem 3.

Let $(\mathcal{H},\mathsf{S})$ be an isometric Hilbert module.

  1. (1)

There exists a unitary module $(\mathcal{K},\mathsf{R})$ containing
$(\mathcal{H},\mathsf{S})$ as an isometrically embedded submodule such that
$\mathsf{R}$ is the minimal extension of $\mathsf{S}$. In fact, $\mathcal{K}$
is the span closure of the following elements:

$\\{U^{m}h:h\in\mathcal{H},\text{ and }m\in\mathbb{Z}\\}.$

Moreover, any operator $X$ acting on $\mathcal{H}$ commutes with $\mathsf{S}$
if and only if $X$ has a unique norm preserving extension $Y$ acting on
$\mathcal{K}$ commuting with $\mathsf{R}$.

  2. (2)

An operator $X$ is in $\mathcal{T}(\mathsf{S})$ if and only if there exists a
unique operator $Y$ in the commutant of the von-Neumann algebra generated by
$\\{R_{1},\dots,R_{d-1},U\\}$ such that $\|X\|=\|Y\|$ and
$X=P_{\mathcal{H}}Y|_{\mathcal{H}}.$

  3. (3)

Let $\mathcal{C}^{*}(\mathsf{S})$ and $\mathcal{C}^{*}(\mathsf{R})$ denote the
unital $\mathcal{C}^{*}$-algebras generated by $\mathsf{S}$ and $\mathsf{R}$
respectively and $\mathcal{I}(\mathsf{S})$ denote the closed ideal of
$\mathcal{C}^{*}(\mathsf{S})$ generated by all the commutators $XY-YX$ for
$X,Y\in\mathcal{C}^{*}(\mathsf{S})\cap\mathcal{T}(\mathsf{S})$. Then there
exists a short exact sequence

$0\rightarrow\mathcal{I}(\mathsf{S})\hookrightarrow\mathcal{C}^{*}(\mathsf{S})\xrightarrow{\pi_{0}}\mathcal{C}^{*}(\mathsf{R})\rightarrow
0$

with a completely isometric cross section, where
$\pi_{0}:\mathcal{C}^{*}(\mathsf{S})\to\mathcal{C}^{*}(\mathsf{R})$ is the
canonical unital $*$-homomorphism which sends the generating set $\mathsf{S}$
to the corresponding generating set $\mathsf{R}$, i.e., $\pi_{0}(P)=U$ and
$\pi_{0}(S_{i})=R_{i}$ for all $1\leq i\leq d-1$.

The proof of this is in Section 3. As an application of the above, we obtain a
version of the commutant lifting theorem in Section 4.

######  Theorem 4.

Let $({\mathcal{H}},\mathsf{S})$ be a contractive Hilbert module. Let
$\mathfrak{J}:{\mathcal{H}}\to{\mathcal{K}}$ be a canonical embedding of
$({\mathcal{H}},\mathsf{S})$ as a submodule of a unitary Hilbert module
$({\mathcal{K}},\mathsf{R})$. Let $X$ be in the commutant of $\mathsf{S}$.
Then the module $({\mathcal{H}},X)$ can be contractively embedded as a
submodule of $({\mathcal{K}},Y)$ for an $Y$ in the commutant of $\mathsf{R}$
and $\|Y\|\leq\|X\|$.

##  2\. Proof of Theorem 1 and relation to quotient modules

Fix a contractive Hilbert module $({\mathcal{H}},\mathsf{S})$ or, in other
words, a $\Gamma_{d}$-contraction $\mathsf{S}=(S_{1},\dots,S_{d-1},P)$. It is
well-known that

$S_{i}-S_{d-i}^{*}P=D_{P}F_{i}D_{P};\;\;i=1,2,\ldots,d-1$

for a certain operator tuple $(F_{1},F_{2},\ldots,F_{d-1})$. The operators
$F_{i}$ are called the fundamental operators. The existence was discovered for
$d=2$ in [3]. For a proof for $d>2$, see [13] or [12]. Alternatively, it is
easily deducible from Proposition 2.5(3) of [14].

######  Lemma 2.1.

For each $i=1,2,\dots d-1$, we have

$\displaystyle{P^{*}}^{j}(S_{d-i}-S_{i}^{*}P)P^{j}\to 0\quad\text{strongly
as}\quad j\to\infty.$ (2.1)

###### Proof.

For every $h\in\mathcal{H}$, we have

$\displaystyle\|P^{*j}(S_{d-i}-S_{i}^{*}P)P^{j}h\|^{2}$ $\displaystyle=$
$\displaystyle\|P^{*j}(D_{P}F_{d-i}D_{P})P^{j}h\|^{2}$ $\displaystyle\leq$
$\displaystyle\|F_{d-i}\|^{2}\|D_{P}P^{j}h\|^{2}=\|F_{d-i}\|^{2}(\|P^{j}h\|^{2}-\|P^{j+1}h\|^{2}).$

The proof follows because the last term in bracket converges to zero as
$j\to\infty$. ∎

To prove Theorem 1, we shall take the path
$(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)\Rightarrow(1)$.

###### Proof of $(1)\Rightarrow(2)$:.

Let there be a non-zero $\mathsf{S}$-Toeplitz operator $A$. This, in
particular, implies that for all $n\geq 0$ we have $A=P^{*n}AP^{n}$ and hence
$\|Ah\|\leq\|A\|\|P^{n}h\|$ for every vector $h$. So, if $P^{n}$ strongly
converges to $0$, then $A=0$ which is a contradiction.

Proof of $(2)\Rightarrow(3)$: Assume $P^{n}\nrightarrow 0$ strongly. As $P$ is
a contraction

$I_{\mathcal{H}}\succeq P^{*}P\succeq
P^{*2}P^{2}\succeq\cdots\succeq{P^{*}}^{n}P^{n}\succeq\cdots\succeq 0.$

This guarantees a positive non-zero contraction $Q$ such that

$\displaystyle Q=\operatorname{SOT-}\lim P^{*n}P^{n}.$ (2.2)

Clearly,

$P^{*}QP=Q.$

Hence we can define an isometry
$V:\overline{\operatorname{Ran}}Q\rightarrow\overline{\operatorname{Ran}}Q$
satisfying

$\displaystyle V:Q^{\frac{1}{2}}h\mapsto Q^{\frac{1}{2}}Ph\text{ for each
}h\in{\mathcal{H}}.$ (2.3)

We now prove that $Q$ is an $\mathsf{S}$-Toeplitz operator. Indeed,

$\displaystyle S_{i}^{*}QP-
QS_{d-i}=\lim_{j}(S_{i}^{*}P^{*j}P^{j}P-P^{*j}P^{j}S_{d-i})=\lim_{j}P^{*j}(S_{i}^{*}P-S_{d-i})P^{j}.$

By (2.1), the limit above is zero and hence $S_{i}^{*}QP=QS_{d-i}$. Define
operators
$T_{j}:\overline{\operatorname{Ran}}Q\rightarrow\overline{\operatorname{Ran}}Q$
for $j=1,2,\dots,d-1$ as

$\displaystyle T_{j}:Q^{\frac{1}{2}}h\mapsto Q^{\frac{1}{2}}S_{j}h.$ (2.4)

It is straightforward to see that each $T_{j}$ is well-defined. The tuple
$(T_{1},\ldots,T_{d-1},V)$ is a commuting tuple too. The computation below
establishes the identities $T_{i}=T_{d-i}^{*}V$ for each $i=1,2,\dots,d-1$:

$\displaystyle\langle
T_{d-i}^{*}VQ^{\frac{1}{2}}h,Q^{\frac{1}{2}}h^{\prime}\rangle=\langle
Q^{\frac{1}{2}}Ph,Q^{\frac{1}{2}}S_{d-i}h^{\prime}\rangle=\langle
QS_{i}h,h^{\prime}\rangle=\langle
T_{i}Q^{\frac{1}{2}}h,Q^{\frac{1}{2}}h^{\prime}\rangle,$

where to obtain the third equality, we use the fact that $Q$ is an
$\mathsf{S}$-Toeplitz operator. We have already noted that $V$ is an isometry.
Thus by Theorem 4.12 of [4], all we need to show to conclude that
$(T_{1},\dots,T_{d-1},V)$ is a $\Gamma_{d}$-isometry is that the $d-1$ tuple
$(\gamma_{1}T_{1},\dots,\gamma_{d-1}T_{d-1})$ is a $\Gamma_{d-1}$-contraction,
where $\gamma_{j}=(d-j)/d$ for each $j=1,2,\dots,d-1$. This readily follows
from the identity

$\xi(\gamma_{1}T_{1},\dots,\gamma_{d-1}T_{d-1})Q^{\frac{1}{2}}=\xi(\gamma_{1}S_{1},\dots,\gamma_{d-1}S_{d-1})Q^{\frac{1}{2}}$

for every polynomial $\xi$ in ${\mathbb{C}}[z_{1},z_{2},\dots,z_{d-1}]$, and
the fact that $(\gamma_{1}S_{1},\dots,\gamma_{d-1}S_{d-1})$ is a
$\Gamma_{d-1}$-contraction where $\gamma_{j}=(d-j)/d$.

Let $\mathsf{R}=(R_{1},\dots,R_{d-1},U)$ acting on ${\mathcal{K}}$ be a
minimal $\Gamma_{d}$-unitary extension of the $\Gamma_{d}$-isometry
$\mathsf{T}=(T_{1},\dots,T_{d-1},V)$. Define a contraction
$\mathfrak{J}:{\mathcal{H}}\to{\mathcal{K}}$ as

$\displaystyle\mathfrak{J}:h\mapsto Q^{\frac{1}{2}}h\text{ for every
}h\in{\mathcal{H}}.$

This is a module homomorphism because

$\displaystyle
R_{j}\mathfrak{J}h=R_{j}Q^{\frac{1}{2}}h=T_{j}Q^{\frac{1}{2}}h=Q^{\frac{1}{2}}S_{j}h=\mathfrak{J}S_{j}h.$
(2.5)

Similarly, $U\mathfrak{J}=\mathfrak{J}P$. Finally, by definition of
$\mathfrak{J}$ and $Q$, it follows that $\mathfrak{J}^{*}\mathfrak{J}$ is the
limit of $P^{*n}P^{n}$ in the strong operator topology.

For the uniqueness part, consider

$(\mathfrak{J},{\mathcal{K}},\mathsf{R})=(R_{1},\dots,R_{d-1},U)$

as above and

$(\mathfrak{J}^{\prime},{\mathcal{K}}^{\prime},{\mathsf{R}^{\prime}})=(R_{1}^{\prime},\dots,R_{d-1}^{\prime},U^{\prime})$

with similar properties. Define the operator
$\tau:\mathcal{K}\to\mathcal{K}^{\prime}$ densely by

$\tau:f(\mathsf{R},\mathsf{R}^{*})\mathfrak{J}h\mapsto
f(\mathsf{R}^{\prime},\mathsf{R}^{\prime*})\mathfrak{J}h$

for every $h\in\mathcal{H}$ and polynomial $f$ in $\bm{z}$ and
$\overline{\bm{z}}$. The map $\tau$ is surjective by minimality. Note that
$\tau$ clearly satisfies $\tau\mathfrak{J}=\mathfrak{J}^{\prime}$. We will be
done if we can show that $\tau$ is an isometry. Let $f$ be a polynomial in
$\bm{z}$ and $\overline{\bm{z}}$ and $\bar{f}f=\sum
a_{\bm{n},\bm{m}}\bm{z}^{\bm{n}}\overline{\bm{z}}^{\bm{m}}$. Then for every
$h\in{\mathcal{H}}$,

$\displaystyle\|f(\mathsf{R},\mathsf{R}^{*})\mathfrak{J}h\|^{2}$
$\displaystyle=\sum
a_{\bm{n},\bm{m}}\langle\mathfrak{J}^{*}\mathsf{R}^{*\bm{m}}\mathsf{R}^{\bm{n}}\mathfrak{J}h,h\rangle$
$\displaystyle=\sum
a_{\bm{n},\bm{m}}\langle\mathsf{S}^{*\bm{m}}\mathfrak{J}^{*}\mathfrak{J}\mathsf{S}^{\bm{n}}h,h\rangle$
$\displaystyle=\sum
a_{\bm{n},\bm{m}}\langle\mathsf{S}^{*\bm{m}}Q\mathsf{S}^{\bm{n}}h,h\rangle.$
(2.6)

Since the last term depends on $\mathsf{S}$ only, $\tau$ is an isometry.

Proof of $(3)\Rightarrow(4)$: Obvious.

Proof of $(4)\Rightarrow(1)$: Let $(\mathcal{M},\mathsf{T})$ be the isometric
module with $\mathsf{T}=(T_{1},\ldots,T_{d-1},V)$. In this case, we have

$P^{*}\mathfrak{J}^{*}\mathfrak{J}P=\mathfrak{J}^{*}V^{*}V\mathfrak{J}=\mathfrak{J}^{*}\mathfrak{J}$

and

$S_{d-j}^{*}\mathfrak{J}^{*}\mathfrak{J}P=\mathfrak{J}^{*}T_{d-j}^{*}V\mathfrak{J}=\mathfrak{J}^{*}T_{j}\mathfrak{J}=\mathfrak{J}^{*}\mathfrak{J}S_{j}.$

This proves that the non-zero operator $\mathfrak{J}^{*}\mathfrak{J}$ belongs
to $\mathcal{T}(\mathsf{S})$. This in particular establishes that (4) implies
(1).

To complete the proof, first note that the proof of $(3)\Rightarrow(1)$ above
implies that $\mathfrak{J}^{\prime*}\mathfrak{J}^{\prime}$ is an
$\mathsf{S}$-Toeplitz operator. In particular,
$\mathfrak{J}^{\prime*}\mathfrak{J}^{\prime}$ is in ${\mathcal{T}}(P)$. This
implies

$\displaystyle\mathfrak{J}^{\prime*}\mathfrak{J}^{\prime}=P^{*n}\mathfrak{J}^{\prime*}\mathfrak{J}^{\prime}P^{n}\leq
P^{*n}P^{n}\text{ for every }n.$

This proves part (i).

For part (ii) we define the operator
$\mathfrak{T}:\mathcal{K}\to{\mathcal{K}}^{\prime}$ densely by

$\mathfrak{T}:f(\mathsf{R},\mathsf{R}^{*})\mathfrak{J}h\mapsto
f(\mathsf{R}^{\prime},\mathsf{R}^{\prime*})\mathfrak{J}^{\prime}h$

for every $h\in\mathcal{H}$ and polynomial $f$ in $\bm{z}$ and
$\overline{\bm{z}}$. Using part (i), a similar computation as done in (2)
yields

$\displaystyle\|f(\mathsf{R}^{\prime},\mathsf{R}^{\prime*})\mathfrak{J}^{\prime}h\|\leq\|f(\mathsf{R},\mathsf{R}^{*})\mathfrak{J}h\|\text{
for every }h\in{\mathcal{H}}.$

This shows that $\mathfrak{T}$ is not only well-defined but also a
contraction. Finally, it readily follows from the definition of $\mathfrak{T}$
that it intertwines $\mathsf{R}$ and $\mathsf{R}^{\prime}$ and that
$\mathfrak{T}\mathfrak{J}=\mathfrak{J}^{\prime}$. ∎

######  Remark 2.2.

It is known that the minimal unitary (or isometric) dilation space of a
contraction acting on a Hilbert space is always infinite dimensional even in
the case of matrices. In contrast, if ${\mathcal{H}}$ is finite dimensional,
the isometric module $(T_{1},\dots,T_{d-1},V)$ defined in (2.4) acts on a
finite dimensional space, viz., $\overline{\operatorname{Ran}}Q$ and hence is
a $\Gamma_{d}$-unitary.

Biswas and Shyam Roy showed in [4] that an isometric Hilbert module
$(\mathcal{M},\mathsf{T})$ always decomposes (up to unitary equivalence) as
the direct sum of a pure isometric Hilbert module and a unitary Hilber module,
i.e., $\mathcal{M}=H^{2}({\mathcal{E}})\oplus{\mathcal{K}}$ where
${\mathcal{E}}$ is a Hilbert space, $T_{i}=M_{A_{i}+A_{n-i}^{*}z}\oplus R_{i}$
for $i=1,2,\ldots,d-1$ and $V=M_{z}\oplus U$ for certain
$A_{i}\in{\mathcal{B}}({\mathcal{E}})$ and $({\mathcal{K}},\mathsf{R})$ is a
unitary Hilbert module, called the unitary part of $(\mathcal{M},\mathsf{T})$.

If a Hilbert module $({\mathcal{H}},\mathsf{S})$ is isomorphic to
$(\mathcal{M},\mathsf{T})$ quotiented by a submodule
$(\mathcal{M}^{\prime},\mathsf{T}|_{\mathcal{M}^{\prime}})$, then
$({\mathcal{H}},\mathsf{S})$ is said to be realized as a quotient module or
equivalently $({\mathcal{H}},\mathsf{S})$ is said to have a co-extension
$(\mathcal{M},\mathsf{T})$ because there is a co-isometric module map
$L:\mathcal{M}\rightarrow{\mathcal{H}}$; see [7]. The module
$(\mathcal{M},\mathsf{T})$ is said to be minimal over
$({\mathcal{H}},\mathsf{S})$ if there is no proper submodule of
$(\mathcal{M},\mathsf{T})$ which contains $({\mathcal{H}},\mathsf{S})$ and
reduces $\mathsf{T}$.

######  Lemma 2.3.

If a contractive Hilbert module $({\mathcal{H}},\mathsf{S})$ can be realized
as a quotient module of a minimal isometric module $(\mathcal{M},\mathsf{T})$
which has a non-trivial unitary part $({\mathcal{K}},\mathsf{R})$, then there
is a (not necessarily canonical) contractive embedding of the adjoint module
$({\mathcal{H}},\mathsf{S}^{*})$ in $({\mathcal{K}},\mathsf{R})$.

###### Proof.

The proof consists of defining $\mathfrak{J}:\mathcal{H}\to{\mathcal{K}}$ as

$\mathfrak{J}:h\to P_{{\mathcal{K}}}h,\quad(h\in\mathcal{H})$

where $P_{\mathcal{K}}$ denotes the orthogonal projection of ${\mathcal{M}}$
onto ${\mathcal{K}}$. ∎

Lemma 2.3 has an interesting, but not a very unexpected, consequence whose
proof is easy and hence omitted.

######  Corollary 2.4.

If a pure contractive Hilbert module $({\mathcal{H}},\mathsf{S})$ can be
realized as a quotient module of an isometric module
$(\mathcal{M},\mathsf{T})$ which is minimal over $({\mathcal{H}},\mathsf{S})$,
then the unitary part in the Biswas - Shyam Roy decomposition of
$(\mathcal{M},\mathsf{T})$ is absent.

##  3\. Algebraic structure of the Toeplitz $C^{*}$-algebra

###  3.1. Proof of Theorem 2.

We begin with a few lemmas. The first lemma below gives us the existence of an
important completely positive map. This is a particular case of Lemma 2.1 in
[15]. The central idea of the proof goes back to Arveson, see Proposition 5.2
in [1]. For a subnormal operator tuple, in the multivariable situation,
Eschmeier and Everard have proven a similar result by direct construction, see
Section 3 of [9].

######  Lemma 3.1.

Let $P$ be a contraction on the Hilbert space $\mathcal{H}$. Then there exists
a completely positive, completely contractive, idempotent linear map
$\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ such that
$\operatorname{Ran}\Phi=\mathcal{T}(P)$. Moreover, if
$A,B\in\mathcal{B}(\mathcal{H})$ satisfy $P^{*}(AXB)P=AP^{*}XPB$ for all
$X\in\mathcal{B}(\mathcal{H})$ then $\Phi(AXB)=A\Phi(X)B$. In addition,

$\Phi(I_{\mathcal{H}})=Q=\lim_{n\to\infty}P^{*n}P^{n}$

where the limit is in the strong operator topology.

Part of Theorem 3.1 of [5] gives us more properties of $\Phi$.

######  Lemma 3.2 (Choi and Effros).

Let $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ be a completely
positive and completely contractive map such that $\Phi\circ\Phi=\Phi$. Then
for all $X$ and $Y$ in $\mathcal{B}(\mathcal{H})$ we have

$\displaystyle\Phi(\Phi(X)Y)=\Phi(X\Phi(Y))=\Phi(\Phi(X)\Phi(Y)).$ (3.1)

The last lemma that we need will play a crucial role. Its proof follows from
Theorem 3.1 in [15]. For someone not familiar with [15], this might create
some opacity and hence the simplified proof in our context is supplied below.

######  Lemma 3.3.

There is an idempotent, completely positive and completely contractive map
$\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ such that

$\displaystyle\operatorname{Ran}\Phi=\\{X\in\mathcal{B}(\mathcal{H}):P^{*}XP=X\\}=\mathcal{T}(P).$
(3.2)

If $(\mathcal{K},\pi,\mathfrak{J})$ is the minimal Stinespring dilation of the
restriction of $\Phi$ to the unital $C^{*}$-algebra
$C^{*}(I_{\mathcal{H}},\mathcal{T}(P))$ generated by $\mathcal{T}(P)$, and if
$Q=$ SOT-$\lim P^{*n}P^{n}$, then the following properties are satisfied.

(${\bf{P_{1}}}$) $U:=\pi(QP)$ is a unitary operator. Moreover,
$\mathfrak{J}P=U\mathfrak{J}$ and $\mathcal{K}$ is the smallest reducing
subspace for $U$ containing $\mathfrak{J}\mathcal{H}$.

(${\bf{P_{2}}}$) The map $\rho:\\{U\\}^{\prime}\to\mathcal{T}(P)$ defined by
$\rho(Y)=\mathfrak{J}^{*}Y\mathfrak{J}$, for all $Y\in\\{U\\}^{\prime}$, is
surjective and a complete isometry.

(${\bf{P_{3}}}$) The Stinespring triple $({\mathcal{K}},\pi,\mathfrak{J})$
satisfies $\pi\circ\rho=I$. In particular,

$\pi(C^{*}(I_{\mathcal{H}},\mathcal{T}(P)))=\\{U\\}^{\prime}.$

(${\bf{P_{4}}}$) The linear map $\Theta:\\{P\\}^{\prime}\to\\{U\\}^{\prime}$
defined by $\Theta(X)=\pi(QX)$ is completely contractive, unital and
multiplicative.

###### Proof.

We restrict $\Phi$ to $\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))$ and
continue to call it $\Phi$.

By definition of minimal Stinepring dilation,

$\Phi(X)=\mathfrak{J}^{*}\pi(X)\mathfrak{J}\text{ for every
$X\in\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))$}.$ (3.3)

Note that
$Q=\Phi(I_{\mathcal{H}})=\mathfrak{J}^{*}\mathfrak{J}=\operatorname{SOT-}\lim_{n\rightarrow\infty}P^{*n}P^{n}.$

The kernel of $\Phi$ is an ideal in $C^{*}(I_{\mathcal{H}},\mathcal{T}(P))$ by
Lemma 3.2 (when $\Phi$ is allowed as a map on whole of
$\mathcal{B}(\mathcal{H})$, its kernel may not be an ideal) and hence it
follows from the construction of the minimal Stinespring dilation that
$\text{Ker }\\!\Phi=\text{Ker }\\!\pi$. Thus

$\pi(X)=\pi(\Phi(X))\text{ for any }X\in C^{*}(I,\mathcal{T}(P)).$ (3.4)

Since $\pi$ is a representation,we get $U^{*}\pi(X)U=\pi(X)\text{ for any
}X\in C^{*}(I,\mathcal{T}(P)).$

Since $\pi$ is unital, we get that $U$ is an isometry. If $P^{\prime}$ is a
projection in the weak* closure of $\pi(C^{*}(I,\mathcal{T}(P)))$, then we
also have $U^{*}P^{\prime}U=P^{\prime}$ and
$U^{*}P^{\prime\perp}U=P^{\prime\perp}$. This shows that
$UP^{\prime}=P^{\prime}U$ and therefore $\pi(X)U=U\pi(X)$ for all $X\in
C^{*}(I,\mathcal{T}(P))$. In particular, it follows that $U$ is a unitary and
$\pi(C^{*}(I_{\mathcal{H}},\mathcal{T}(P)))\subseteq\\{U\\}^{\prime}.$

We now prove an identity which will be used in this proof as well as later.
The identity is

$\pi(QX)\mathfrak{J}=\mathfrak{J}X$ (3.5)

for any $X\in\mathcal{B}(\mathcal{H})$ that commutes with $P$. The proof of
(3.5) follows from two computations. For every $h,h^{\prime}\in\mathcal{H}$,
we have

$\displaystyle\langle\pi(QX)\mathfrak{J}h,\mathfrak{J}h^{\prime}\rangle$
$\displaystyle=\langle\mathfrak{J}^{*}\pi(QX)\mathfrak{J}h,k\rangle$
$\displaystyle=\langle\Phi(QX)h,h^{\prime}\rangle$ $\displaystyle=\langle
QXh,h^{\prime}\rangle\quad[\text{because }\mathcal{T}(P)\text{ is fixed by
}\Phi]$ $\displaystyle=\langle\mathfrak{J}Xh,\mathfrak{J}h^{\prime}\rangle$

showing that
$P_{\overline{\operatorname{Ran}}\mathfrak{J}}\pi(QX)\mathfrak{J}=\mathfrak{J}X$.
On the other hand,

$\displaystyle\|\pi(QX)\mathfrak{J}h\|^{2}$
$\displaystyle=\langle\mathfrak{J}^{*}\pi(X^{*}Q^{2}X)\mathfrak{J}h,h\rangle$
$\displaystyle=\langle\Phi(X^{*}Q^{2}X)h,h\rangle$ $\displaystyle=\langle
X^{*}\Phi(Q^{2})Xh,h\rangle\quad[\text{by Lemma \ref{L:PJFA}}]$
$\displaystyle=\langle X^{*}QXh,h\rangle\quad[\text{by Lemma \ref{L:CE}}]$
$\displaystyle=\|\mathfrak{J}Xh\|^{2}.$

Hence, (3.5) is proved.

A trivial consequence of (3.5) is that $U\mathfrak{J}=\mathfrak{J}P$. To
complete the proof of ${\bf{P_{1}}}$, it is required to establish that
$\mathcal{K}$ is the smallest reducing subspace for $U$ containing
$\mathfrak{J}\mathcal{H}$. To that end, we consider a map $\delta$ from
$\operatorname{Ran}\pi$ into $\mathcal{T}(P)$ given by

$\delta(\pi(X))=\mathfrak{J}^{*}\pi(X)\mathfrak{J}=\Phi(X)\text{ for all }X\in
C^{*}(I,\mathcal{T}(P)).$

It is injective because $\text{Ker }\\!\Phi=\text{Ker }\\!\pi$.

Since $\delta\circ\pi=\Phi$, we have $\delta\circ\pi$ to be idempotent and
this coupled with the injectivity of $\delta$ gives us $\pi\circ\delta=I$ on
$\pi\\{C^{*}(I,\mathcal{T}(P))\\}$. This immediately implies that $\delta$ is
a complete isometry.

Let $\mathcal{K}_{0}\subseteq\mathcal{K}$ be the smallest reducing subspace
for $U$ containing $\mathfrak{J}\mathcal{H}$. Let $P_{\mathcal{K}_{0}}$ be the
projection in $\mathcal{B}(\mathcal{K})$ onto the space $\mathcal{K}_{0}$.
Consider the vector space

$P_{\mathcal{K}_{0}}\\{U\\}^{\prime}P_{\mathcal{K}_{0}}:=\\{P_{\mathcal{K}_{0}}XP_{\mathcal{K}_{0}}:X\in\\{U\\}^{\prime}\\}=\\{P_{\mathcal{K}_{0}}X|_{\mathcal{K}_{0}}\oplus
0_{\mathcal{K}_{0}^{\perp}}:X\in\\{U\\}^{\prime}\\}.$

and the map
$\delta^{\prime}:P_{\mathcal{K}_{0}}\\{U\\}^{\prime}P_{\mathcal{K}_{0}}\to\mathcal{T}(P)\subseteq\mathcal{B}(\mathcal{H})$
defined by $X\mapsto\mathfrak{J}^{*}X\mathfrak{J}$. This is injective.

Indeed, it is easy to check that
$\mathfrak{J}^{*}X\mathfrak{J}\in\mathcal{T}(P)$ for $X\in\\{U\\}^{\prime}$.
Now if $\mathfrak{J}^{*}X\mathfrak{J}=0$ for some $X\in\\{U\\}^{\prime}$ then
using the identity $\mathfrak{J}P=U\mathfrak{J}$, we get that

$\langle Xf(U,U^{*})\mathfrak{J}h,g(U,U^{*})\mathfrak{J}k\rangle=0$

for any two variable polynomials $f$ and $g$ and $h,k\in\mathcal{H}$. This
shows that $P_{\mathcal{K}_{0}}XP_{\mathcal{K}_{0}}=0$ and therefore,
$\delta^{\prime}$ is injective. For any $Y\in
P_{\mathcal{K}_{0}}\\{U\\}^{\prime}P_{\mathcal{K}_{0}}$,

$\displaystyle\delta^{\prime}(P_{\mathcal{K}_{0}}\pi(\mathfrak{J}^{*}Y\mathfrak{J})P_{\mathcal{K}_{0}}-Y)=\mathfrak{J}^{*}\pi(\mathfrak{J}^{*}Y\mathfrak{J})\mathfrak{J}-\mathfrak{J}^{*}Y\mathfrak{J}=\Phi(\mathfrak{J}^{*}Y\mathfrak{J})-\mathfrak{J}^{*}Y\mathfrak{J}=0.$

Thus, by the injectivity of $\delta^{\prime}$, we have
$P_{\mathcal{K}_{0}}\pi(C^{*}(I,\mathcal{T}(P)))P_{\mathcal{K}_{0}}=P_{\mathcal{K}_{0}}\\{U\\}^{\prime}P_{\mathcal{K}_{0}}$.
In other words, we have a surjective complete contraction

$\tilde{C}_{\mathcal{K}_{0}}:\pi(C^{*}(I,\mathcal{T}(P)))\to
P_{\mathcal{K}_{0}}\\{U\\}^{\prime}P_{\mathcal{K}_{0}}=\\{P_{\mathcal{K}_{0}}X|_{\mathcal{K}_{0}}\oplus
0_{\mathcal{K}_{0}^{\perp}}:X\in\\{U\\}^{\prime}\\},$

defined by $X\mapsto P_{\mathcal{K}_{0}}XP_{\mathcal{K}_{0}}$. Since
$\delta=\delta^{\prime}\circ\tilde{C}_{\mathcal{K}_{0}}$ and $\delta$ is a
complete isometry, $\tilde{C}_{\mathcal{K}_{0}}$ is a complete isometry. Then
the induced compression map

$C_{\mathcal{K}_{0}}:\pi(C^{*}(I,\mathcal{T}(P)))\to\\{P_{\mathcal{K}_{0}}U|_{\mathcal{K}_{0}}\\}^{\prime}\subseteq\mathcal{B}(\mathcal{K}_{0}),\quad
X\mapsto P_{\mathcal{K}_{0}}X|_{\mathcal{K}_{0}}$

is a unital complete isometry and therefore a $C^{*}$-isomorphism by a result
of Kadison ([10]). Hence by the minimality of the Stinespring representation
$\pi$ we have $\mathcal{K}=\mathcal{K}_{0}$ and therefore
$\pi(C^{*}(I,\mathcal{T}(P)))=\\{U\\}^{\prime}$. This completes the proofs of
${\bf{P_{1}}}$, ${\bf{P_{2}}}$ and ${\bf{P_{3}}}$.

To prove ${\bf{P_{4}}}$, first note that $\Theta$ is completely contractive
and unital as $\pi(Q)=I$. We have also proved that
$\Theta(X)\mathfrak{J}=\mathfrak{J}X$ for all $X\in\\{P\\}^{\prime}$. Since,
for $X,Y\in\\{P\\}^{\prime}$,

$\delta(\Theta(XY)-\Theta(X)\Theta(Y))=\mathfrak{J}^{*}\mathfrak{J}XY-\mathfrak{J}^{*}\Theta(X)\Theta(Y)\mathfrak{J}=0,$

then by injectivity of $\delta$, we have $\Theta$ to be multiplicative. Hence
${\bf{P_{4}}}$ is proved. ∎

With these properties of the Stinespring dilation of $\Phi$ at hand, we start
the proof of the theorem. Define

$\displaystyle R_{i}:=\pi(QS_{i})\mbox{ for }1\leq i\leq
d-1\quad\text{and}\quad U=\pi(QP).$

We first show that $(\mathcal{K},\mathsf{R})$ is a unitary module. To that
end, we shall use Theorem 4.2 of [4]. Let $\gamma_{i}=(d-i)/d$ for
$i=1,2,\ldots,d-1$. Since $\Theta$ in $({\bf{P_{4}}})$ is multiplicative, the
tuple $(R_{1},\dots,R_{d-1},U)$ is commuting. Note that
$\pi\circ\Phi(X)=\pi(X)$, for every
$X\in\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))$, which follows from the
facts that $\rho\circ\pi(X)=\Phi(X)$ for all
$X\in\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))$ and $\pi\circ\rho=I$.

Also note that for each $1\leq i\leq d-1$,

$R_{i}^{*}U=\pi(S_{i}^{*}Q^{2}P)=\pi(\Phi({S_{i}}^{*}Q^{2}P))=\pi({S_{i}}^{*}\Phi(Q^{2})P)=\pi(S_{i}^{*}QP)=\pi(QS_{d-i})=R_{d-i}.$

It remains to show that the tuple
$(\gamma_{1}R_{1},\dots,\gamma_{d-1}R_{d-1})$ is a $\Gamma_{d-1}$-contraction.
Since $(S_{1},\dots,S_{d-1},P)$ is a $\Gamma_{d}$-contraction, by Lemma 2.7 in
[4], $(\gamma_{1}S_{1},\dots,\gamma_{d-1}S_{d-1})$ is a
$\Gamma_{d-1}$-contraction. Since $\Theta$ in $({\bf{P_{4}}})$ is
multiplicative, $(\gamma_{1}R_{1},\dots,\gamma_{d-1}R_{d-1})$ is also a
$\Gamma_{d-1}$-contraction. It follows that the tuple
$(R_{1},\dots,R_{d-1},U)$ is a $\Gamma_{d}$-unitary.

That $(\mathcal{H},\mathsf{S})$ is canonically embedded as a submodule of the
unitary module $(\mathcal{K},\mathsf{R})$ by $\mathfrak{J}$ follows from the
operator identity $\pi(QX)\mathfrak{J}=\mathfrak{J}X$ for every
$X\in\\{P\\}^{\prime}$ which was proved in the paragraphs following (3.5).

Minimality follows from $({\bf{P_{1}}})$, which says that ${\mathcal{K}}$ is
actually equal to

$\displaystyle\overline{\operatorname{span}}\\{U^{m}\mathfrak{J}h:h\in{\mathcal{H}}\text{
and }m\in\mathbb{Z}\\}.$

Let $\rho$ be as in $(\bf P_{2})$ above. Consider the restriction of $\rho$ to
$\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ and continue to denote it by $\rho$.
Since complete isometry is a hereditary property, to prove part (1), all we
have to show is that $\rho(Y)$ lands in ${\mathcal{T}}(\mathsf{S})$, whenever
$Y$ is in $\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ and $\rho$ is surjective. To
that end, let $Y\in\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$. Then for each
$i=1,2,\dots,d-1$, we see that

$\displaystyle S_{i}^{*}\rho(Y)P$ $\displaystyle=$ $\displaystyle
S_{i}^{*}\mathfrak{J}^{*}Y\mathfrak{J}P=\mathfrak{J}^{*}R_{i}^{*}YU\mathfrak{J}=\mathfrak{J}^{*}R_{d-i}Y\mathfrak{J}$
$\displaystyle=$
$\displaystyle\mathfrak{J}^{*}YR_{d-i}\mathfrak{J}=\mathfrak{J}^{*}Y\mathfrak{J}S_{d-i}=\rho(Y)S_{d-i}.$

Thus $\rho$ maps $\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ into
$\mathcal{T}(\mathsf{S})$. For proving surjectivity of $\rho$, pick
$X\in\mathcal{T}(\mathsf{S})$. Then by $({\bf{P_{2}}})$ above there exists a
$Y$ in $\\{U\\}^{\prime}$ such that $\rho(Y)=\mathfrak{J}^{*}Y\mathfrak{J}=X$.
We have to show that $Y$ commutes with each $R_{i}$. Since $X$ is in
$\mathcal{T}(\mathsf{S})$, we have
$S_{i}^{*}\mathfrak{J}^{*}Y\mathfrak{J}P=\mathfrak{J}^{*}Y\mathfrak{J}S_{d-i}$.
Therefore by the intertwining property of $\mathfrak{J}$ we have
$\mathfrak{J}^{*}R_{i}^{*}YU\mathfrak{J}=\mathfrak{J}^{*}YR_{d-i}\mathfrak{J}$,
which is the same as
$\mathfrak{J}^{*}R_{d-i}^{*}Y\mathfrak{J}=\mathfrak{J}^{*}YR_{d-i}\mathfrak{J}$.
This implies that for each $i=1,2,\dots,d-1$,

$\rho(YR_{d-i}-R_{d-i}Y)=\mathfrak{J}^{*}(YR_{d-i}-R_{d-i}Y)\mathfrak{J}=0.$

This establishes the commutativity of $Y$ with each $R_{i}$, since $\rho$ is
an isometry. This completes the proof of part (1).

Part (2) of the theorem follows from the content of $({\bf{P_{3}}})$ if we
restrict $\pi$ to ${C^{*}(I,\mathcal{T}(\mathsf{S}))}$ and continue to call it
$\pi$.

For the last part of theorem, let us take $\Theta$ as in $({\bf{P_{4}}})$,
i,e.,

$\Theta(X)=\pi(QX)$

for every $X$ in $\\{P\\}^{\prime}$. Restrict $\Theta$ to
${\\{S_{1},\dots,S_{d-1},P\\}^{\prime}}$ and continue to call it $\Theta$. The
aim is to show that $\Theta(X)\in\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ if
$X\in\\{S_{1},\dots,S_{d-1},P\\}^{\prime}$. For this we first observe that if
$X$ commutes with each $S_{j}$, then $QX$ is in ${\mathcal{T}}(\mathsf{S})$.
Now the rest of the proof follows from part (2) of the theorem and (3.5). ∎

###  3.2. Proof of Theorem 3.

Let $Q$, $\mathfrak{J}$, $\pi$, $\rho$ and $\mathsf{R}=(R_{1}\dots,R_{d-1},U)$
be as in Theorem 2. We first note that $\mathfrak{J}$ is an isometry because
$\mathfrak{J}^{*}\mathfrak{J}=Q=\text{SOT}-\lim_{j}P^{*j}P^{j}=I$. We shall
identify the module $(\mathcal{H},\mathsf{S})$ with the submodule
$(\mathfrak{J}\mathcal{H},\mathfrak{J}\mathsf{S}\mathfrak{J}^{*}$) of
$(\mathcal{K},\mathsf{R})$. Thus, we get from part $(1)$ of Theorem 2 that
$(\mathcal{K},\mathsf{R})$ is a minimal unitary extension of
$(\mathcal{H},\mathsf{S})$. Now let $X$ be an operator on $\mathcal{H}$ which
commutes with $\mathsf{S}$. Set $Y:=\pi(X)$. Then by part $(3)$ of Theorem 2,
it follows that $Y$ commutes with $\mathsf{R}$ and $Y|_{\mathcal{H}}=X$, that
is $Y$ is an extension of $X$. Also since the map $\rho$, as in part (2) of
Theorem 2 is an isometry, $Y$ is a unique norm preserving extension of $X$.
Thus part (1) follows.

Part (2) follows straightforward by setting $Y:=\pi(X)$ and remembering that
$\pi$ is the minimal Stinespring dilation of the CP map
$\Phi_{0}=\Phi|_{\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))}:\mathcal{C}^{*}(I_{\mathcal{H}},\mathcal{T}(P))\to\mathcal{B}(\mathcal{H})$
where $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ is an
idempotent CP map with range $\mathcal{T}(P)$.

To prove part (3), set $\pi_{0}$ to be the restriction of $\pi$ to
$\mathcal{C}^{*}(\mathsf{S})$. The representation $\pi$ maps the generating
set $\mathsf{S}$ of $\mathcal{C}^{*}(\mathsf{S})$ to the generating set
$\mathsf{R}$ of $\mathcal{C}^{*}(\mathsf{R})$. Since
$\pi_{0}(\mathsf{S})=\mathsf{R}$, the range of $\pi_{0}$ is
$\mathcal{C}^{*}(\mathsf{R})$. Therefore to prove that the following sequence

$0\rightarrow\mathcal{I}(\mathsf{S})\hookrightarrow\mathcal{C}^{*}(\mathsf{S})\xrightarrow{\pi_{0}}\mathcal{C}^{*}(\mathsf{R})\rightarrow
0$

is a short exact sequence, all we need to show is that
ker$\pi_{0}=\mathcal{I}(\mathsf{S})$.

Since $\pi_{0}(\mathcal{C}^{*}(\mathsf{S}))$ is abelian, we have $XY-YX$ in
the kernel of $\pi_{0}$, for any
$X,Y\in\mathcal{C}^{*}(\mathsf{S})\cap\mathcal{T}(\mathsf{S})$. Hence
$\mathcal{I}(\mathsf{S})\subseteq$ ker$\pi_{0}$. To prove the other inclusion,
let $Z_{1}$ be a finite product of members of $\mathsf{S}^{*}$ and $Z_{2}$ be
a finite product of members of $\mathsf{S}$ and call $Z=Z_{1}Z_{2}$. Since
$Z\in\mathcal{T}(\mathsf{S})\subseteq\mathcal{T}(P)$, we have $\Phi_{0}(Z)=Z$.
Note that $\Phi_{0}(Z)=P_{\mathcal{H}}\pi_{0}(Z)|_{\mathcal{H}}$, for every
$Z\in\mathcal{C}^{*}(\mathsf{S})$. Now let $Z$ be any arbitrary finite product
of members from $\mathsf{S}$ and $\mathsf{S}^{*}$. Since
$\pi_{0}(\mathsf{S})=\mathsf{R}$, which is a family of normal operators, we
obtain, by the Fuglede-Putnam Theorem that, action of $\Phi_{0}$ on $Z$ has
all the members from $\mathsf{S}^{*}$ at the left and all the members from
$\mathsf{S}$ at the right. It follows from $\text{ker}\pi=\text{ker}\Phi_{0}$
and idempotence of $\Phi_{0}$ that
ker$\pi_{0}=\\{Z-\Phi_{0}(Z):Z\in\mathcal{C}^{*}(\mathsf{S})\\}$.

Because of the above action of $\Phi_{0}$, if $Z$ is a finite product of
elements from $\mathsf{S}$ and $\mathsf{S}^{*}$ then a simple commutator
manipulation shows that $Z-\Phi_{0}(Z)$ belongs to the ideal generated by all
the commutators $XY-YX$, where
$X,Y\in\mathcal{C}^{*}(\mathsf{S})\cap\mathcal{T}(\mathsf{S})$. This shows
that ker$\pi_{0}=\mathcal{I}(\mathsf{S})$.

In order to find a completely isometric cross section, set
$\rho_{0}:=\rho|_{\pi(\mathcal{C}^{*}(\mathsf{S}))}$. Then by the definition
of $\rho$ and the action of $\Phi_{0}$, it follows that
$\rho_{0}(\pi(X))=\mathfrak{J}^{*}\pi(X)\mathfrak{J}=\Phi_{0}(X)\in\mathcal{C}^{*}(\mathsf{S})$
for all $X\in\mathcal{C}^{*}(\mathsf{S})$. Thus
$\operatorname{\operatorname{Ran}}\rho_{0}\subseteq\mathcal{C}^{*}(\mathsf{S})$
and therefore is a completely isometric cross section. This completes the
proof of the theorem.

##  4\. An application

In this section, we prove Theorem 4. Let $Q$ be the limit as in (2.2).
Consider the isometric module $(\overline{\operatorname{Ran}}Q,\mathsf{T})$
constructed in the proof of Theorem 1, see equations (2.3) and (2.4). We shall
obtain a bounded operator $\tilde{X}$ acting on
$\overline{\operatorname{Ran}}Q$ with the following properties:

  1. (1)

$\tilde{X}$ would commute with $\mathsf{T}=(T_{1},\dots,T_{d-1},V)$ and

  2. (2)

$\|\tilde{X}\|\leq\|X\|$.

We shall then apply the commutant extension theorem established in part (1) of
Theorem 3.

To that end, we first do a simple inner product computation. For every
$h\in{\mathcal{H}}$

$\displaystyle\|Q^{\frac{1}{2}}Xh\|^{2}=\langle
X^{*}QXh,h\rangle=\lim_{n}\langle
P^{*n}X^{*}XP^{n}h,h\rangle\leq\|X\|^{2}\langle Qh,h\rangle.$

Thus there is a bounded operator
$\tilde{X}:\overline{\operatorname{Ran}}Q\to\overline{\operatorname{Ran}}Q$
such that

$\tilde{X}:Q^{\frac{1}{2}}h\mapsto Q^{\frac{1}{2}}Xh.$

with norm at most $\|X\|$. Let $j=1,2,\dots,d-1$ and let $T_{j}$ be the
operators defined in (2.4). Then for each $h\in{\mathcal{H}}$, we have

$\displaystyle\tilde{X}T_{j}Q^{\frac{1}{2}}h=\tilde{X}Q^{\frac{1}{2}}S_{j}h=Q^{\frac{1}{2}}XS_{j}h=Q^{\frac{1}{2}}S_{j}Xh=T_{j}Q^{\frac{1}{2}}Xh=T_{j}\tilde{X}Q^{\frac{1}{2}}h$

showing that $\tilde{X}$ commutes with $T_{j}$ for all $j=1,\dots,d-1$. A
similar computation also establishes that $\tilde{X}V=V\tilde{X}$. We are now
ready to apply the technique of commutant extension mentioned above.

Consider $\mathsf{R}=(R_{1},\dots,R_{d-1},U)$ acting on ${\mathcal{K}}$ as in
(2.5). Recall that $({\mathcal{K}},\mathsf{R})$ is a unitary module and there
is a contraction $\mathfrak{J}:{\mathcal{H}}\to{\mathcal{K}}$ defined as
$\mathfrak{J}h=Q^{\frac{1}{2}}h$ such that (2.5) holds. Now by the moreover
part of item (1) in Theorem 3, there exists an operator $Y$ in the commutant
of $\mathsf{R}$ such that $Y|_{\overline{\operatorname{Ran}}Q}=\tilde{X}$ and
$\|Y\|=\|\tilde{X}\|\leq\|X\|$. Finally, we note that for every
$h\in{\mathcal{H}}$,

$\displaystyle\mathfrak{J}Xh=Q^{\frac{1}{2}}Xh=\tilde{X}Q^{\frac{1}{2}}h=YQ^{\frac{1}{2}}h=Y\mathfrak{J}h.$

This completes the proof.

The following analogue of the intertwining lifting theorem is easily obtained
as a corollary to Theorem 4 by a $2\times 2$ operator matrix trick.

######  Corollary 4.1.

Let $({\mathcal{H}},\mathsf{S})$ and
$({\mathcal{H}}^{\prime},\mathsf{S}^{\prime})$ be two contractive modules. Let
$(\mathfrak{J},{\mathcal{K}},\mathsf{R})$ and
$(\mathfrak{J}^{\prime},{\mathcal{K}}^{\prime},\mathsf{R}^{\prime})$ be as
obtained by part (3) of Theorem 1 corresponding to
$({\mathcal{H}},\mathsf{S})$ and
$({\mathcal{H}}^{\prime},\mathsf{S}^{\prime})$ respectively. Then
corresponding to any operator $X:{\mathcal{H}}\to{\mathcal{H}}^{\prime}$
intertwining $\mathsf{S}$ and $\mathsf{S}^{\prime}$ there exists another
operator $Y:{\mathcal{K}}\to{\mathcal{K}}^{\prime}$ such that $Y$ intertwines
$\mathsf{R}$ and $\mathsf{R}^{\prime}$, $Y\mathfrak{J}=\mathfrak{J}^{\prime}X$
and $\|Y\|\leq\|X\|$.

Acknowledgement: The research work of the first named author is supported by a
J C Bose National Fellowship JCB/2021/000041 and that of the second and third
named authors are supported by DST-INSPIRE Faculty Fellowships
DST/INSPIRE/04/2015/001094 and DST/INSPIRE/04/2018/002458 respectively.

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# Bardeen spacetime as a quantum corrected Schwarzschild black hole:
Quasinormal modes and Hawking radiation

R. A. Konoplya<EMAIL_ADDRESS>D. Ovchinnikov
<EMAIL_ADDRESS>Research Centre for Theoretical Physics and
Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo
náměstí 13, CZ-74601 Opava, Czech Republic  B. Ahmedov<EMAIL_ADDRESS>Ulugh Beg Astronomical Institute, Astronomy St. 33, Tashkent 100052,
Uzbekistan  Institute of Fundamental and Applied Research, National Research
University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan  National
University of Uzbekistan, Tashkent 100174, Uzbekistan

###### Abstract

The Bardeen black hole holds historical significance as the first model of a
regular black hole. Recently, there have been proposed interpretations of the
Bardeen spacetime as quantum corrections to the Schwarzschild solution. Our
study focuses on investigating the quasinormal modes and Hawking radiation of
the Bardeen black hole. We have observed that previous studies on the
quasinormal modes for the Bardeen black hole suffer from inaccuracies that
cannot be neglected. Therefore, we propose accurate calculations of the
quasinormal modes for scalar, electromagnetic, and neutrino fields in the
Bardeen spacetime. Additionally, we have computed the grey-body factors and
analyzed the emission rates of Hawking radiation. Even when the quantum
correction is small and the fundamental mode only slightly differs from its
Schwarzschild value, the first several overtones deviate at an increasingly
stronger rate. This deviation leads to the appearance of overtones with very
small real oscillation frequencies. This outburst of overtones is closely
linked to the fact that the quantum-corrected black hole differs from its
classical limit primarily near the event horizon. Moreover, the intensity of
the Hawking radiation is significantly suppressed (up to three orders of
magnitude) by the quantum correction.

##  I Introduction

The problem of a central singularity in the Schwarzschild solution represents
a limitation of Einstein’s theory of gravity. Consequently, the development of
regular black hole models within various approaches and theories of gravity
has become an important endeavor. The Bardeen black hole stands as the first
historically significant ad hoc model describing a regular black hole [1].
Subsequently, an interpretation of the Bardeen metric was proposed in the
context of a specific nonlinear electrodynamics in curved spacetime, positing
it as a giant gravitating magnetic monopole [2]. However, if the weak field
limit is reduced to the usual Maxwell electrodynamics, the electric charge
must vanish, as per the Bronnikov theorem [3]. Despite such an exotic
interpretation of the Bardeen spacetime, numerous studies have been dedicated
to exploring various effects around magnetic monopoles, particularly their
characteristic oscillation frequencies known as quasinormal modes [4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

Quasinormal modes not only serve as fundamental observables that characterize
black holes [25], but also play a crucial role in gauge/gravity duality for
describing strongly coupled quantum systems [26], as well as in determining
the (in)stability of black holes [27].

However, a significant portion of the aforementioned quasinormal mode
calculations for the Bardeen black hole suffer from considerable numerical
inaccuracies due to the utilization of various approximations. Consequently,
it becomes necessary to reexamine some of these results using accurate methods
based on converging procedures, such as the Leaver methods [28].

Nevertheless, the focus of our investigation here is on the Bardeen spacetime
not within the context of an exotic nonlinear electrodynamics, but rather as a
quantum-corrected solution to the Schwarzschild spacetime. In this framework,
the parameter that formerly represented the magnetic charge now controls the
magnitude of the quantum correction for a neutral black hole. Remarkably,
stringy corrections to Schwarzschild black hole spacetimes arising from string
T-duality were proposed in [29]. As an initial step, the static Newtonian
potential was derived by exploiting the relationship between T-duality and
path integral duality. Subsequently, it was demonstrated that the intrinsic
non-perturbative nature of stringy corrections introduces an ultraviolet
cutoff known as the zero-point length, resulting in a regular static
potential. This finding was employed to derive a consistent black hole metric
for a spherically symmetric, electrically neutral regular black hole, which is
equivalent to the Bardeen spacetime after redefinition of constants. It is
noteworthy that a similar duality is utilized to probe the quantum
fluctuations of the background spacetime [30]. Ultimately, the Bardeen
spacetime can also be deduced as an effective metric that reproduces the
thermodynamics of a black hole within the Generalized Uncertainty Principle
[31], which represents another, more speculative approach to uncovering
quantum corrections.

In this paper, we will investigate the quasinormal modes of scalar,
electromagnetic, and neutrino fields in the background of the Bardeen
spacetime, considering its aforementioned interpretation as a neutral quantum-
corrected black hole. Our aim is to demonstrate the inaccuracies present in
some of the previous calculations within certain parameter ranges. Here, we
present precise calculations not only for the fundamental mode but also for
the initial overtones with $\ell\leq n$, where $\ell$ represents the multipole
number and $n$ denotes the overtone number. While it is commonly believed that
the fundamental mode suffices as a fingerprint of the black hole, this is not
entirely accurate. The fundamental mode is insensitive to the geometry of the
event horizon and is primarily determined by the geometry near the peak of the
effective potential. Consequently, if a black hole were to be replaced by a
wormhole with the same geometry near the potential’s peak, the fundamental
mode would exhibit minimal change [32]. To ascertain the near-horizon
behavior, it is necessary to consider multiple overtones [33], which also
characterize the early phase of the ringdown [34]. Therefore, studying the
initial few overtones will enable us to probe the geometry of the black hole
in the vicinity of the event horizon.

We will demonstrate that the overtones deviate from their Schwarzschild limits
at a significantly higher rate than the fundamental mode, and this deviation
stems from the deformation of the metric near the event horizon [33]. Similar
outbursts have recently been observed in the context of black holes in higher-
curvature corrected theories and asymptotically safe gravity [35, 36, 37, 38].

Quantum-corrected black holes are expected to undergo intense Hawking
evaporation, making it crucial to investigate Hawking radiation in the
vicinity of Bardeen black holes. To the best of our knowledge, a comprehensive
study of Hawking evaporation in this context has not yet been proposed.

In this paper, we will explore Hawking radiation of massless Standard Model
fields around Bardeen black holes and demonstrate that the quantum correction
significantly reduces the energy emission rate of the test fields (up to three
orders of magnitude). To determine the emission rates, we will solve the
problem of field scattering around the black hole, specifically by calculating
the grey-body factors that account for the reflection from the effective
potential and diminish the total emission rate.

The paper is organized as follows. In Section II, we provide a brief review of
the Bardeen solution and the corresponding wave-like equations with effective
potentials. In Section III, we describe the numerical methods employed to
calculate quasinormal modes and grey-body factors, including the WKB approach,
accurate Frobenius method, and time-domain integration. Section IV is
dedicated to discussing the obtained quasinormal modes and the outburst of
overtones. In Section V, we present the results of the calculations for grey-
body factors and energy emission rates in Hawking radiation. Finally, in the
Conclusions section, we summarize the obtained results and mention some open
problems.

##  II Bardeen spacetime and the wavelike equations

The Bardeen spacetime is described by the following line element

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (1)

where

$f(r)=1-\frac{2Mr^{2}}{(r^{2}+l_{0}^{2})^{3/2}}.$ (2)

For $l_{0}\neq 0$, the space-time in eq.(5) has horizons only if,

$|l_{0}|\leq\frac{4M}{3\sqrt{3}},$

as was shown in [39]. The parameter $l_{0}$ is related to the Regge slope as
follows:

$\sqrt{\alpha^{\prime}}=\frac{l_{0}}{2\pi}\approx 0.117l_{P},$ (3)

where $l_{P}$ is the Plank mass, so that, as in [30], we have: $l_{0}\sim
l_{P}$. Stringy effects produce a de Sitter core at the origin, because

$f(r)\rightarrow 1-\frac{\Lambda_{eff}r^{2}}{3},\qquad r\ll l_{0},$ (4)

with an effective cosmological constant $\Lambda_{eff}=6M/l_{0}^{3}$. This
repulsive effect at small distances stabilizes the matter configuration
against collapse.

The general relativistic equations for the scalar ($\Phi$), electromagnetic
($A_{\mu}$), and Dirac ($\Upsilon$) fields can be written as follows:

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)$
$\displaystyle=$ $\displaystyle 0,$ (5a)
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(F_{\rho\sigma}g^{\rho\nu}g^{\sigma\mu}\sqrt{-g}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (5b)
$\displaystyle\gamma^{\alpha}\left(\frac{\partial}{\partial
x^{\alpha}}-\Gamma_{\alpha}\right)\Upsilon$ $\displaystyle=$ $\displaystyle
0,$ (5c)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic tensor, $\gamma^{\alpha}$ are gamma matrices and
$\Gamma_{\alpha}$ are spin connections in the tetrad formalism. After
separation of the variables the above equations (5) take the Schrödinger wave-
like form:

$\dfrac{d^{2}\Psi}{dr_{*}^{2}}+(\omega^{2}-V(r))\Psi=0,$ (6)

where the “tortoise coordinate” $r_{*}$ is defined as follows:

$dr_{*}\equiv\frac{dr}{f(r)}.$ (7)

The effective potentials for the scalar ($s=0$) and electromagnetic ($s=1$)
fields have the form

$V(r)=f(r)\frac{\ell(\ell+1)}{r^{2}}+\left(1-s\right)\cdot\frac{f(r)}{r}\frac{df(r)}{dr},$
(8)

where $\ell=s,s+1,s+2,\ldots$ are the multipole numbers. For the Dirac field
($s=1/2$) one has two iso-spectral potentials

$V_{\pm}(r)=W^{2}\pm\frac{dW}{dr_{*}},\quad
W\equiv\left(\ell+\frac{1}{2}\right)\frac{\sqrt{f(r)}}{r}.$ (9)

The iso-spectral wave functions can be transformed one into another by the
Darboux transformation

$\Psi_{+}=q\left(W+\dfrac{d}{dr_{*}}\right)\Psi_{-},\quad q=const,$ (10)

so that it is sufficient to calculate quasinormal modes and grey-body factors
for only one of the effective potentials.

Figure 1: The effective potential for the $\ell=1$ perturbation of the Maxwell
field for the Schwarzschild (bottom) and near extreme Bardeen $l_{0}=0.7$
(top) black holes; $r_{0}=1$.

##  III  Methods for finding of quasinormal modes and grey-body factors

Quasinormal modes represent eigenvalues of the wave-like equations mentioned
earlier, corresponding to purely outgoing waves at spatial infinity and purely
ingoing waves at the event horizon. Since methods for determining quasinormal
frequencies have been extensively discussed in numerous papers, we will
provide a brief summary of the three methods employed in this study: the WKB
approach with Padé approximants, the Leaver method, and time-domain
integration. Furthermore, we will discuss the ordinary WKB method’s
application in finding grey-body factors.

###  III.1 WKB approach

We will utilize the semi-analytic WKB approach developed by Will and Schutz
[40] to determine the quasinormal modes. The Will-Schutz formula has been
extended to higher orders [41, 42, 43] and improved further by implementing
Padé approximants [43, 44].

The general form of the WKB formula, as derived by Konoplya [45], is as
follows:

$\displaystyle\omega^{2}$ $\displaystyle=$ $\displaystyle V_{0}+A_{2}({\cal
K}^{2})+A_{4}({\cal K}^{2})+A_{6}({\cal K}^{2})+\ldots$ $\displaystyle-$
$\displaystyle i{\cal K}\sqrt{-2V_{2}}\left(1+A_{3}({\cal K}^{2})+A_{5}({\cal
K}^{2})+A_{7}({\cal K}^{2})+\ldots\right),$

where ${\cal K}=n+1/2$ is a half-integer. The corrections $A_{k}({\cal
K}^{2})$, of order $k$, in the eikonal formula are polynomials of ${\cal
K}^{2}$ with rational coefficients. These corrections depend on the values of
higher derivatives of the potential $V(r)$ at its maximum. To enhance the
accuracy of the WKB formula, we will follow the approach of Matyjasek and
Opala [43] by utilizing Padé approximants. Specifically, we will employ the
sixth-order WKB method with $\tilde{m}=4,5$, where $\tilde{m}$ is defined in
[43, 45], as this choice yields the best accuracy in the Schwarzschild limit
and is also appropriate for the Bardeen black hole, as confirmed by
comparisons with accurate data.

It is important to note that the WKB series converges only asymptotically and
does not guarantee improved accuracy at each order. Therefore, it is necessary
to cross-validate the WKB results with the convergent Frobenius method.
Additionally, the outburst of overtones, occurring when $\ell<n$, cannot be
captured by the WKB formula in principle. Hence, in this study, we primarily
employ the WKB method as an additional check and solely for calculating the
fundamental modes.

###  III.2 Frobenius method

To obtain precise values of quasinormal modes, including overtones with
$\ell<n$, we will utilize the Frobenius method, as initially employed by
Leaver for calculating quasinormal modes [28]. The wave-like equation always
exhibits a regular singularity at the horizon $r=r_{0}$ and an irregular
singularity at spatial infinity $r=\infty$. To address this, we introduce a
new function:

$\Psi(r)=P(r,\omega)\left(1-\frac{r_{0}}{r}\right)^{-i\omega/F^{\prime}(r_{0})}y(r),$
(12)

Here, the factor $P$ is chosen such that $y(r)$ is regular for $r_{0}\leq
r<\infty$, ensuring that $\Psi(r)$ corresponds to a purely outgoing wave at
spatial infinity and a purely ingoing wave at the event horizon. Consequently,
we can express $y(r)$ in terms of a Frobenius series:

$y(r)=\sum_{k=0}^{\infty}a_{k}\left(1-\frac{r_{0}}{r}\right)^{k}.$ (13)

By employing Gaussian elimination in the recurrence relation for the expansion
coefficients, we reduce the problem to solving an algebraic equation. To
enhance convergence speed, we will implement the Nollert improvement [46] in
its general form for the n-term recurrence relation [47]. When the singular
points of the wavelike equation appear within the unit circle $|x|<1$, we
employ a sequence of positive real midpoints as described in [48].

###  III.3 Time domain integration

To determine the quasinormal modes and asymptotic tails, we will employ the
time-domain integration method. We integrate the wave-like equation using the
null-cone variables $u=t-r_{*}$ and $v=t+r_{*}$. For the discretization
scheme, we will apply the Gundlach-Price-Pullin method [49], which can be
expressed as:

$\Psi\left(N\right)=\Psi\left(W\right)+\Psi\left(E\right)-\Psi\left(S\right)$
$-\Delta^{2}V\left(S\right)\frac{\Psi\left(W\right)+\Psi\left(E\right)}{4}+{\cal
O}\left(\Delta^{4}\right).$ (14)

Here, the points are denoted as follows:
$N\equiv\left(u+\Delta,v+\Delta\right)$, $W\equiv\left(u+\Delta,v\right)$,
$E\equiv\left(u,v+\Delta\right)$, and $S\equiv\left(u,v\right)$. Gaussian
initial data are imposed on the two null surfaces $u=u_{0}$ and $v=v_{0}$. The
dominant quasinormal frequencies can be extracted from the time-domain
profiles using the Prony method [27]. While extracting the frequency from the
time-domain profile with high precision can be challenging, the accuracy for
the fundamental mode is guaranteed to be sufficiently good.

###  III.4 Scattering problem

To investigate Hawking radiation, we will address a different boundary problem
that pertains to the scattering of fields in the vicinity of the black hole.
Grey-body factors play a crucial role in determining the proportion of initial
radiation that gets reflected back to the event horizon by the potential
barrier nearby. Subsequently, by employing the Hawking semiclassical formula
along with the grey-body factor, we can calculate the amount of radiation that
reaches an observer in the far zone. It is worth noting that while temperature
is typically the dominant factor in estimating the intensity of Hawking
radiation, there are cases where grey-body factors can be equally important
[50].

We will examine the wave equations under boundary conditions that permit
incoming waves from infinity. Due to the scattering symmetry, this is
equivalent to considering the scattering of a wave originating from the
horizon. The boundary conditions for this scattering problem are as follows:

$\begin{array}[]{ccll}\Psi&=&e^{-i\omega r_{*}}+Re^{i\omega
r_{*}},&r_{*}\to+\infty,\\\\[5.0pt] \Psi&=&Te^{-i\omega
r_{*}},&r_{*}\to-\infty,\\\ \end{array}$ (15)

Here, $R$ and $T$ represent the reflection and transmission coefficients,
respectively.

The effective potentials for test Maxwell and Dirac fields have the form of
potential barriers which decrease monotonically towards both infinities 111One
of the two effective potentials for the Dirac perturbations have a negative
gap, but it is iso-spectral to the other, which is positive definite.,
allowing for application of the WKB approach [40, 41, 42] to determine $R$ and
$T$. As $\omega^{2}$ is real for the scattering problem, the WKB values for
$R$ and $T$ will be real as well [40, 41, 42], and

$|T|^{2}+|R|^{2}=1.$ (16)

Once the reflection coefficient is obtained, we can find the transmission
coefficient for each multipole number $\ell$,

$|{\pazocal A}_{\ell}|^{2}=1-|R_{\ell}|^{2}=|T_{\ell}|^{2},$ (17)

where ${\pazocal A}_{\ell}$ is called the grey-body factor.

Here we will use the higher order WKB formula [42] for a relatively accurate
calculation of the grey-body factors. However, this formula is not suitable
for very small values of $\omega$, which correspond to almost complete wave
reflection and have negligible contributions to the overall energy emission
rate. For this mode, we employed extrapolation of the WKB results at a given
order to smaller $\omega$ [52]. According to [40, 41], the reflection
coefficient can be expressed as follows:

$R=(1+e^{-2i\pi K})^{-1/2},$ (18)

where $K$ is determined by solving the equation

$K-i\frac{(\omega^{2}-V_{\max})}{\sqrt{-2V^{\prime\prime}_{\max}}}-\sum_{i=2}^{i=6}\Lambda_{i}(K)=0,$
(19)

involving the maximum effective potential $V_{\max}$, its second-order
derivative $V^{\prime\prime}_{\max}$ with respect to the tortoise coordinate,
and higher order WKB corrections $\Lambda_{i}$. As mentioned above, the WKB
series does not guarantee convergence in each order, but only asymptotically,
so that usually there is an optimal moderate order at which the accuracy is
the best. This order depends on the form of the effective potential. Here we
used 6th order for Maxwell perturbations and 3rd order for the Dirac field
with a plus-potential, because these orders provide the best accuracy in the
Schwarzschild limit, and we expect that this will also take place for the
Bardeen black hole. This approach was used in a few papers [45, 53, 54, 55],
showing usually reasonably good concordance with the numerical integration.
Unfortunately, the Pade approximant, which greatly improve the accuracy when
finding quasinormal modes, cannot be used when finding the grey-body factors,
because of the non-uniquness of such procedure.

Figure 2: Time-domain profile of the scalar field perturbations $\ell=0$ for
the quasi-extremal Bardeen black hole $l_{0}=0.707107$; $r_{0}=1$. Left: Semi-
logarithmic plot; Right: logarithmic plot with the line $\sim t^{-3}$.
---
Figure 3: Quasinormal frequencies for the scalar perturbations ($\ell=0$) and
$n$ from $0$ to $6$.

##  IV Quasinormal modes and the outburst of overtones

As a general rule, analytical solutions for quasinormal frequencies of four-
and higher-dimensional black holes are not available. However, they can often
be obtained in the regime of large multipole numbers $\ell$. By utilizing the
first-order WKB formula and expanding the position of the maximum of the
effective potential, we can derive the following WKB expression for $\omega$
in terms of $1/L$ and $l_{0}$, where $L=\ell+\frac{1}{2}$:

$r_{max}=3M-\frac{5l_{0}^{2}}{6M}-\frac{65l_{0}^{4}}{216M^{3}}+\mathcal{O}(l_{0}^{6})$
(20)
$\omega=\frac{L}{3\sqrt{3}M}-\frac{i(2n+1)}{6\sqrt{3}M}+l_{0}^{2}\left(\frac{L}{18\sqrt{3}M^{3}}+\frac{i(2n+1)}{54\sqrt{3}M^{3}}\right)+$
$l_{0}^{4}\left(\frac{17L}{648\sqrt{3}M^{5}}+\frac{7i(2n+1)}{324\sqrt{3}M^{5}}\right)+\mathcal{O}\left(\frac{1}{L},l_{0}^{6}\right).$
(21)

It is evident that the above expressions are accurate in the limit of large
$\ell$, while still providing reasonable accuracy even at moderate values of
$\ell$. When $l_{0}=0$, these analytical formulas reduce to the well-known
expressions for the Schwarzschild black hole [56] (see [57, 58] and references
therein for various generalizations of these relations). Furthermore, the
aforementioned eikonal formula can be derived through the correspondence
between eikonal quasinormal modes and null geodesics [59, 60, 61]. However, it
should be noted that this correspondence has several limitations, as described
in [60, 61, 62].

From this point onwards in this section, we will compute quasinormal
frequencies in units of the fixed radius of the event horizon, $r_{0}=1$,
rather than fixing the mass of the black hole, $M$. The mass and radius of the
event horizon are related by the following equation:

$M=\frac{(r_{0}^{2}+l_{0}^{2})^{3/2}}{2r_{0}^{2}}.$ (22)

This equation allows us to switch between the units $r_{0}=1$ and $M=1$. The
units of the fixed event horizon are more convenient for the application of
the Frobenius method, which requires a precise understanding of the singular
points of the differential equation under consideration.

The fundamental quasinormal modes for the test scalar field ($\ell=n=0$),
electromagnetic field ($\ell=1$, $n=0$), and Dirac field ($\ell=1/2$, $n=0$)
are presented in Tables I-III. The accurate values are obtained using the
convergent Frobenius method [28] for scalar and electromagnetic fields, while
for the Dirac field, which has a non-polynomial wave equation, we were limited
to using the WKB method. A comparison of these values with earlier published
results [4, 7] reveals that the ordinary 6th order WKB data (without Padé
approximants) is inaccurate, with a relative error reaching approximately
$10\%$. However, the advanced WKB method with Padé approximants considered
here [43] shows much better agreement with accurate quasinormal frequencies,
reducing the relative error to a small fraction of one percent. The Frobenius
method can only be applied to polynomial forms of the metric function and
effective potentials. To meet this requirement, we expand the metric function
in terms of small $l_{0}$ (see Appendix for the explicit form of the
expansion) and apply the Frobenius method to this expanded metric. We observe
that the quasinormal frequencies calculated using the metric function expanded
up to the 20th and 24th orders in $l_{0}$ are in excellent agreement (see
Table I-II). An example of the convergence in terms of $l_{0}$ is shown in
Figure 5. The higher the overtone $n$ one needs to find, the higher the order
of expansion of the metric that should be used.

Integration of the wave equation in the time domain and subsequent extraction
of dominant frequencies from the profile using the Prony method also confirm
the accuracy of the Frobenius and WKB (with Padé) data. For instance, for the
quasi-extremal state $\ell_{0}=0.707107$ and $\ell=0$ scalar perturbations,
the frequency $\omega=0.12958-0.09393i$ is extracted from the time-domain
profile. This is in close agreement with the Frobenius method for the metric
expanded up to the 24th order, which gives $\omega=0.129521-0.0939448i$, and
the 6th order WKB method with Padé approximants, which gives
$\omega=0.129879-0.093637i$. On the other hand, the usual 6th order WKB method
produces an approximately $10\%$ error in the damping rate.

At asymptotically late times, when the quasinormal ringing is suppressed by
power-law tails, the Price decay law for the test scalar and gravitational
fields [63] is fulfilled:

$|\Psi|\sim t^{-(2\ell+3)},\quad t\rightarrow\infty.$ (23)

This can be observed in Figure 2.

The first several overtones exhibit the most interesting feature. They deviate
from their Schwarzschild values at a significantly higher rate compared to the
fundamental mode, and this rate increases with the overtone number (see Figure
3). While the fundamental mode undergoes only a slight change, limited to
moderate values of the quantum deformation parameter $l_{0}$ (as shown in
Figure 3), the real oscillation frequency of the sixth overtone decreases by
more than twice its Schwarzschild limit.

This outburst of overtones occurs because the Bardeen spacetime differs from
the Schwarzschild spacetime mainly in a region near the event horizon. This
characteristic could potentially enable us to discern the quantum corrections
near the horizon through the early phase of the quasinormal ringing, which
includes contributions from the overtones [33, 34]. It is important to note
that if we consider the Bardeen spacetime as a deformation of the
Schwarzschild black hole, this deformation is not highly localized and,
therefore, is not related to the so-called ”high-frequency perturbations”
(which actually refer to ”highly localized deformation”) discussed in [64] and
subsequent works. The deformation of the Schwarzschild spacetime introduced by
the parameter $l_{0}$ is smooth and extends over a neighborhood of the event
horizon, gradually decaying as we move away from the black hole.

Table 1:  Fundamental quasinormal modes for the scalar perturbations ($n=0$, $\ell=0$). | WKB | Frobenius
---|---|---
$l_{0}$ | 6th | 6th (Pade 5) | 6th (Pade 6) | 20th order | 24th order
0 | $0.220934-0.201633i$ | $0.221584-0.209367i$ | $0.221357-0.208847i$ | $0.220910-0.209791i$ | $0.220910-0.209791i$
0.07698 | $0.220288-0.199177i$ | $0.220818-0.206828i$ | $0.220585-0.206324i$ | $0.220146-0.207248i$ | $0.220146-0.207248i$
0.15396 | $0.218078-0.192489i$ | $0.218398-0.199312i$ | $0.218146-0.198876i$ | $0.217717-0.199765i$ | $0.217717-0.199765i$
0.23094 | $0.214096-0.182285i$ | $0.213780-0.187430i$ | $0.213541-0.187065i$ | $0.213239-0.187801i$ | $0.213239-0.187801i$
0.30792 | $0.209605-0.165605i$ | $0.205935-0.172526i$ | $0.205928-0.172054i$ | $0.206151-0.172156i$ | $0.206151-0.172156i$
0.3849 | $0.203168-0.140512i$ | $0.195809-0.154749i$ | $0.195900-0.154227i$ | $0.195821-0.154042i$ | $0.195821-0.154042i$
0.46188 | $0.186592-0.119218i$ | $0.183120-0.134872i$ | $0.182642-0.134402i$ | $0.181717-0.135270i$ | $0.181717-0.135270i$
0.53886 | $0.166413-0.106640i$ | $0.164673-0.118464i$ | $0.164432-0.118343i$ | $0.164121-0.118732i$ | $0.164119-0.118735i$
0.61584 | $0.149688-0.096004i$ | $0.147642-0.106578i$ | $0.147526-0.106419i$ | $0.146968-0.106725i$ | $0.146995-0.106719i$
0.69282 | $0.134525-0.086147i$ | $0.132643-0.095711i$ | $0.132524-0.095546i$ | $0.132241-0.0959643i$ | $0.132129-0.0958409i$
0.707107 | $0.131840-0.084424i$ | $0.129995-0.093798i$ | $0.129879-0.093637i$ | $0.129702-0.0941100i$ | $0.129521-0.0939448i$
Table 2:  Fundamtenal quasinormal modes for the electromagnetic perturbations ($n=0$, $\ell=1$). | WKB | Frobenius
---|---|---
$l_{0}$ | 6th | 6th (Pade 5) | 6th (Pade 6) | 20th order | 24th order
0 | $0.496383-0.185274i$ | $0.496509-0.184993i$ | $0.496509-0.184961i$ | $0.496527-0.184975i$ | $0.496527-0.184975i$
0.069282 | $0.494945-0.183682i$ | $0.495092-0.183337i$ | $0.495086-0.183305i$ | $0.495104-0.183318i$ | $0.495104-0.183318i$
0.138564 | $0.490564-0.178948i$ | $0.490767-0.178429i$ | $0.490754-0.178398i$ | $0.490772-0.178411i$ | $0.490772-0.178411i$
0.207846 | $0.483055-0.171211i$ | $0.483353-0.170467i$ | $0.483338-0.170434i$ | $0.483353-0.170452i$ | $0.483353-0.170452i$
0.277128 | $0.472162-0.160742i$ | $0.472577-0.159796i$ | $0.472563-0.159759i$ | $0.472572-0.159784i$ | $0.472572-0.159784i$
0.34641 | $0.457592-0.147991i$ | $0.458107-0.146925i$ | $0.458095-0.146889i$ | $0.458099-0.146918i$ | $0.458099-0.146918i$
0.415692 | $0.439053-0.133649i$ | $0.439614-0.132573i$ | $0.439603-0.132545i$ | $0.439605-0.132570i$ | $0.439605-0.132570i$
0.484974 | $0.416343-0.118720i$ | $0.416890-0.117721i$ | $0.416882-0.117704i$ | $0.416881-0.117720i$ | $0.416881-0.117720i$
0.554256 | $0.389608-0.104530i$ | $0.390103-0.103633i$ | $0.390099-0.103626i$ | $0.390100-0.103635i$ | $0.390096-0.103636i$
0.623538 | $0.359764-0.092365i$ | $0.360192-0.091561i$ | $0.360190-0.091559i$ | $0.360256-0.0915651i$ | $0.360201-0.0915712i$
0.69282 | $0.328400-0.082759i$ | $0.328778-0.082042i$ | $0.328774-0.082038i$ | $0.329444-0.0819892i$ | $0.328945-0.0820359i$
0.707107 | $0.321891-0.081076i$ | $0.322262-0.080374i$ | $0.322258-0.080369i$ | $0.323296-0.0802851i$ | $0.322542-0.0803561i$
---
Figure 4: Quasinormal frequencies for the electromagnetic perturbations ($\ell=1$) and $n$ from $0$ to $6$. Table 3:  Fundamental quasinormal modes for the Dirac perturbations ($n=0$, $\ell=1/2$). | WKB
---|---
$l_{0}$ | 6th | 6th (Pade 5) | 6th (Pade 6)
0 | $0.366162-0.194105i$ | $0.365943-0.193956i$ | $0.365813-0.193991i$
0.07698 | $0.364600-0.191742i$ | $0.364392-0.191630i$ | $0.364267-0.191664i$
0.15396 | $0.359889-0.184810i$ | $0.359702-0.184821i$ | $0.359589-0.184851i$
0.23094 | $0.351950-0.173766i$ | $0.351761-0.174019i$ | $0.351660-0.174036i$
0.30792 | $0.340642-0.159328i$ | $0.340367-0.159983i$ | $0.340279-0.159978i$
0.3849 | $0.325739-0.142490i$ | $0.325228-0.143677i$ | $0.325166-0.143648i$
0.46188 | $0.306947-0.124700i$ | $0.306169-0.126350i$ | $0.306135-0.126322i$
0.53886 | $0.284334-0.107994i$ | $0.283432-0.109828i$ | $0.283418-0.109814i$
0.61584 | $0.259273-0.094155i$ | $0.258320-0.095939i$ | $0.258314-0.095930i$
0.69282 | $0.233822-0.083456i$ | $0.232929-0.085144i$ | $0.232909-0.085107i$
0.707107 | $0.229173-0.081764i$ | $0.228298-0.083422i$ | $0.228278-0.083383i$
---
Figure 5: An example of the convergence of the scalar field QNMs $\omega$
calculated by the Frobenius method as the order of the expansion $\mathcal{N}$
in terms of small $l_{0}$ is increased ($n=0$, $\ell=0$) and $l_{0}=0.46669$.

The significant decrease in the real oscillation frequencies of the overtones
with increasing deformation parameter $l_{0}$ suggests the possibility of
purely imaginary quasinormal modes in the spectrum at larger values of
$l_{0}$. However, due to the computational limitations, we were unable to find
such modes as they would require much longer computation times. It is
important to note that the quantum correction parameter is intended to
represent a relatively small or at most moderate correction to the
Schwarzschild spacetime. Therefore, the metric should remain sufficiently far
from its extreme state, ensuring that the corrections remain within a
reasonable range.

Figure 6: Grey-body factors for the Maxwell field as function of $\omega$ for
the Schwarzschild solution (left panel) and the near extreme ($l_{0}=0.7$)
Bardeen black hole (right panel) computed by the 6th order WKB approach for
the first five multipoles $\ell=1,2,..5$ from left to right.

Figure 7: Grey-body factors for the Dirac field as function of $\omega$ for
the Schwarzschild solution (left panel) and the near extreme ($l_{0}=0.7$)
Bardeen black hole (right panel) computed by the 3d order WKB approach for the
first five multipoles $\ell=1,2,..5$ from left to right.

Figure 8: Energy emission rates per unit of frequency for the Maxwell field
for the Schwarzschild spacetime (left plot) and for the near extreme Bardeen
black hole $l_{0}=0.7$ (right plot). One can see that the contribution
consists almost completely from the first multipole $\ell=1$ (blue, top) and
$\ell=2$ (yellow, bottom), while higher multipoles could neglected. The black
line is for the total emission rate summed over the first five multipoles.

##  V Hawking radiation

It is essential to note that radiation from test fields surpasses that of
gravitons: as is known using Schwarzschild black holes as an example,
gravitons account for less than $2\%$ of the total radiation flux, as
illustrated in [65] and also summarized in Table I of [50].

Here we will assume that the black hole is in a state of thermal equilibrium
with its surroundings. This means that the temperature of the black hole
remains constant between the emission of two consecutive particles. According
to this assumption, the system can be characterized by the canonical ensemble,
which is extensively discussed in the literature (see, e.g., [66]).
Consequently, the well-known Hawking formula for the energy emission rate [67]
can be applied:

$\displaystyle\frac{\text{d}E}{\text{d}t}=\sum_{\ell}N_{\ell}|\pazocal{A}_{\ell}|^{2}\frac{\omega}{\exp\left(\omega/T_{\text{H}}\right)\pm
1}\cdot\frac{\text{d}\omega}{2\pi},$ (24)

where $T_{H}$ is the Hawking temperature, $\pazocal{A}_{\ell}$ are grey-body
factors, and $N_{\ell}$ are the multiplicity factors which depend on the
number of species of particles and $\ell$. The Hawking temperature is [67]

$T=\left.\frac{f^{\prime}(r)}{(4\pi)}\right|_{r=r_{0}}.$ (25)

The dependence of the temperature on the parameter $l_{0}$ is shown in fig. 6.

Figure 9: Temperature of the Bardeen black hole at fixed mass $M=1$ as a
function of the deforamtion parameter $l_{0}$.

As long as we are considering the evaporation stage when the black hole is
large enough, one can neglect emission of massive particles [68] as compared
to massless ones. Accordingly, we will here consider only massless particles
of the Standard Model, and, as was argued above, we neglect the emission of
gravitons. Once the black hole is small enough, or, alternatively, the
deformation parameter $l_{0}/r_{0}$ is large enough, massive particles, such
as electrons and positrons, are emitted in an ultrarelativistic mode, that is,
roughly at the same rate as massless neutrinos for each helicity.

The multiplicity factors for the 4D spherically symmetrical black holes
consist of the number of degenerate $m$-modes (which are
$m=-\ell,-\ell+1,....-1,0,1,...\ell$, that is $2\ell+1$ modes) multiplied by
the number of species of particles, which in turn depends also on the number
of polarizations and helicities of particles. Therefore, we have

$\displaystyle N_{\ell}=2(2\ell+1)\qquad(\rm Maxwell),$ (26) $\displaystyle
N_{\ell}=8\ell\qquad(\rm Dirac),$ (27)

The multiplicity factor for the Dirac field is fixed taking into account both
the “plus” and “minus” potentials related by Darboux transformations, which
leads to an isospectral problem and the same grey-body factors for both
chiralities. We will use here the “plus” potential, because the WKB results
are more accurate for that case in the Schwarzschild limit.

The energy emitted makes the black hole mass decrease at the following rate
[65]:

$\frac{dM}{dt}=-\frac{\hbar
c^{4}}{G^{2}}\frac{\alpha_{0}}{M^{2}}\approx-1.91\times 10^{29}\alpha_{0}\
\frac{\rm g}{\rm s}\cdot\bigg{(}\frac{\rm g^{2}}{M_{0}^{2}}\bigg{)},$ (28)

where we have restored the dimensional constants. Here $\alpha_{0}=dE/dt$ is
taken in units used in Table IV for a given initial mass $M_{0}$. Indeed, this
can be seen by comparison of our data in the Schwarzschild limit with [65]. A
relatively small difference with the Page’s results (less than two percents)
is due to the systematic error of the WKB method [45] and can be neglected,
taking into account that the overall effect of deviation of the energy
emission rate from the Schwarzschild limit achieves a few orders, i.e. the
relative error is a few orders smaller than the effect.

$l_{0}/M$ | Maxwell | Dirac
---|---|---
0 ($\ell=1$) | $0.0000335107$ | $0.00015963$
0 ($\ell=2$) | $6.67916\cdot 10^{-7}$ | $5.8691\cdot 10^{-6}$
0 ($\ell=3$) | $1.00693\cdot 10^{-8}$ | $1.16695\cdot 10^{-7}$
0 (total) | $0.0000341888$ | $0.000165617$
0.1 ($\ell=1$) | $0.0000323629$ | $0.000155663$
0.1 ($\ell=2$) | $6.28544\cdot 10^{-7}$ | $5.58669\cdot 10^{-6}$
0.1 ($\ell=3$) | $9.23148\cdot 10^{-9}$ | $1.08274\cdot 10^{-7}$
0.1 (total) | $0.0000330008$ | $0.00016136$
0.2 ($\ell=1$) | $0.0000290019$ | $0.000144034$
0.2 ($\ell=2$) | $5.1916\cdot 10^{-7}$ | $4.78308\cdot 10^{-6}$
0.2 ($\ell=3$) | $7.02324\cdot 10^{-9}$ | $8.55366\cdot 10^{-8}$
0.2 (total) | $0.0000295282$ | $0.000148904$
0.3 ($\ell=1$) | $0.0000236953$ | $0.00012464$
0.3 ($\ell=2$) | $3.64925\cdot 10^{-7}$ | $3.5915\cdot 10^{-6}$
0.3 ($\ell=3$) | $4.24242\cdot 10^{-9}$ | $5.53788\cdot 10^{-8}$
0.3 (total) | $0.0000240645$ | $0.000128288$
0.4 ($\ell=1$) | $0.0000169636$ | $0.000102356$
0.4 ($\ell=2$) | $2.03653\cdot 10^{-7}$ | $2.23439\cdot 10^{-6}$
0.4 ($\ell=3$) | $1.84218\cdot 10^{-9}$ | $2.70179\cdot 10^{-8}$
0.4 (total) | $0.0000171691$ | $0.000104618$
0.5 ($\ell=1$) | $9.71567\cdot 10^{-6}$ | $0.0000658481$
0.5 ($\ell=2$) | $7.69549\cdot 10^{-8}$ | $1.01073\cdot 10^{-6}$
0.5 ($\ell=3$) | $4.57759\cdot 10^{-10}$ | $8.10308\cdot 10^{-9}$
0.5 (total) | $9.79308\cdot 10^{-6}$ | $0.000066867$
0.6 ($\ell=1$) | $3.45285\cdot 10^{-6}$ | $0.000031204$
0.6 ($\ell=2$) | $1.26319\cdot 10^{-8}$ | $2.34735\cdot 10^{-7}$
0.6 ($\ell=3$) | $3.44314\cdot 10^{-11}$ | $8.70239\cdot 10^{-10}$
0.6 (total) | $3.46552\cdot 10^{-6}$ | $0.00003144$
0.7 ($\ell=1$) | $2.34399\cdot 10^{-7}$ | $4.29524\cdot 10^{-6}$
0.7 ($\ell=2$) | $1.18349\cdot 10^{-10}$ | $4.79822\cdot 10^{-9}$
0.7 ($\ell=3$) | $4.36355\cdot 10^{-14}$ | $2.55798\cdot 10^{-12}$
0.7 (total) | $2.34517\cdot 10^{-7}$ | $4.30004\cdot 10^{-6}$
Table 4: Energy emission rate $dE/dt$ for the Bardeen black hole

From Table 4, it is evident that the energy emission rate $dE/dt$ decreases by
three orders of magnitude as the deformation parameter $l_{0}$ approaches the
quasi-extreme limit. An opposite effect has been observed in another model of
quantum correction [55], based on the non-perturbative Kazakov-Solodukhin
solution [69], where the Hawking temperature of the quantum corrected black
hole is the same as for the Schwarzschild case. The effect observed in our
current research can be attributed to two factors. The primary and dominant
factor is the decrease in the Hawking temperature of the black hole when the
quantum correction is introduced (see fig. 6). The second factor is the grey-
body coefficient. Analyzing the shape of the effective potentials of the
Bardeen black hole shown in fig. 1, it is evident that the height of the
potential barrier increases with the inclusion of the quantum correction.
Consequently, a taller effective potential reflects a larger portion of the
initial radiation flow, contributing to the suppression of Hawking radiation.

From fig. 8, we observe that while the emission rate of the Maxwell field per
unit frequency decreases significantly when the near-extreme quantum
correction is applied, the peak of the radiation occurs at approximately the
same frequency $\omega$. Similar behavior is observed for the Dirac particles.

We have calculated the contributions of the first five multipoles $\ell$.
However, it is noteworthy that the contributions of only the first two
multipoles are significant for the total energy emission rate. Therefore,
higher multipoles can be safely neglected.

##  VI Conclusions

The Bardeen spacetime characterizes a regular black hole featuring a de Sitter
core in place of a central singularity. By interpreting this spacetime as the
metric of a quantum-corrected black hole within the framework of string
T-duality [29], we conducted a comprehensive analysis of the quasinormal
spectrum for scalar, electromagnetic, and Dirac fields in this background. Our
investigation revealed that some previous studies focusing on the fundamental
mode exhibited inaccuracies, allowing for a relative error of up to $10\%$.
However, our primary findings pertain to the calculations of overtones, which
probe the geometry near the event horizon and thus play a crucial role in
discerning potential quantum corrections through observations of classical
radiation emitted by black holes. Notably, we observed that the initial few
overtones deviate from their Schwarzschild values at a significantly higher
rate compared to the fundamental mode. This deviation stems from the quantum
correction-induced deformation of the Schwarzschild spacetime.

Additionally, we derived an analytical expression for the quasinormal modes in
the eikonal regime and established the decay law at asymptotically late times.

Furthermore, we computed the grey-body factors and energy emission rates for
massless fields surrounding the Bardeen black hole. Our analysis revealed that
the quantum correction leads to a significant suppression of the Hawking
radiation intensity by several orders of magnitude. This suppression arises
from two factors: a decrease in the Hawking temperature and a smaller grey-
body factor.

Our study on the Hawking evaporation of the Bardeen black hole holds potential
significance in estimating the number of primordial black holes formed during
the early Universe. Specifically, for $M\lesssim 10^{15}g$, the evaporation of
primordial black holes is predominantly constrained by big bang
nucleosynthesis [70]. Therefore, the quantum correction examined in this study
could serve as one of the mechanisms, alongside those considered in [71], that
chnages the speed of the black hole evaporation, potentially influencing the
constraints imposed by big bang nucleosynthesis on the abundance of primordial
black holes.

There are several avenues for extending our work. Firstly, one could
investigate the Hawking radiation of massive particles to obtain a more
accurate understanding of the radiation process during the later stages of
black hole evaporation. Secondly, the decreasing real oscillation frequencies
of the overtones at moderate values of $l_{0}$ suggest that, at a sufficiently
large deformation parameter, some of these overtones may become purely
imaginary, corresponding to non-oscillatory modes. However, due to the slow
convergence of the procedure in this regime, we were unable to explore this
phenomenon within a reasonable computation time. It is possible that
alternative methods, such as the Bernstein polynomial method [72, 73], which
are more effective at finding purely imaginary modes, could offer a solution
to this challenge.

##  VII Appendix

The expansion of the metric function $f(r)$ in terms of small $l_{0}$ has the
following form:

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}\right)+\frac{3l_{0}^{2}M}{r^{3}}-\frac{15l_{0}^{4}M}{4r^{5}}+\frac{35l_{0}^{6}M}{8r^{7}}$
(29) $\displaystyle-$
$\displaystyle\frac{315l_{0}^{8}M}{64r^{9}}+\frac{693l_{0}^{10}M}{128r^{11}}-\frac{3003l_{0}^{12}M}{512r^{13}}+\frac{6435l_{0}^{14}M}{1024r^{15}}$
$\displaystyle-$
$\displaystyle\frac{109395l_{0}^{16}M}{16384r^{17}}+\frac{230945l_{0}^{18}M}{32768r^{19}}-\frac{969969l_{0}^{20}M}{131072r^{21}}$
$\displaystyle+$
$\displaystyle\frac{2028117l_{0}^{22}M}{262144r^{23}}-\frac{16900975l_{0}^{24}M}{2097152r^{25}}+O\left(l_{0}^{26}\right).$

###### Acknowledgements.

The authors would like to acknowledge useful discussions with Alexander
Zhidenko. They would also like to acknowledge the support of the Research
Centre for Theoretical Physics and Astrophysics, Institute of Physics,
Silesian University in Opava. D. O. thanks the financial support of the
internal grant of the Silesian University in Opava SGS/30/2023. B. A.
acknowledges support by grants F-FA- 2021-432, and MRB-2021-527 from the
Agency for Innovative Development of Uzbekistan and Erasmus+ project towards
his stay at the Silesian University in Opava.

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