# Symmetric matrix representations of truncated Toeplitz operators on finite
dimensional spaces

Ryan O’Loughlin
E-mail address<EMAIL_ADDRESS>School of Mathematics, University of
Leeds, Leeds, LS2 9JT, U.K.

###### Abstract

In this paper, we study matrix representations of truncated Toeplitz operators
with respect to orthonormal bases which are invariant under a canonical
conjugation map. In particular, we determine necessary and sufficient
conditions for when a 3-by-3 symmetric matrix is the matrix representation of
a truncated Toeplitz operator with respect to a given conjugation invariant
orthonormal basis. We specialise our result to the case when the conjugation
invariant orthonormal basis is a modified Clark basis. As a corollary to this
specialisation, we answer a previously stated open conjecture in the negative,
and show that not every unitary equivalence between a complex symmetric matrix
and a truncated Toeplitz operator arises from a modified Clark basis
representation. Finally, we show that a given 3-by-3 symmetric matrix is the
matrix representation of a truncated Toeplitz operator with respect to a
conjugation invariant orthonormal basis if and only if a specified system of
polynomial equations is satisfied with a real solution.

Keywords: Hardy space, truncated Toeplitz operator, model space.
MSC: 30H10, 47B35, 46E20.

##  1 Introduction

Since the seminal work of Sarason in 2007 [23], there has been a surge of
interest in truncated Toeplitz operators (which we will abbreviate to TTOs)
[3, 2, 22, 15, 5, 7, 6, 24, 20, 21]. One area of interest comes from the link
between TTOs and complex symmetric operators.

In finite dimensions, complex symmetric operators are the operators which are
unitarily equivalent to a complex symmetric matrix. Complex symmetric
operators have wide reaching applications in numerous fields such as quantum
reaction dynamics, electrical power modelling, graph theory, thermoelastic
wave propagation, low-dimensional symplectic gravity models, atomic collision
theory as well as many others. We refer the reader to [17] and references
thereafter for a detailed discussion of these applications.

One noteworthy domain where complex symmetric operators appear naturally is
non-Hermitian Quantum Physics. Non-Hermitian Quantum Physics does not require
that the Hamiltonian be a self adjoint operator, but that the Hamiltonian is a
symmetric operator which commutes with another so called $\mathcal{PT}$
operator. We refer the reader to [4] for a introductory background on this
topic, which clearly demonstrates how complex symmetric operators appear in
this field. From an operator theoretic point of view, when one wants to
decompose a self adjoint operator (or Hermitian Hamiltonian in the Quantum
Physics terminology) one can use the celebrated spectral theorem to show such
an operator is unitarily equivalent to a multiplication operator. However,
when one considers non-Hermitian Quantum Physics, there is no comparable
theorem which decomposes symmetric operators. Indeed, it was asked in [12] if
one can develop a model theory for complex symmetric operators and it was
noted that TTOs may play a role in this endeavour.

It is well known that for an inner function $\theta$, every TTO on the model
space $K_{\theta}$ is $C$-symmetric with the canonical conjugation
$C=C_{\theta}$, where $C_{\theta}:K_{\theta}\to K_{\theta}$ is defined by
$C_{\theta}(f)=\theta\overline{zf}$. As a result of this fact, one can deduce
every TTO is unitarily equivalent to a complex symmetric matrix (see Section 2
for details). As part of the ongoing investigation into whether TTOs serve as
a model operator for the complex symmetric operators, an open question posed
in [18] asks whether every complex symmetric matrix is unitarily equivalent to
a direct sum of TTOs. The study of the unitary equivalence classes of complex
symmetric matrices has generated research interest from fields within, and
outside of, Operator Theory [1, 16, 25].

There is a growing list of complex symmetric operators which have shown to be
unitarily equivalent to a TTO. Section 5 of [7] shows normal operators, rank
one operators and $2$-by-$2$ matrices are all unitarily equivalent to a TTO,
and the $k$-fold inflation of a TTO for $k=1,2...,\infty$ was shown to be
unitarily equivalent to a TTO in [24].

When one wants to obtain a symmetric matrix representation of a TTO the
simplest way of doing this is to compute the matrix representation of the TTO
with respect to a $C_{\theta}$-invariant orthonormal basis (or
$C_{\theta}$-real basis for short). The most notable class of
$C_{\theta}$-real bases are the modified Clark bases (see Section 8 in [11]).

In this vein, the purpose of this paper is to study the matrix representations
of TTOs with respect to a $C_{\theta}$-real basis. In particular, we determine
a necessary and sufficient condition for when a given 3-by-3 symmetric matrix
is the matrix representation of a TTO with respect to some fixed
$C_{\theta}$-real basis. We specialise this result to the case when the
$C_{\theta}$-real basis is a modified Clark basis, and as a corollary to this,
we answer the following question in the negative.

######  Question 1.1.

Suppose that $M$ is a complex symmetric matrix. If $M$ is unitarily equivalent
to a TTO, does there exist an inner function $u$, and a modified Clark basis
for the model space ${K}_{u}$ such that $M$ is the matrix representation of a
TTO on $K_{u}$ with respect to this basis? In other words, do all such unitary
equivalences between complex symmetric matrices and TTOs arise from modified
Clark basis representations?

Question 1.1 was posed in Section 7 of [15], and again in Section 9 of [18].

In Section 2 we provide the necessary preliminary background in order to study
the matrix representations of TTOs with respect to $C_{\theta}$-real bases. In
particular, we include a brief discussion on the construction of the modified
Clark bases.

We split Section 3 into two subsections. In the first subsection, we show that
a given 3-by-3 symmetric matrix is the the matrix representation of a TTO on
$K_{\theta}$ with respect to a given $C_{\theta}$-real basis for $K_{\theta}$
if and only if a corresponding matrix has determinant 0. In the second
subsection we specialise the results of the first subsection, and show a given
3-by-3 symmetric matrix is the the matrix representation of a TTO on
$K_{\theta}$ with respect to a given modified Clark basis for $K_{\theta}$ if
and only if the off diagonal entries satisfy a particular linear relation.
With this condition, we then answer Question 1.1 in the negative.

In Section 4 we show that every matrix representation of a TTO with respect to
some $C_{\theta}$-real basis is orthogonally equivalent to the matrix
representation of the same TTO with respect to a modified Clark basis. With
this realisation we can then show that a given 3-by-3 symmetric matrix is the
matrix is the matrix representation of a TTO on a model space $K_{\theta}$
with respect to a $C_{\theta}$-real basis for $K_{\theta}$ if and only if a
specified system of polynomial equations is satisfied with a real solution.

##  2 Preliminaries

We denote the open unit disc by $\mathbb{D}$ and we denote the unit circle by
$\mathbb{T}$. We denote the classical Hardy space on the unit circle by
$H^{2}$ and we refer the reader to [10, 19] for a detailed background on the
Hardy space. Throughout we let $\theta$ be an inner function (i.e. a function
in $H^{2}$ which is unimodular on $\mathbb{T}$). A famous theorem of Beurling
characterises all the forward shift invariant subspaces of $H^{2}$. It states
that all nontrivial closed forward shift invariant subspaces are of the form
$IH^{2}$, where $I$ is some inner function. The backward shift is the adjoint
of the forward shift on $H^{2}$ and is given by the map
$f(z)\mapsto\frac{f-f(0)}{z}$. It follows from Beurling’s Theorem that all
nontrivial closed backward shift invariant subspaces of $H^{2}$ are of the
form $K_{I}:=H^{2}\ominus IH^{2}=H^{2}\cap I\overline{zH^{2}}$, where the
multiplication $I\overline{zH^{2}}$ is understood as functions on the unit
circle. The space $K_{I}$ is referred to as a model space, and we refer the
reader to [8] for a detailed background on model spaces.

Model spaces are reproducing kernel Hilbert spaces, where the reproducing
kernel at $\lambda\in\mathbb{D}$ is given by

$k_{\lambda}(z)=\frac{1-\overline{\theta(\lambda)}\theta(z)}{1-\bar{\lambda}z},$
(1)

(see Section 5.5 of [13]). A model space $K_{\theta}$ is finite dimensional if
and only if $\theta$ is a finite Blaschke product (Proposition 5.19 in [13]),
and in this case the order of $\theta$ (i.e. the number of zeros of $\theta$
in the disc) is equal to the dimension of $K_{\theta}$. In the case when
$\theta$ is a finite Blaschke product, for every $\zeta\in\mathbb{T}$, every
$f\in K_{\theta}$ has a nontangential limit at $\zeta$, and so in this case by
taking radial limits we see every $f\in K_{\theta}$ can be defined everywhere
on $\mathbb{T}$. When every $f\in K_{\theta}$ is defined at a point
$\zeta\in\mathbb{T}$, we can consider the reproducing kernel at $\zeta$, which
is given by

$k_{\zeta}(z)=\frac{1-\overline{\theta(\zeta)}\theta(z)}{1-\bar{\zeta}z}\in
K_{\theta}.$

For vectors $x,y\in K_{\theta}$ we use the notation $x\otimes y:K_{\theta}\to
K_{\theta}$ to mean the map $f\mapsto\langle f,y\rangle x$.

If $\mathcal{H}$ is a Hilbert space, then we say that $C$ is a conjugation
operator on $\mathcal{H}$ if the following conditions hold:
(a) $C$ is antilinear:

$C\left(a_{1}f_{1}+a_{2}f_{2}\right)=\overline{a_{1}}Cf_{1}+\overline{a_{2}}Cf_{2}$

for all $a_{1},a_{2}\in\mathbb{C}$ and $f_{1},f_{2}\in\mathcal{H}$.
(b) $C$ is isometric:

$\langle Cf,Cg\rangle=\langle g,f\rangle$ (2)

for all $f,g\in\mathcal{H}$.
(c) $C$ is involutive:

$C^{2}=I.$ (3)

An operator $F$ on a Hilbert space $\mathcal{H}$ is said to be $C$-symmetric
if $CFC=F^{*}$, and an operator is complex symmetric if it is $C$-symmetric
with respect to some conjugation map $C$.

As previously mentioned, the space $K_{\theta}$ carries a canonical
conjugation operator given by $C_{\theta}f=\theta\overline{zf}$. This model
space conjugation operator is discussed at length in [11]. Furnishing an
orthonormal basis for $K_{\theta}$ which is invariant under $C_{\theta}$ turns
out to be a non-trivial task. Such a basis is called a $C_{\theta}$-real
basis, and Lemma 2.6 in [11] shows every model space $K_{\theta}$ admits a
$C_{\theta}$-real basis.

Although a detailed explanation of how to construct a $C_{\theta}$-real basis
for a model space is given by Section 8 in [11], we outline the approach here.
In the case when the inner function $\theta$ is a finite Blaschke product, a
$C_{\theta}$-real basis is given by the unit eigenvectors of a generalised
Clark operator $U_{t,\alpha}:K_{\theta}\to K_{\theta}$,

$U_{t,\alpha}=S_{t}+(\alpha+\theta(t))\left(k_{t}\otimes
C_{\theta}k_{t}\right),$ (4)

where $t\in\mathbb{D}$, $\alpha\in\mathbb{T}$ and
$S_{t}f=P_{\theta}\left(\frac{z-t}{1-\bar{t}z}f\right)$ for
$P_{\theta}:L^{2}\to K_{\theta}$, the orthogonal projection. We refer to a
$C_{\theta}$-real basis of this form as a modified Clark basis. Explicitly the
unit eigenvectors of $U_{t,\alpha}$ are given by the formula

$\textbf{cb}_{i}(z)=\left(\overline{\eta_{i}}\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}\right)^{\frac{1}{2}}k_{\eta_{i}},$
(5)

where we take the convention that the above square root is defined by taking
$\delta_{1}=\arg(\overline{\eta_{i}})\in[0,2\pi)$,
$\delta_{2}=\arg\left(\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}\right)\in[0,2\pi)$
and setting
$\arg\left(\left(\overline{\eta_{i}}\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}\right)^{\frac{1}{2}}\right)=\frac{\delta_{1}+\delta_{2}}{2}$
and where $\eta_{i}$ are the points such that

$\theta\left(\eta_{i}\right)=\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}.$
(6)

If the order of $\theta$ is $n$, then Theorem 3.4.10 in [14] ensures that
there are precisely $n$ distinct $\eta_{i}$ satisfying (6) and each $\eta_{i}$
lies on $\mathbb{T}$. We remark that either choice of the square root will
give a $C_{\theta}$-real basis, however our choice of square root will make
our working in later sections easier to follow.

For $\phi\in L^{2}$, we densely define the truncated Toeplitz operator (which
we have abbreviated to TTO) $A_{\phi}$ on bounded functions from $K_{\theta}$
by

$A_{\phi}(f)=P_{\theta}(\phi f),\hskip 56.9055ptf\in K_{I}\cap L^{\infty}.$

We will often suppress the $\phi$ subscript and just write $A$ instead of
$A_{\phi}$. Although one can study unbounded TTOs, in this article we will
only consider TTOs which have a bounded extension to $K_{\theta}$.

Throughout we denote $\mathcal{T}_{\theta}$ to be the space of all TTOs on the
model space $K_{\theta}$. For $A\in\mathcal{T}_{\theta}$ and a
$C_{\theta}$-real basis for $K_{\theta}$ given by $v_{j}$ where $j=1,...,\dim
K_{\theta}$, we denote $[A]_{v_{j}}$ to be the matrix representation of $A$
with respect to $v_{j}$. Lemma 2.7 in [11] shows that the matrix
representation of a $C$-symmetric operator with respect to a $C$-real basis is
a complex symmetric matrix (i.e. a complex matrix which is self transpose) and
so as $A\in\mathcal{T}_{\theta}$ is $C$-symmetric with respect to the
canonical conjugation $C_{\theta}$ (see Lemma 2.1 in [23]), we must have that
$[A]_{{v}_{j}}$ is a complex symmetric matrix. For this reason, when we are
trying to realise which matrices are the matrix representation of a TTO with
respect to a $C_{\theta}$-real basis we in fact only need to consider which
complex symmetric matrices are the representation of a TTO with respect to a
$C_{\theta}$-real basis.

Throughout this article we will mostly only consider the model spaces
$K_{\theta}$ where $\theta$ is a Blaschke product of order 3. Thus in this
case the dimension of $K_{\theta}$ will be 3, and so the the matrix
representation of a $A\in\mathcal{T}_{\theta}$ with respect an orthonormal
basis will be a 3-by-3 matrix.

##  3 Matrix representations of TTOs with respect to $C_{\theta}$-real bases

###  3.1 General $C_{\theta}$-real basis

The following is a generalisation of Theorem 7.1 (b) in [23].

######  Lemma 3.1.

Let $\theta$ be a Blaschke product of order 3, let
$t_{1},t_{2},t_{3}\in\mathbb{T}$ and $\lambda_{4},\lambda_{5}\in\mathbb{D}$
then

$k_{t_{1}}\otimes k_{t_{1}}\quad k_{t_{2}}\otimes k_{t_{2}}\quad
k_{t_{3}}\otimes k_{t_{3}}\quad k_{\lambda_{4}}\otimes
C_{\theta}k_{\lambda_{4}}\quad k_{\lambda_{5}}\otimes
C_{\theta}k_{\lambda_{5}}$ (7)

is a basis for $\mathcal{T}_{\theta}$.

###### Proof.

Theorem 5.1 in [23] shows that the specified functions are in indeed in
$\mathcal{T}_{\theta}$ and Theorem 7.1(a) in [23] shows the dimension of
$\mathcal{T}_{\theta}$ is 5, so it suffices to show that the functions given
by (7) are linearly independent.

We first show that any 3 distinct reproducing kernels in $K_{\theta}$ are
linearly independent. The well known result, which can be found as Corollary
5.18 in [13], shows that if the zeroes of $\theta$ are given by $z_{n}$
(allowing for repeated zeroes) then

$K_{\theta}=\left\\{\frac{a_{0}+a_{1}z+a_{2}z^{2}}{(1-\overline{z_{1}}z)(1-\overline{z_{2}}z)(1-\overline{z_{3}}z)}\,:\,a_{1},a_{2},a_{3}\in\mathbb{C}\right\\}.$

Thus for any three distinct points $x_{1},x_{2},x_{3}$ in
$\mathbb{D}\cup\mathbb{T}$, we can can define a function

$f(z):=\frac{(z-x_{2})(z-x_{3})}{(1-\overline{z_{1}}z)(1-\overline{z_{2}}z)(1-\overline{z_{3}}z)}\in
K_{\theta},$

such that $f(x_{2})=f(x_{3})=0$, but $f(x_{1})\neq 0$. So if for any three
distinct points $x_{1},x_{2},x_{3}$ in $\mathbb{D}\cup\mathbb{T}$ we have

$\sum_{i=1}^{3}\mu_{i}k_{x_{i}}=0,$

for some constants $\mu_{i}$, then for the $f$ defined above we have

$0=\langle
f,\sum_{i=1}^{3}\mu_{i}k_{x_{i}}\rangle=\overline{\mu_{1}}f(x_{1}),$

which implies $\mu_{1}=0$. We can repeat a similar argument to show
$\mu_{2},\mu_{3}$ are both 0.

So if we have complex constants $\alpha_{1},...,\alpha_{5}$ such that

$\sum_{i=1}^{3}\alpha_{i}(k_{t_{i}}\otimes
k_{t_{i}})+\sum_{i=4}^{5}\alpha_{i}(k_{\lambda_{i}}\otimes
C_{\theta}k_{\lambda_{i})}=0,$

then for $f\in K_{\theta}$ such that $f(t_{2})=f(t_{3})=0$ and $f(t_{1})\neq
0$, we have

$0=\left(\sum_{i=1}^{3}\alpha_{i}(k_{t_{i}}\otimes
k_{t_{i}})+\sum_{i=4}^{5}(\alpha_{i}k_{\lambda_{i}}\otimes
C_{\theta}k_{\lambda_{i}})\right)(f)={\alpha_{1}}f(t_{1})k_{t_{1}}+\sum_{i=4}^{5}{\alpha_{i}}k_{\lambda_{i}}\langle
f,C_{\theta}k_{\lambda_{i}}\rangle.$ (8)

Now linear independence of $k_{t_{1}},k_{\lambda_{4}},k_{\lambda_{5}}$
guarantees $\alpha_{1}=0$.

A similar argument shows $\alpha_{2},\alpha_{3}$ are $0$ and from here the
linear independence of $k_{\lambda_{4}}\otimes
C_{\theta}k_{\lambda_{4}},k_{\lambda_{5}}\otimes C_{\theta}k_{\lambda_{5}}$
,which is shown by Theorem 7.1 in [23], ensures $\alpha_{4},\alpha_{5}$ are 0.
(Alternatively one can show $\alpha_{4},\alpha_{5}$ are $0$, by noticing that
for $f\in K_{\theta}$ such that $f(\lambda_{4})=0$, $f(\lambda_{5})\neq 0$ we
have $0=\left(\sum_{i=4}^{5}(\alpha_{i}k_{\lambda_{i}}\otimes
C_{\theta}k_{\lambda_{i}})\right)(C_{\theta}(f))=\alpha_{5}\overline{f(\lambda_{5})}k_{\lambda_{5}}$,
and so $\alpha_{5}=0$.)

∎

######  Theorem 3.2.

Let $\theta$ be a Blaschke product of order 3, let ${v}_{1},{v}_{2},{v}_{3}$
be a $C_{\theta}$-real basis for $K_{\theta}$, let $t_{1},t_{2},t_{3}$ be
distinct points on $\mathbb{T}$ and $\lambda_{4},\lambda_{5}$ be distinct
points in $\mathbb{D}$. Then a complex symmetric matrix

$S=\begin{pmatrix}s_{1}&s_{4}&s_{5}\\\ s_{4}&s_{2}&s_{6}\\\
s_{5}&s_{6}&s_{3}\end{pmatrix},$

is a matrix representation of a TTO on $K_{\theta}$ with respect to $v_{j}$ if
and only if

$\det[c_{1},c_{2},c_{3},c_{4},c_{5},\tilde{S}]=0,$

where

$c_{i}=\begin{pmatrix}v_{1}(t_{i})\overline{v_{1}(t_{i})}\\\
v_{2}(t_{i})\overline{v_{2}(t_{i})}\\\ v_{3}(t_{i})\overline{v_{3}(t_{i})}\\\
v_{1}(t_{i})\overline{v_{2}(t_{i})}\\\ v_{1}(t_{i})\overline{v_{3}(t_{i})}\\\
v_{2}(t_{i})\overline{v_{3}(t_{i})}\end{pmatrix}\text{ for }i=1,2,3\quad
c_{i}=\begin{pmatrix}\overline{v_{1}(\lambda_{i})v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})v_{2}(\lambda_{i})}\\\
\overline{v_{3}(\lambda_{i})v_{3}(\lambda_{i})}\\\
\overline{v_{1}(\lambda_{i})v_{2}(\lambda_{i})}\\\
\overline{v_{1}(\lambda_{i})v_{3}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})v_{3}(\lambda_{i})}\end{pmatrix}\text{ for
}i=4,5,$ (9)

and

$\tilde{S}=\begin{pmatrix}s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\\\ s_{5}\\\
s_{6}\end{pmatrix}.$ (10)

######  Remark.

This theorem may easily be altered so that
$t_{1},t_{2},t_{3},\lambda_{4},\lambda_{5}$ may take any values in
$\mathbb{D}\,\cup\,\mathbb{T}$. The reason we phrase the theorem so that
$t_{1},t_{2},t_{3}\in\mathbb{T}$ and $\lambda_{4},\lambda_{5}\in\mathbb{D}$ is
because this will allow us to easily specialise the result to the case when
$v_{j}$ is a modified Clark basis in the next subsection.

###### Proof.

We first compute $[k_{t_{i}}\otimes k_{t_{i}}]_{v_{j}}$ for $i=1,2,3$ and
$[k_{\lambda_{i}}\otimes C_{\theta}k_{\lambda_{i}}]_{v_{j}}$ for $i=4,5$. For
$i=1,2,3$ we have

$k_{t_{i}}=\sum_{j=1}^{3}\langle k_{t_{i}},v_{j}\rangle
v_{j}=\sum_{j=1}^{3}\overline{v_{j}(t_{i})}v_{j},$

and so

$[k_{t_{i}}\otimes
k_{t_{i}}]_{v_{j}}=\begin{pmatrix}v_{1}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}&\,v_{2}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}&\,v_{3}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}\end{pmatrix}.$

Similarly for $i=4,5$ as each $v_{j}$ is $C_{\theta}$-invariant using
properties (2) (3) we have $\langle
v_{j},C_{\theta}k_{\lambda_{i}}\rangle=\overline{v_{j}(\lambda_{i})}$, and so

$[k_{\lambda_{i}}\otimes
C_{\theta}k_{\lambda_{i}}]_{v_{j}}=\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{2}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{3}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}\end{pmatrix}.$

By the previous lemma we can say that $[k_{t_{i}}\otimes k_{t_{i}}]_{v_{j}}$
for $i=1,2,3$ $[k_{\lambda_{i}}\otimes C_{\theta}k_{\lambda_{i}}]_{v_{j}}$ for
$i=4,5$ is a basis for the space of matrix representations of
$\mathcal{T}_{\theta}$ with respect to $v_{j}$ (i.e. a basis for
$[\mathcal{T}_{\theta}]_{v_{j}}=\\{[A]_{v_{j}}\text{ such that
}A\in\mathcal{T}_{\theta}\\}$).

So for a matrix

$S=\begin{pmatrix}s_{1}&s_{4}&s_{5}\\\ s_{4}&s_{2}&s_{6}\\\
s_{5}&s_{6}&s_{3}\end{pmatrix},$

there exists an $A\in\mathcal{T}_{\theta}$ such that $[A]_{v_{j}}=S$ if and
only if there exists $\mu_{1},...,\mu_{5}\in\mathbb{C}$ such that

$\sum_{i=1}^{3}\mu_{i}\begin{pmatrix}v_{1}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}&v_{2}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}&v_{3}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}\end{pmatrix}$ (11) $+\newline
\sum_{i=4}^{5}\mu_{i}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{2}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{3}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}\end{pmatrix}=S.$

As noted in the previous section, each summand in the above expression must be
a complex symmetric matrix. So using the symmetry of the matrices, we see when
considering whether there exists $\mu_{1},...,\mu_{5}$ such that (11) holds,
we in fact only need to consider whether there are $\mu_{1},...,\mu_{5}$ such
that

$\sum_{i=1}^{3}\mu_{i}\begin{pmatrix}v_{1}(t_{i})\begin{pmatrix}\overline{v_{1}(t_{i})}\\\
\overline{v_{2}(t_{i})}\\\ \overline{v_{3}(t_{i})}\\\
\end{pmatrix}&v_{2}(t_{i})\begin{pmatrix}0\\\ \overline{v_{2}(t_{i})}\\\
\overline{v_{3}(t_{i})}\\\ \end{pmatrix}&v_{3}(t_{i})\begin{pmatrix}0\\\ 0\\\
\overline{v_{3}(t_{i})}\\\ \end{pmatrix}\end{pmatrix}$ (12) $+\newline
\sum_{i=4}^{5}\mu_{i}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\begin{pmatrix}\overline{v_{1}(\lambda_{i})}\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{2}(\lambda_{i})}\begin{pmatrix}0\\\
\overline{v_{2}(\lambda_{i})}\\\ \overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}&\overline{v_{3}(\lambda_{i})}\begin{pmatrix}0\\\ 0\\\
\overline{v_{3}(\lambda_{i})}\\\
\end{pmatrix}\end{pmatrix}=\begin{pmatrix}s_{1}&0&0\\\ s_{4}&s_{2}&0\\\
s_{5}&s_{6}&s_{3}\end{pmatrix}.$

Rewriting this in terms of column vectors, we see that this is actually
equivalent to the existence of $\mu_{1},...,\mu_{5}\in\mathbb{C}$ such that
$\sum_{i=1}^{5}\mu_{i}c_{i}=\tilde{S}$, or equivalently (as
$c_{1},c_{2},c_{3},c_{4},c_{5}$ are linearly independent)

$\det[c_{1},c_{2},c_{3},c_{4},c_{5},\tilde{S}]=0.$

∎

###  3.2 When the $C_{\theta}$-real basis is a modified Clark basis

######  Theorem 3.3.

Let $\theta$ be a Blaschke product of order 3 and $\textbf{cb}_{i}$ be
modified Clark basis for $K_{\theta}$. Then a complex symmetric matrix

$S=\begin{pmatrix}s_{1}&s_{4}&s_{5}\\\ s_{4}&s_{2}&s_{6}\\\
s_{5}&s_{6}&s_{3}\end{pmatrix},$

is a matrix representation of a TTO on $K_{\theta}$ with respect to
$\textbf{cb}_{i}$ if and only if

$s_{6}=\frac{1}{\eta_{3}-\eta_{2}}\left(s_{4}\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{3}}^{\frac{1}{2}}}\right)(\eta_{1}-\eta_{2})+s_{5}\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{2}}^{\frac{1}{2}}}\right)(\eta_{3}-\eta_{1})\right),$
(13)

where the $\eta_{i}$ are given by (6) and where
$\overline{\eta_{j}}^{\frac{1}{2}}$ is defined such that if
$\arg(\overline{\eta_{j}}):=\gamma\in[0,2\pi)$, then
$\arg(\overline{\eta_{j}}^{\frac{1}{2}})=\frac{\gamma}{2}\in[0,\pi)$.

###### Proof.

The proof of this result just involves specialising Theorem 3.2. With the same
notations as Theorem 3.2, we set $v_{i}=\mathbf{cb}_{i}$, set $t_{i}=\eta_{i}$
for $i=1,2,3$, set $\lambda_{4}=0$ and keep
$\mathbb{D}\ni\lambda_{5}:=\lambda\neq 0$ arbitrary. Then for $i\neq j$, as
$\textbf{cb}_{i}(\eta_{j})$ is a constant multiplied by
$\langle\textbf{cb}_{i},\textbf{cb}_{j}\rangle=0$, we must have
$\textbf{cb}_{i}(\eta_{j})=0$. So $[c_{1},c_{2},c_{3},c_{4},c_{5},\tilde{S}]$
simplifies to become a matrix of the form

$\begin{pmatrix}D&A_{1}\\\ 0&A_{2}\end{pmatrix},$

where $D$ is a 3-by-3 diagonal matrix with each diagonal entry non-zero, 0 is
the 3-by-3 zero matrix and $A_{1},A_{2}$ are $3$-by-$3$ matrices. So in this
case $\det[c_{1},c_{2},c_{3},c_{4},c_{5},\tilde{S}]=\det D\det A_{2}$, and so
$\det[c_{1},c_{2},c_{3},c_{4},c_{5},\tilde{S}]=0$ if and only if $\det
A_{2}=0$. We now argue $\det A_{2}=0$ if and only if (13) holds.

If we write $\textbf{cb}_{i}=b_{i}k_{\eta_{i}}$ for $b_{i}\in\mathbb{C}$
(where $b_{i}$ is explicitly given in (5)), then
$\overline{\textbf{cb}_{i}(\lambda)}=\overline{b_{i}}\overline{\langle
k_{\eta_{i}},k_{\lambda}\rangle}=\overline{b_{i}}k_{\lambda}(\eta_{i})$, and
so we can compute $A_{2}$ to be given by

$\begin{pmatrix}x_{4}&y_{4}&s_{4}\\\ x_{5}&y_{5}&s_{5}\\\
x_{6}&y_{6}&s_{6}\end{pmatrix},$

where

$\displaystyle
x_{4}={\overline{b_{2}}{k_{0}(\eta_{2})}}{{\overline{b_{1}}{k_{0}(\eta_{1})}}},\quad
x_{5}={\overline{b_{3}}{k_{0}(\eta_{3})}{{\overline{b_{1}}{k_{0}(\eta_{1})}}}},$
(14) $\displaystyle
x_{6}={\overline{b_{3}}{k_{0}(\eta_{3})}}{{\overline{b_{2}}{k_{0}(\eta_{2})}}},\quad
y_{4}={\overline{b_{2}}{k_{\lambda}(\eta_{2})}}{{\overline{b_{1}}{k_{\lambda}(\eta_{1})}}},$
(15) $\displaystyle
y_{5}={\overline{b_{3}}{k_{\lambda}(\eta_{3})}}{{\overline{b_{1}}{k_{\lambda}(\eta_{1})}}},\quad
y_{6}={\overline{b_{3}}{k_{\lambda}(\eta_{3})}}{{\overline{b_{2}}{k_{\lambda}(\eta_{2})}}}.$
(16)

For ease of notation we define the constant
$B_{\lambda}=1-\overline{\theta(\lambda)}\theta(\eta_{1})$. Now noting that as
$\theta(\eta_{1})=\theta(\eta_{2})=\theta(\eta_{3})$, by (1) we must have
$k_{0}(\eta_{1})=k_{0}(\eta_{2})=k_{0}(\eta_{3})=B_{0}$, and similarly
${k_{\lambda}(\eta_{i}))}{k_{\lambda}(\eta_{j}))}=\frac{B_{{\lambda}}^{2}}{{(1-\overline{\lambda}\eta_{i})(1-\overline{\lambda}\eta_{j})}}$,
and so from the above we obtain

$\displaystyle x_{4}=$ $\displaystyle\overline{b_{2}b_{1}}B_{0}^{2},\quad
y_{4}=\overline{b_{2}b_{1}}\frac{B_{{\lambda}}^{2}}{{(1-\overline{\lambda}\eta_{2})(1-\overline{\lambda}\eta_{1})}},$
(18) $\displaystyle x_{5}=$ $\displaystyle\overline{b_{3}b_{1}}B_{0}^{2},\quad
y_{5}=\overline{b_{3}b_{1}}\frac{B_{{\lambda}}^{2}}{{(1-\overline{\lambda}\eta_{3})(1-\overline{\lambda}\eta_{1})}},$
(19) $\displaystyle x_{6}=$ $\displaystyle\overline{b_{3}b_{2}}B_{0}^{2},\quad
y_{6}=\overline{b_{3}b_{2}}\frac{B_{{\lambda}}^{2}}{{(1-\overline{\lambda}\eta_{3})(1-\overline{\lambda}\eta_{2})}}.$
(20)

Expanding the determinant of $A_{2}=\begin{pmatrix}x_{4}&y_{4}&s_{4}\\\
x_{5}&y_{5}&s_{5}\\\ x_{6}&y_{6}&s_{6}\end{pmatrix}$ via the third column we
see that when $x_{4}y_{5}-y_{4}x_{5}\neq 0$, $\det A_{2}=0$ is equivalent to

$s_{6}=\frac{s_{4}(y_{5}x_{6}-x_{5}y_{6})+s_{5}(x_{4}y_{6}-y_{4}x_{6})}{x_{4}y_{5}-y_{4}x_{5}}.$
(21)

Using the values of $x_{4},x_{5},x_{6},y_{4},y_{5},y_{6}$ found in (18) (19)
(20) we can write

$\displaystyle y_{5}x_{6}-x_{5}y_{6}$
$\displaystyle=\overline{b_{3}}^{2}\overline{b_{1}b_{2}}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{3}}\left(\frac{1}{1-\overline{\lambda}\eta_{1}}-\frac{1}{1-\overline{\lambda}\eta_{2}}\right)$
(22)
$\displaystyle=\overline{b_{3}}^{2}\overline{b_{1}b_{2}}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{3}}\left(\frac{\overline{\lambda}(\eta_{1}-\eta_{2})}{(1-\overline{\lambda}\eta_{1})(1-\overline{\lambda}\eta_{2})}\right),$
(23) $\displaystyle x_{4}y_{6}-y_{4}x_{6}$
$\displaystyle=\overline{b_{3}b_{1}}\overline{b_{2}}^{2}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{2}}\left(\frac{1}{1-\overline{\lambda}\eta_{3}}-\frac{1}{1-\overline{\lambda}\eta_{1}}\right)$
(24)
$\displaystyle=\overline{b_{3}b_{1}}\overline{b_{2}}^{2}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{2}}\left(\frac{\overline{\lambda}(\eta_{3}-\eta_{1})}{(1-\overline{\lambda}\eta_{3})(1-\overline{\lambda}\eta_{1})}\right),$
(25) $\displaystyle x_{4}y_{5}-y_{4}x_{5}$
$\displaystyle=\overline{b_{3}}\overline{b_{1}}^{2}\overline{b_{2}}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{1}}\left(\frac{1}{1-\overline{\lambda}\eta_{3}}-\frac{1}{1-\overline{\lambda}\eta_{2}}\right)$
(26)
$\displaystyle=\overline{b_{3}}\overline{b_{1}}^{2}\overline{b_{2}}B_{\lambda}^{2}B_{0}^{2}\frac{1}{1-\overline{\lambda}\eta_{1}}\left(\frac{\overline{\lambda}(\eta_{3}-\eta_{2})}{(1-\overline{\lambda}\eta_{3})(1-\overline{\lambda}\eta_{2})}\right).$
(27)

At this stage we may now see that $x_{4}y_{5}-y_{4}x_{5}\neq 0$ as if
$x_{4}y_{5}-y_{4}x_{5}=0$, this would imply $\eta_{3}=\eta_{2}$, which can
never be the case as pointed out in Section 2. Substituting the above values
into (21) gives

$s_{6}=\frac{1}{\eta_{3}-\eta_{2}}\left(s_{4}\overline{\left(\frac{{b_{3}}}{{b_{1}}}\right)}(\eta_{1}-\eta_{2})+s_{5}\overline{\left(\frac{{b_{2}}}{{b_{1}}}\right)}(\eta_{3}-\eta_{1})\right).$
(28)

Finally as each $b_{i}\in\mathbb{T}$, for $i,j\in\\{1,2,3\\}$ we have
$\overline{\left(\frac{{b_{i}}}{{b_{j}}}\right)}=\frac{b_{j}}{b_{i}}$ and
using the explicit description of $b_{i}$ given in (5) we see

$\frac{b_{j}}{b_{i}}=\frac{\left(\overline{\eta_{j}}\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}\right)^{\frac{1}{2}}}{\left(\overline{\eta_{i}}\frac{\alpha+\theta(t)}{1+\overline{\theta(t)}\alpha}\right)^{\frac{1}{2}}}=\frac{\overline{\eta_{j}}^{\frac{1}{2}}}{\overline{\eta_{i}}^{\frac{1}{2}}},$

where $\overline{\eta_{j}}^{\frac{1}{2}}$ is defined such that if
$\arg(\overline{\eta_{j}}):=\gamma\in[0,2\pi)$, then
$\arg(\overline{\eta_{j}}^{\frac{1}{2}})=\frac{\gamma}{2}$. So (28) simplifies
to

$s_{6}=\frac{1}{\eta_{3}-\eta_{2}}\left(s_{4}\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{3}}^{\frac{1}{2}}}\right)(\eta_{1}-\eta_{2})+s_{5}\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{2}}^{\frac{1}{2}}}\right)(\eta_{3}-\eta_{1})\right).$

∎

######  Remark.

The above theorem may be viewed as a 3-by-3 version of a generalisation of
Theorem 1.15 in [9]. In [9] the authors only consider the case when $t=0$ in
(4), whereas our result holds for all $t\in\mathbb{D}$. Furthermore we note
that our condition for $S$ to be a matrix representation of a TTO with respect
to $\textbf{cb}_{i}$ eliminates all variables apart from $\eta_{i}$ for
$i=1,2,3$.

One can make a corollary to the above theorem which provides numerous examples
of matrices which show Question 1.1 to be negative.

######  Corollary 3.4.

Consider a symmetric matrix, $S$, of the form

$S=\begin{pmatrix}a&0&1\\\ 0&b&0\\\ 1&0&c\end{pmatrix},\quad
S=\begin{pmatrix}a&1&0\\\ 1&b&0\\\ 0&0&c\end{pmatrix}\text{ or
}S=\begin{pmatrix}a&0&0\\\ 0&b&1\\\ 0&1&c\end{pmatrix}$ (29)

where $a,b,c\in\mathbb{R}$. Then $S$ is unitarily equivalent to a TTO, but is
not the matrix representation of a TTO with respect to a modified Clark basis.

###### Proof.

In every case we have that $S$ is a normal matrix and so is unitarily
equivalent to a TTO by Theorem 5.4 in [7]. Clearly by the previous theorem $S$
does not satisfy (13) for any choice of distinct
$\eta_{1},\eta_{2},\eta_{3}\in\mathbb{T}$, and so $S$ can not be the matrix
representation of a TTO with respect to a modified Clark basis. ∎

Naturally, the above working leads us to consider another conjecture which is
a modification of Question 1.1.

######  Conjecture 3.1.

Suppose that $M$ is a complex symmetric matrix. If $M$ is unitarily equivalent
to a TTO, does there exist an inner function $u$, and a $C_{\theta}$-real
basis for the model space ${K}_{u}$ such that $M$ is the matrix representation
of a TTO on $K_{u}$ with respect to this basis? In other words, do all such
unitary equivalences between complex symmetric matrices and TTOs arise from
$C_{\theta}$-real matrix representations?

Finally, we remark that if the above conjecture is true then by Proposition
4.1 (see below) every complex symmetric matrix $M$ which is unitarily
equivalent to a TTO is orthogonally equivalent to the matrix representation of
a TTO with respect to a modified Clark basis.

##  4 A geometric approach

Although function theory was used to construct one class of $C_{\theta}$-real
bases, the modified Clark bases, it turns out that when we have the
description of one $C_{\theta}$-real basis for a model space $K_{\theta}$, we
can use a purely algebraic method to generate all $C_{\theta}$-real bases for
$K_{\theta}$. Furthermore, one can describe the matrix representation of a TTO
with respect to given $C_{\theta}$-real basis by relating this to the matrix
representation of the TTO with respect to a modified Clark basis. We show this
with the following proposition.

######  Remark.

The term orthogonal is typically used for a real matrix $A$ such that
$AA^{T}=A^{T}A=I_{d}$, however we note that orthogonal matrices are just the
real unitary matrices.

######  Proposition 4.1.

Let the order of $\theta$ be $n$. Given a $C_{\theta}$-real basis for
$K_{\theta}$, $v_{i}$, and a modified Clark basis for $K_{\theta}$,
$\mathbf{cb}_{i}$, there exists an orthogonal matrix $U_{r}$ such that
$v_{i}=TU_{r}e_{i}$, where $T:\mathbb{C}^{n}\to K_{\theta}$,
$e_{i}\mapsto\mathbf{cb}_{i}$. Conversely, given any orthogonal $U_{r}$ and
any modified Clark basis $\mathbf{cb}_{i}$, $TU_{r}e_{i}$ is a
$C_{\theta}$-real basis for $K_{\theta}$.

Furthermore for $v_{i}=TU_{r}e_{i}$ and $A\in\mathcal{T}_{\theta}$ we have

$[A]_{v_{i}}=U_{r}^{-1}[A]_{\mathbf{cb}_{i}}U_{r}.$

###### Proof.

Given a $C_{\theta}$-real basis $v_{i}$, and a modified Clark basis
$\mathbf{cb}_{i}$ there clearly exists constants $r_{ij}$ such that
$v_{i}=r_{i1}\mathbf{cb}_{1}+...+r_{in}\mathbf{cb}_{n}$. As $v_{i}$ and
$\mathbf{cb}_{i}$ are $C_{\theta}$-invariant, we must have

$r_{i1}\mathbf{cb}_{1}+...+r_{in}\mathbf{cb}_{n}=v_{i}=C_{\theta}(v_{i})=\overline{r_{i1}}\mathbf{cb}_{1}+...+\overline{r_{in}}\mathbf{cb}_{n},$

and so for every $i,\,j\in\\{1,...,n\\}$ we must have
$r_{ij}=\overline{r_{ij}}\in\mathbb{R}$. This establishes that
$v_{i}=TU_{r}e_{i}$, where $U_{r}$ is a real matrix. To show $U_{r}$ is
unitary we just have to observe that $TU_{r}$ is unitary, as it sends an
orthonormal basis to an orthonormal basis, and so is $T$, so $U_{r}$ must also
be unitary.

To show the converse part of the proposition one can readily see that each
$TU_{r}e_{i}$, being a real linear combination of a $C_{\theta}$-real basis,
is itself $C_{\theta}$-invariant, and $TU_{r}e_{1},...,TU_{r}e_{n}$ is an
orthonormal basis because $e_{1},...,e_{n}$ is an orthonormal basis and
$TU_{r}$ is unitary. In order to prove the second statement of the proposition
we simply observe that

$[A]_{v_{i}}=U_{r}^{-1}T^{-1}ATU_{r}=U_{r}^{-1}[A]_{\mathbf{cb}_{i}}U_{r}.$

∎

######  Remark.

By the above proposition we may observe that if a 3-by-3 matrix $M$ is such
that there exist an orthogonal matrix $U_{r}$ with $U_{r}MU_{r}^{-1}=D$ where
$D$ is a diagonal matrix with entries $d_{1},d_{2},d_{3}$, then $M$ is a
representation of a TTO with respect to a $C_{\theta}$-real basis. Indeed, if
we consider $A:=\sum_{i=1}^{3}d_{i}\mathbf{cb}_{i}\otimes\mathbf{cb}_{i}$, for
some modified Clark basis $\mathbf{cb}_{i}$ then $[A]_{\mathbf{cb}_{i}}=D$,
and so for the basis $v_{i}=TU_{r}e_{i}$ we have
$[A]_{v_{i}}=U_{r}^{-1}[A]_{\mathbf{cb}_{i}}U_{r}=U_{r}^{-1}DU_{r}=M$. With
this observation we see that although the matrices in (29) can not be a
modified Clark basis representation of a TTO, they are a matrix representation
of a TTO with respect to some $C_{\theta}$-real basis.

In this section we will exploit the above proposition and show that the
problem of determining whether a given 3-by-3 complex symmetric matrix, $M$,
is a matrix representation of a TTO on some fixed model space with respect to
a $C_{\theta}$-real basis can actually be rephrased purely as a geometric
problem.

By Proposition 4.1 we can see that a given complex symmetric matrix

$M=\begin{pmatrix}m_{1}&m_{4}&m_{5}\\\ m_{4}&m_{2}&m_{6}\\\
m_{5}&m_{6}&m_{3}\end{pmatrix},$

is a matrix representation of a TTO on $K_{\theta}$ with respect to a
$C_{\theta}$-real basis if and only if for a modified Clark basis for
$K_{\theta}$, $\mathbf{cb}_{i}$, there exists a orthogonal matrix $U_{r}$ and
a $A\in\mathcal{T}_{\theta}$ such that

$U_{r}MU_{r}^{T}=[A]_{\mathbf{cb}_{i}}.$ (30)

In fact, this is actually equivalent to the existence of a special orthogonal
matrix $U_{s}$ such that $U_{s}MU_{s}^{T}=[A]_{\mathbf{cb}_{i}}$, as if (30)
holds for some orthogonal $U_{r}$ with $\det U_{r}=-1$ then setting
$U_{s}=-U_{r}$ we see $U_{s}MU_{s}^{T}=[A]_{\mathbf{cb}_{i}}$.

Given matrices $M$ and $S$, when trying to solve a matrix equation of the form
$XMX^{T}=S$ for $X$, we may actually rephrase this equation as a problem in
algebraic geometry. In the case of (30) we have the following.

######  Theorem 4.2.

Let $\theta$ be a Blaschke product of order 3, let $\mathbf{cb}_{i}$ be a
modified Clark basis for $K_{\theta}$ with corresponding points on the
boundary $\eta_{i}$ given by (6). Consider the system of polynomials with
variables $r_{1},...,r_{9}$, defined by

$r_{1}^{2}+r_{4}^{2}+r_{7}^{2}=1,\quad r_{2}^{2}+r_{5}^{2}+r_{8}^{2}=1,\quad
r_{3}^{2}+r_{6}^{2}+r_{9}^{2}=1,$ (31)
$r_{1}r_{2}+r_{4}r_{5}+r_{7}r_{8}=0,\quad
r_{2}r_{3}+r_{5}r_{6}+r_{8}r_{9}=0,\quad r_{1}r_{3}+r_{4}r_{6}+r_{7}r_{9}=0,$
(32)
$(\eta_{3}-\eta_{2})A_{6}-\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{3}}^{\frac{1}{2}}}\right)(\eta_{1}-\eta_{2})A_{4}-\left(\frac{\overline{\eta_{1}}^{\frac{1}{2}}}{\overline{\eta_{2}}^{\frac{1}{2}}}\right)(\eta_{3}-\eta_{1})A_{5}=0,$
(33)

where the square root is given by the same convention as in (13) and where
$A_{6}=m_{1}r_{4}r_{7}+m_{4}r_{5}r_{7}+m_{5}r_{6}r_{7}+m_{4}r_{4}r_{8}+m_{2}r_{5}r_{8}+m_{6}r_{6}r_{8}+m_{5}r_{4}r_{9}+m_{6}r_{5}r_{9}+m_{3}r_{6}r_{9}$,
$A_{4}=m_{1}r_{1}r_{4}+m_{4}r_{2}r_{4}+m_{5}r_{3}r_{4}+m_{4}r_{1}r_{5}+m_{2}r_{2}r_{5}+m_{6}r_{3}r_{5}+m_{5}r_{1}r_{6}+m_{6}r_{2}r_{6}+m_{3}r_{3}r_{6}$
$A_{5}=m_{1}r_{1}r_{7}+m_{4}r_{2}r_{7}+m_{5}r_{3}r_{7}+m_{4}r_{1}r_{8}+m_{2}r_{2}r_{8}+m_{6}r_{3}r_{8}+m_{5}r_{1}r_{9}+m_{6}r_{2}r_{9}+m_{3}r_{3}r_{9}$.

Then

$M=\begin{pmatrix}m_{1}&m_{4}&m_{5}\\\ m_{4}&m_{2}&m_{6}\\\
m_{5}&m_{6}&m_{3}\end{pmatrix}$

is a matrix representation of a TTO on $K_{\theta}$ with respect to a
$C_{\theta}$-real basis if and only if equations (31) (32) and (33) are
simultaneously satisfied with a real solution.

###### Proof.

As shown by (30), we see that $M$ is a matrix representation of a TTO on
$K_{\theta}$ with respect to a $C_{\theta}$-real basis if and only if there
exists orthogonal matrix $\left(\begin{array}[]{lll}r_{1}&r_{2}&r_{3}\\\
r_{4}&r_{5}&r_{6}\\\ r_{7}&r_{8}&r_{9}\end{array}\right)$ and a
$A\in\mathcal{T}_{\theta}$ such that

$\left(\begin{array}[]{lll}r_{1}&r_{2}&r_{3}\\\ r_{4}&r_{5}&r_{6}\\\
r_{7}&r_{8}&r_{9}\end{array}\right)M\left(\begin{array}[]{lll}r_{1}&r_{2}&r_{3}\\\
r_{4}&r_{5}&r_{6}\\\
r_{7}&r_{8}&r_{9}\end{array}\right)^{T}=[A]_{\mathbf{cb}_{i}}.$ (34)

Now equations (31) (32) being satisfied (with a real solution) are equivalent
to

$\left(\begin{array}[]{lll}r_{1}&r_{2}&r_{3}\\\ r_{4}&r_{5}&r_{6}\\\
r_{7}&r_{8}&r_{9}\end{array}\right)$

being a orthogonal matrix. Furthermore, by multiplying out the left hand side
of (34), we see that the left hand side of (34) is a representation of a TTO
on $K_{\theta}$ with respect to $\mathbf{cb}_{i}$ (i.e. that the condition
from Theorem 3.3 is satisfied) if and only if (33) is satisfied. ∎

######  Remark.

We notice that in the above theorem the condition that $M$ is a matrix
representation of a TTO on $K_{\theta}$ with respect to a $C_{\theta}$-real
basis is independent of our original choice of $\mathbf{cb}_{i}$.

######  Remark.

In the above theorem, there is a procedure to find a choice of
$\eta_{1},\eta_{2},\eta_{3}$, which is the following. Consider the modified
Clark basis for $K_{\theta}$ with parameters in (6) (5) given by $\alpha=1$
and $t$ such that $\theta(t)=0$. Then in order to find $\eta_{i}$ one must
solve
$\theta(z)=\prod_{i=1}^{3}\frac{z-\lambda_{i}}{1-\overline{\lambda_{i}}z}=1$,
or equivalently, after multiplying both sides by
$\prod_{i=1}^{3}1-\overline{\lambda_{i}}z$ and rearranging

$z^{3}K_{3}-z^{2}K_{2}+zK_{1}-K_{0}=0,$

where
$K_{3}=1+\overline{\lambda_{1}\lambda_{2}\lambda_{3}},\,K_{2}=\lambda_{1}+\lambda_{2}+\lambda_{3}+\overline{\lambda_{1}\lambda_{2}}+\overline{\lambda_{1}\lambda_{3}}+\overline{\lambda_{2}\lambda_{3}},\,K_{1}=\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}+\overline{\lambda_{1}}+\overline{\lambda_{2}}+\overline{\lambda_{3}},\,K_{0}=\lambda_{1}\lambda_{2}\lambda_{3}+1$.

## Acknowledgements

The author is grateful to EPSRC for financial support.
The author is grateful to Dr Ben Sharp for his advice and guidance during the
preparation of this paper.

## References

  * [1] L. Balayan and S. R. Garcia.  Unitary equivalence to a complex symmetric matrix: geometric criteria.  Oper. Matrices, 4(1):53–76, 2010.
  * [2] A. Baranov, R. Bessonov, and V. Kapustin.  Symbols of truncated Toeplitz operators.  J. Funct. Anal., 261(12):3437–3456, 2011.
  * [3] A. Baranov, I. Chalendar, E. Fricain, J. Mashreghi, and D. Timotin.  Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators.  Journal of Functional Analysis, 259(10):2673–2701, 2010.
  * [4] C. M. Bender.  Introduction to PT-symmetric quantum theory.  Contemp.Phys., 46(4):277–292, 2005.
  * [5] R. V. Bessonov.  Fredholmness and compactness of truncated Toeplitz and Hankel operators.  Integral Equations Operator Theory, 82(4):451–467, 2015.
  * [6] M. C. Câmara and J. R. Partington.  Spectral properties of truncated Toeplitz operators by equivalence after extension.  J. Math. Anal. Appl., 433(2):762–784, 2016.
  * [7] J. A. Cima, S. R. Garcia, W. T. Ross, and W. R. Wogen.  Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity.  Indiana Univ. Math. J., 59(2):595–620, 2010.
  * [8] J. A. Cima and W. T. Ross.  The backward shift on the Hardy space.  American Mathematical Soc., 2000.
  * [9] J. A. Cima, W. T. Ross, and W. R. Wogen.  Truncated Toeplitz operators on finite dimensional spaces.  Oper. Matrices, 2(3):357–369, 2008.
  * [10] P. L. Duren.  Theory of ${H}^{p}$ spaces.  Pure Appl. Math, 38:74, 1970.
  * [11] S. R. Garcia.  Conjugation and Clark operators.  In Recent advances in operator-related function theory, volume 393 of Contemp. Math., pages 67–111. Amer. Math. Soc., Providence, RI, 2006\.
  * [12] S. R. Garcia.  Three questions about complex symmetric operators.  Integral Equations Operator Theory, 72(1):3–4, 2012.
  * [13] S. R. Garcia, J. Mashreghi, and W. T. Ross.  Introduction to model spaces and their operators, volume 148 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2016.
  * [14] S. R. Garcia, J. Mashreghi, and W. T. Ross.  Finite Blaschke products and their connections.  Springer, Cham, 2018.
  * [15] S. R. Garcia, D. E. Poore, and W. T. Ross.  Unitary equivalence to a truncated Toeplitz operator: analytic symbols.  Proc. Amer. Math. Soc., 140(4):1281–1295, 2012.
  * [16] S. R. Garcia, D. E. Poore, and J. E. Tener.  Unitary equivalence to a complex symmetric matrix: low dimensions.  Linear Algebra Appl., 437(1):271–284, 2012.
  * [17] S. R. Garcia, E. Prodan, and M. Putinar.  Mathematical and physical aspects of complex symmetric operators.  J. Phys. A, 47(35):353001, 54, 2014.
  * [18] S. R. Garcia and W. T. Ross.  Recent progress on truncated Toeplitz operators.  Blaschke Products and Their Applications, 65:265–319, 2013.
  * [19] N. K. Nikolski.  Operators, Functions, and Systems-An Easy Reading: Hardy, Hankel, and Toeplitz, volume 1.  American Mathematical Soc., 2002.
  * [20] R. O’Loughlin.  Nearly invariant subspaces with applications to truncated Toeplitz operators.  Complex Anal. Oper. Theory, 14, 2020, no.8, 86.
  * [21] R. O’Loughlin.  Multidimensional Toeplitz and truncated Toeplitz operators.  https://etheses.whiterose.ac.uk/29284/, University of Leeds, 2021\.
  * [22] R. O’Loughlin.  Symbols of compact truncated Toeplitz operators.  Journal of Mathematical Analysis and Applications, 507(2):125819, 2022.
  * [23] D. Sarason.  Algebraic properties of truncated Toeplitz operators.  Oper. Matrices, 1(4):491–526, 2007.
  * [24] E. Strouse, D. Timotin, and M. Zarrabi.  Unitary equivalence to truncated Toeplitz operators.  Indiana Univ. Math. J., 61(2):525–538, 2012.
  * [25] J. Vermeer.  Orthogonal similarity of a real matrix and its transpose.  Linear Algebra Appl., 428(1):382–392, 2008.


* Arous et al., (2019) Arous, G. B., Mei, S., Montanari, A., and Nica, M. (2019).  The landscape of the spiked tensor model.  Communications on Pure and Applied Mathematics, 72(11):2282–2330.
  * Bai and Li, (2012) Bai, J. and Li, K. (2012).  Statistical analysis of factor models of high dimension.  The Annals of Statistics, 40(1):436–465.
  * Bai and Wang, (2016) Bai, J. and Wang, P. (2016).  Econometric analysis of large factor models.  Annual Review of Economics, 8:53–80.
  * Bai and Ding, (2012) Bai, Z. and Ding, X. (2012).  Estimation of spiked eigenvalues in spiked models.  Random Matrices: Theory and Applications, 1(02):1150011.
  * Balzano et al., (2018) Balzano, L., Chi, Y., and Lu, Y. M. (2018).  Streaming PCA and subspace tracking: The missing data case.  Proceedings of the IEEE, 106(8):1293–1310.
  * Bao et al., (2022) Bao, Z., Ding, X., Wang, J., and Wang, K. (2022).  Statistical inference for principal components of spiked covariance matrices.  The Annals of Statistics, 50(2):1144–1169.
  * Bao et al., (2021) Bao, Z., Ding, X., and Wang, K. (2021).  Singular vector and singular subspace distribution for the matrix denoising model.  The Annals of Statistics, 49(1):370–392.
  * Cai et al., (2021) Cai, C., Li, G., Chi, Y., Poor, H. V., and Chen, Y. (2021).  Subspace estimation from unbalanced and incomplete data matrices: $\ell_{2,\infty}$ statistical guarantees.  The Annals of Statistics, 49(2):944–967.
  * (14) Cai, C., Li, G., Poor, H. V., and Chen, Y. (2022a).  Nonconvex low-rank tensor completion from noisy data.  Operations Research, 70(2):1219–1237.
  * Cai et al., (2023) Cai, C., Poor, H. V., and Chen, Y. (2023).  Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality.  IEEE Transactions on Information Theory, 69(1):407–452.
  * Cai et al., (2010) Cai, J.-F., Candès, E. J., and Shen, Z. (2010).  A singular value thresholding algorithm for matrix completion.  SIAM Journal on optimization, 20(4):1956–1982.
  * (17) Cai, J.-F., Li, J., and Xia, D. (2022b).  Provable tensor-train format tensor completion by Riemannian optimization.  Journal of Machine Learning Research, 23(123):1–77.
  * Cai and Zhang, (2018) Cai, T. T. and Zhang, A. (2018).  Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics.  The Annals of Statistics, 46(1):60–89.
  * Candès and Recht, (2009) Candès, E. J. and Recht, B. (2009).  Exact matrix completion via convex optimization.  Foundations of Computational mathematics, 9(6):717.
  * Cape et al., (2019) Cape, J., Tang, M., and Priebe, C. E. (2019).  The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics.  The Annals of Statistics, 47(5):2405–2439.
  * Chatterjee, (2015) Chatterjee, S. (2015).  Matrix estimation by universal singular value thresholding.  The Annals of Statistics, 43(1):177–214.
  * Chen et al., (2015) Chen, P. H., Chen, J., Yeshurun, Y., Hasson, U., Haxby, J. V., and Ramadge, P. J. (2015).  A reduced-dimension fMRI shared response model.  Advances in Neural Information Processing Systems, 2015:460–468.
  * Chen et al., (2022) Chen, S., Liu, S., and Ma, Z. (2022).  Global and individualized community detection in inhomogeneous multilayer networks.  The Annals of Statistics, 50(5):2664–2693.
  * (24) Chen, Y., Cheng, C., and Fan, J. (2021a).  Asymmetry helps: Eigenvalue and eigenvector analyses of asymmetrically perturbed low-rank matrices.  Annals of statistics, 49(1):435.
  * (25) Chen, Y., Chi, Y., Fan, J., and Ma, C. (2021b).  Spectral methods for data science: A statistical perspective.  Foundations and Trends® in Machine Learning, 14(5):566–806.
  * Chen et al., (2020) Chen, Y., Chi, Y., Fan, J., Ma, C., and Yan, Y. (2020).  Noisy matrix completion: Understanding statistical guarantees for convex relaxation via nonconvex optimization.  SIAM journal on optimization, 30(4):3098–3121.
  * (27) Chen, Y., Fan, J., Ma, C., and Wang, K. (2019a).  Spectral method and regularized MLE are both optimal for top-K ranking.  Annals of statistics, 47(4):2204.
  * (28) Chen, Y., Fan, J., Ma, C., and Yan, Y. (2019b).  Inference and uncertainty quantification for noisy matrix completion.  Proceedings of the National Academy of Sciences, 116(46):22931–22937.
  * (29) Chen, Y., Fan, J., Ma, C., and Yan, Y. (2021c).  Bridging convex and nonconvex optimization in robust PCA: Noise, outliers and missing data.  The Annals of Statistics, 49(5):2948–2971.
  * (30) Chen, Y., Fan, J., Wang, B., and Yan, Y. (2021d).  Convex and nonconvex optimization are both minimax-optimal for noisy blind deconvolution under random designs.  Journal of the American Statistical Association, pages 1–11.
  * Cheng et al., (2021) Cheng, C., Wei, Y., and Chen, Y. (2021).  Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise.  IEEE Transactions on Information Theory, 67(11):7380–7419.
  * (32) De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000a).  A multilinear singular value decomposition.  SIAM journal on Matrix Analysis and Applications, 21(4):1253–1278.
  * (33) De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000b).  On the best rank-$1$ and rank-$(r_{1},r_{2},...,r_{n})$ approximation of higher-order tensors.  SIAM journal on Matrix Analysis and Applications, 21(4):1324–1342.
  * Deshpande et al., (2017) Deshpande, Y., Abbe, E., and Montanari, A. (2017).  Asymptotic mutual information for the balanced binary stochastic block model.  Information and Inference: A Journal of the IMA, 6(2):125–170.
  * Dobriban and Owen, (2019) Dobriban, E. and Owen, A. B. (2019).  Deterministic parallel analysis: an improved method for selecting factors and principal components.  Journal of the Royal Statistical Society Series B, 81(1):163–183.
  * Donoho and Gavish, (2014) Donoho, D. and Gavish, M. (2014).  Minimax risk of matrix denoising by singular value thresholding.  The Annals of Statistics, 42(6):2413–2440.
  * Donoho et al., (2018) Donoho, D. L., Gavish, M., and Johnstone, I. M. (2018).  Optimal shrinkage of eigenvalues in the spiked covariance model.  Annals of statistics, 46(4):1742.
  * Eldridge et al., (2018) Eldridge, J., Belkin, M., and Wang, Y. (2018).  Unperturbed: spectral analysis beyond davis-kahan.  In Algorithmic Learning Theory, pages 321–358. PMLR.
  * Elsener and van de Geer, (2019) Elsener, A. and van de Geer, S. (2019).  Sparse spectral estimation with missing and corrupted measurements.  Stat, 8(1):e229.
  * Fan et al., (2020) Fan, J., Li, R., Zhang, C.-H., and Zou, H. (2020).  Statistical foundations of data science.  Chapman and Hall/CRC.
  * Fan et al., (2016) Fan, J., Liao, Y., and Wang, W. (2016).  Projected principal component analysis in factor models.  Annals of statistics, 44(1):219.
  * Fan et al., (2021) Fan, J., Wang, K., Zhong, Y., and Zhu, Z. (2021).  Robust high dimensional factor models with applications to statistical machine learning.  Statistical science: a review journal of the Institute of Mathematical Statistics, 36(2):303.
  * Fan et al., (2018) Fan, J., Wang, W., and Zhong, Y. (2018).  An $\ell_{\infty}$ eigenvector perturbation bound and its application to robust covariance estimation.  Journal of Machine Learning Research, 18(207):1–42.
  * Feng et al., (2022) Feng, O. Y., Venkataramanan, R., Rush, C., Samworth, R. J., et al. (2022).  A unifying tutorial on approximate message passing.  Foundations and Trends® in Machine Learning, 15(4):335–536.
  * Florescu and Perkins, (2016) Florescu, L. and Perkins, W. (2016).  Spectral thresholds in the bipartite stochastic block model.  In Conference on Learning Theory, pages 943–959. PMLR.
  * Gavish and Donoho, (2014) Gavish, M. and Donoho, D. L. (2014).  The optimal hard threshold for singular values is $4/\sqrt{3}$.  IEEE Transactions on Information Theory, 60(8):5040–5053.
  * Gavish and Donoho, (2017) Gavish, M. and Donoho, D. L. (2017).  Optimal shrinkage of singular values.  IEEE Transactions on Information Theory, 63(4):2137–2152.
  * (48) Han, R., Luo, Y., Wang, M., Zhang, A. R., et al. (2022a).  Exact clustering in tensor block model: Statistical optimality and computational limit.  Journal of the Royal Statistical Society Series B, 84(5):1666–1698.
  * (49) Han, R., Willett, R., and Zhang, A. R. (2022b).  An optimal statistical and computational framework for generalized tensor estimation.  The Annals of Statistics, 50(1):1–29.
  * Han and Zhang, (2022) Han, Y. and Zhang, C.-H. (2022).  Tensor principal component analysis in high dimensional CP models.  IEEE Transactions on Information Theory.
  * Hong et al., (2016) Hong, D., Balzano, L., and Fessler, J. A. (2016).  Towards a theoretical analysis of PCA for heteroscedastic data.  In 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 496–503. IEEE.
  * (52) Hong, D., Balzano, L., and Fessler, J. A. (2018a).  Asymptotic performance of PCA for high-dimensional heteroscedastic data.  Journal of multivariate analysis, 167:435–452.
  * (53) Hong, D., Fessler, J. A., and Balzano, L. (2018b).  Optimally weighted PCA for high-dimensional heteroscedastic data.  arXiv preprint arXiv:1810.12862.
  * Hopkins et al., (2015) Hopkins, S. B., Shi, J., and Steurer, D. (2015).  Tensor principal component analysis via sum-of-square proofs.  In Conference on Learning Theory, pages 956–1006. PMLR.
  * Johnstone, (2001) Johnstone, I. M. (2001).  On the distribution of the largest eigenvalue in principal components analysis.  The Annals of statistics, 29(2):295–327.
  * Johnstone and Paul, (2018) Johnstone, I. M. and Paul, D. (2018).  PCA in high dimensions: An orientation.  Proceedings of the IEEE, 106(8):1277–1292.
  * Keshavan et al., (2010) Keshavan, R. H., Montanari, A., and Oh, S. (2010).  Matrix completion from noisy entries.  The Journal of Machine Learning Research, 11:2057–2078.
  * Kolda and Bader, (2009) Kolda, T. G. and Bader, B. W. (2009).  Tensor decompositions and applications.  SIAM review, 51(3):455–500.
  * Koltchinskii and Giné, (2000) Koltchinskii, V. and Giné, E. (2000).  Random matrix approximation of spectra of integral operators.  Bernoulli, 6(1):113–167.
  * Koltchinskii et al., (2011) Koltchinskii, V., Lounici, K., and Tsybakov, A. B. (2011).  Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion.  The Annals of Statistics, 39(5):2302–2329.
  * Koltchinskii and Xia, (2016) Koltchinskii, V. and Xia, D. (2016).  Perturbation of linear forms of singular vectors under Gaussian noise.  In High Dimensional Probability VII, pages 397–423. Springer.
  * Kritchman and Nadler, (2008) Kritchman, S. and Nadler, B. (2008).  Determining the number of components in a factor model from limited noisy data.  Chemometrics and Intelligent Laboratory Systems, 94(1):19–32.
  * Kritchman and Nadler, (2009) Kritchman, S. and Nadler, B. (2009).  Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory.  IEEE Transactions on Signal Processing, 57(10):3930–3941.
  * Lawley and Maxwell, (1962) Lawley, D. N. and Maxwell, A. E. (1962).  Factor analysis as a statistical method.  Journal of the Royal Statistical Society. Series D (The Statistician), 12(3):209–229.
  * Li et al., (2023) Li, G., Fan, W., and Wei, Y. (2023).  Approximate message passing from random initialization with applications to $\mathbb{Z}_{2}$ synchronization.  arXiv preprint arXiv:2302.03682.
  * Li and Wei, (2022) Li, G. and Wei, Y. (2022).  A non-asymptotic framework for approximate message passing in spiked models.  arXiv preprint arXiv:2208.03313.
  * Ling, (2022) Ling, S. (2022).  Near-optimal performance bounds for orthogonal and permutation group synchronization via spectral methods.  Applied and Computational Harmonic Analysis, 60:20–52.
  * Löffler et al., (2021) Löffler, M., Zhang, A. Y., and Zhou, H. H. (2021).  Optimality of spectral clustering in the Gaussian mixture model.  The Annals of Statistics, 49(5):2506–2530.
  * Loh and Wainwright, (2012) Loh, P.-L. and Wainwright, M. J. (2012).  High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity.  The Annals of Statistics, 40(3):1637–1664.
  * Lounici, (2014) Lounici, K. (2014).  High-dimensional covariance matrix estimation with missing observations.  Bernoulli, 20(3):1029–1058.
  * Ma et al., (2020) Ma, C., Wang, K., Chi, Y., and Chen, Y. (2020).  Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution.  Foundations of Computational Mathematics, 20(3):451–632.
  * Montanari and Sun, (2018) Montanari, A. and Sun, N. (2018).  Spectral algorithms for tensor completion.  Communications on Pure and Applied Mathematics, 71(11):2381–2425.
  * Montanari and Venkataramanan, (2021) Montanari, A. and Venkataramanan, R. (2021).  Estimation of low-rank matrices via approximate message passing.  The Annals of Statistics, 49(1):321–345.
  * Montanari and Wu, (2022) Montanari, A. and Wu, Y. (2022).  Fundamental limits of low-rank matrix estimation with diverging aspect ratios.  arXiv preprint arXiv:2211.00488.
  * Nadakuditi, (2014) Nadakuditi, R. R. (2014).  Optshrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage.  IEEE Transactions on Information Theory, 60(5):3002–3018.
  * Ndaoud, (2022) Ndaoud, M. (2022).  Sharp optimal recovery in the two component Gaussian mixture model.  The Annals of Statistics, 50(4):2096–2126.
  * Ndaoud et al., (2021) Ndaoud, M., Sigalla, S., and Tsybakov, A. B. (2021).  Improved clustering algorithms for the bipartite stochastic block model.  IEEE Transactions on Information Theory, 68(3):1960–1975.
  * Paul, (2007) Paul, D. (2007).  Asymptotics of sample eigenstructure for a large dimensional spiked covariance model.  Statistica Sinica, pages 1617–1642.
  * Pavez and Ortega, (2020) Pavez, E. and Ortega, A. (2020).  Covariance matrix estimation with non uniform and data dependent missing observations.  IEEE Transactions on Information Theory, 67(2):1201–1215.
  * Perry et al., (2020) Perry, A., Wein, A. S., and Bandeira, A. S. (2020).  Statistical limits of spiked tensor models.  Annales de l’Institut Henri Poincaré-Probabilités et Statistiques, 56(1):230–264.
  * Perry et al., (2018) Perry, A., Wein, A. S., Bandeira, A. S., and Moitra, A. (2018).  Optimality and sub-optimality of PCA I: Spiked random matrix models.  The Annals of Statistics, 46(5):2416–2451.
  * Richard and Montanari, (2014) Richard, E. and Montanari, A. (2014).  A statistical model for tensor PCA.  In Advances in Neural Information Processing Systems, pages 2897–2905.
  * Rohe et al., (2011) Rohe, K., Chatterjee, S., and Yu, B. (2011).  Spectral clustering and the high-dimensional stochastic blockmodel.  The Annals of Statistics, 39(4):1878–1915.
  * Srivastava et al., (2022) Srivastava, P. R., Sarkar, P., and Hanasusanto, G. A. (2022).  A robust spectral clustering algorithm for sub-Gaussian mixture models with outliers.  Operations Research.
  * Tong et al., (2021) Tong, T., Ma, C., and Chi, Y. (2021).  Accelerating ill-conditioned low-rank matrix estimation via scaled gradient descent.  The Journal of Machine Learning Research, 22(1):6639–6701.
  * Tong et al., (2022) Tong, T., Ma, C., Prater-Bennette, A., Tripp, E., and Chi, Y. (2022).  Scaling and scalability: Provable nonconvex low-rank tensor estimation from incomplete measurements.  Journal of Machine Learning Research, 23(163):1–77.
  * Tropp et al., (2015) Tropp, J. A. et al. (2015).  An introduction to matrix concentration inequalities.  Foundations and Trends® in Machine Learning, 8(1-2):1–230.
  * Vannieuwenhoven et al., (2012) Vannieuwenhoven, N., Vandebril, R., and Meerbergen, K. (2012).  A new truncation strategy for the higher-order singular value decomposition.  SIAM Journal on Scientific Computing, 34(2):A1027–A1052.
  * Vershynin, (2010) Vershynin, R. (2010).  Introduction to the non-asymptotic analysis of random matrices.  arXiv preprint arXiv:1011.3027.
  * Wang and Fan, (2017) Wang, W. and Fan, J. (2017).  Asymptotics of empirical eigenstructure for high dimensional spiked covariance.  Annals of statistics, 45(3):1342.
  * Xia, (2021) Xia, D. (2021).  Normal approximation and confidence region of singular subspaces.  Electronic Journal of Statistics, 15(2):3798–3851.
  * Xia et al., (2022) Xia, D., Zhang, A. R., and Zhou, Y. (2022).  Inference for low-rank tensors—no need to debias.  The Annals of Statistics, 50(2):1220–1245.
  * Xu et al., (2023) Xu, X., Shen, Y., Chi, Y., and Ma, C. (2023).  The power of preconditioning in overparameterized low-rank matrix sensing.  arXiv preprint arXiv:2302.01186.
  * Yan et al., (2021) Yan, Y., Chen, Y., and Fan, J. (2021).  Inference for heteroskedastic PCA with missing data.  arXiv preprint arXiv:2107.12365.
  * Yang and Ma, (2022) Yang, Y. and Ma, C. (2022).  Optimal tuning-free convex relaxation for noisy matrix completion.  arXiv preprint arXiv:2207.05802.
  * Zhang and Xia, (2018) Zhang, A. and Xia, D. (2018).  Tensor SVD: Statistical and computational limits.  IEEE Transactions on Information Theory, 64(11):7311–7338.
  * Zhang et al., (2022) Zhang, A. R., Cai, T. T., and Wu, Y. (2022).  Heteroskedastic PCA: Algorithm, optimality, and applications.  The Annals of Statistics, 50(1):53–80.
  * Zhang and Zhou, (2022) Zhang, A. Y. and Zhou, H. H. (2022).  Leave-one-out singular subspace perturbation analysis for spectral clustering.  arXiv preprint arXiv:2205.14855.
  * Zhao et al., (1986) Zhao, L., Krishnaiah, P. R., and Bai, Z. (1986).  On detection of the number of signals in presence of white noise.  Journal of multivariate analysis, 20(1):1–25.
  * Zhong and Boumal, (2018) Zhong, Y. and Boumal, N. (2018).  Near-optimal bounds for phase synchronization.  SIAM Journal on Optimization, 28(2):989–1016.
  * Zhou et al., (2022) Zhou, Y., Zhang, A. R., Zheng, L., and Wang, Y. (2022).  Optimal high-order tensor SVD via tensor-train orthogonal iteration.  IEEE Transactions on Information Theory, 68(6):3991–4019.
  * Zhu et al., (2019) Zhu, Z., Wang, T., and Samworth, R. J. (2019).  High-dimensional principal component analysis with heterogeneous missingness.  arXiv preprint arXiv:1906.12125.


#  Parametric analysis of COVID-19 expansion in European countries in the
period of February to June 2020

Martin Spousta<EMAIL_ADDRESS>Spousta  Institute of Particle and
Nuclear Physics, Charles University, Prague, Czech Republic

###### Abstract

The data on number of registered cases of COVID-19 disease in twenty European
countries is analyzed by the least-squares fitting procedure with generic
analytic functions. Three regimes of the expansion of the disease are
identified and quantified – early exponential expansion, damped exponential,
and linear expansion. Differences among countries in the early expansion
period are quantified. The velocity of the expansion in the exponential regime
lies within one standard deviation from the average value for 11 countries.
The number of infected individuals at the initial time is excessively high for
Italy, 7 standard deviations from the average value.

Method for predicting the expansion based on extrapolation in the parametric
space is presented. One week predictions based on extrapolations have average
precision of 18% and 29% during the later period of the damped exponential
expansion for the case of Italy and Czechia, respectively. The method based on
extrapolations in the parametric space may provide an elementary method to
quantify the impact of restrictive measures on the spreading of the disease.

###### keywords:

Outbreak, Infection, SARS-CoV-2, COVID-19, Parametric analysis, European
countries

##  1 Introduction

The outbreak of new coronavirus SARS-CoV-2, causing severe respiratory tract
infection in humans, known as COVID-19, is a global health concern.
Restrictive measures were adopted in many countries to mitigate the impact of
the spread of the disease on the public health system [1].

The spread of an infectious disease is usually modeled using compartmental
epidemiological models, see e.g. review in Ref. [2] or examples of their
recent implementation in Refs. [3, 4, 5]. These models are often based on the
numerical solution of a system of coupled linear differential equations for
the number of susceptible (S), infectious (I), and recovered (R) cases (SIR
models). For the simplest SIR model, a set of three differential equation
characterizes the transitions among the three classes.

While it is not possible to obtain an analytic solution for a complex system
of the SIR model equations, it is possible to construct generic analytic
functions describing the data and quantify the robustness of such a
description based on statistical tests. This approach is the backbone of this
analysis. The main goal of this analysis is to identify and quantify robust
features in the dynamics of spread of the disease and present a method that
may help quantify the impact of restrictive measures on the spreading of the
disease.

##  2 Method and dataset

In the simple SIR model, the equation which describes the change in the number
of infectious cases reads $\mathrm{d}I/\mathrm{d}t=(c_{1}S/N-c_{2})I$ where
$N=S+I+R$ and $c_{1},c_{2}$ are constants (see e.g. [6] for details). At the
beginning of the outbreak when no restrictive measures are applied and the
susceptibility of the population to the virus is 100%, one has $S\gg I$ and
$R\rightarrow 0$. This leads to $dI/dt\propto I$ implying an exponential
increase of the number of infected individuals with time,

$I=I_{0}\exp[at],$ (1)

where $I_{0}$ and $a$ are constants, combinations of $c_{1},N$, and $S$. When
restrictive measures are imposed, $c_{1}$ is not a constant but a function
$c_{1}=c_{1}(t,...)$ introducing more complexity to the set of SIR model
equations. Alternatively, the evolution under restrictive measures can be
described by exponential dependence with a time-dependent parameter, that is,
by the function

$F(t)=N_{0}\exp[a(t)t]=N_{0}\exp[b_{1}t+b_{2}t^{2}+b_{3}t^{3}+...],$ (2)

where the $r.h.s.$ of (2) represents a Taylor expansion of the time dependent
parameter $a$ and $N_{0}$ reflects number of infected individuals at $t=0$.
Full description of the data can be achieved with $n$ $b_{n}$ parameters where
$n$ is a large number. The $r.h.s.$ of (2) characterizes the basic exponential
dependence plus any smooth deviation from this dependence.

A large number of free parameters in (2) may be reduced and interpreted. At
the beginning of the outbreak, the number of cases is described by a pure
exponential function ($b_{1}\neq 0,b_{n>1}=0$) as argued in Eq. (1). This
hypothesis is tested in Section 3.1 and the parameters of the exponential
function are compared among European countries. Once the non-zero $b_{2}$ is
needed to describe the data, the expansion starts to slow down with respect to
the purely exponential propagation. This is quantified in Section 3.2. In an
idealized case, only one $b_{2}$ coefficient along with previously fitted
$b_{1}$ coefficient would be needed to describe the data. However, such a
simple two-component characterization is shown to be invalid in Section 3.2,
and higher-order terms are needed or alternatively time-dependent $b_{2}$ and
$N_{0}$ coefficients are needed to describe the data. The latter approach is
used in this analysis. The evolution of parameters with time is studied, and
predictability of the spread based on extrapolations is quantified. After a
certain period, the description by a linear function outperforms the
description by (2). The transition to a linear spread is quantified in Section
3.3.

The cumulative number of registered cases of COVID-19 disease in European
countries is used as input data to this analysis. Specifically, the analysis
uses data from 20 European countries, 17 countries of the European Union
exceeding 5 million of inhabitants and data from Norway, Switzerland, and the
United Kingdom. Data are taken from information system [7, 8] for a period of
15th of February to 1st of June 2020. Analysis uses statistical tools
implemented in ROOT 6.1 [9].

##  3 Results

###  3.1 Period of exponential expansion

During the first period, when there is no implementation of restrictive
measures and the susceptibility of the population to the virus is 100%, the
number of all registered cases exponentially grows. This is described by
Equation (2) with $b_{n>1}=0$ and $b_{1}$ quantifying the velocity of the
spread of the disease, here in units day-1. To quantify the expansion during
this period we perform the least-squares fitting of the number of all
registered cases by an exponential function in the time interval of
$t_{min}-t_{max}$. The default lower bound $t_{min}$ was selected by a
requirement of at least 50 cases to reduce fluctuations due to too small
statistical sample. The upper bound $t_{max}$ was selected individually for
each country based on comparing with an alternative description allowing for
non-zero $b_{2}$ coefficient for each day when the number of cases exceeds
500. The comparison is based on a $\chi^{2}$ test. The value of $t_{max}$
represents a first day when fit with non-zero $b_{2}$ provides a better
description than a fit with zero $b_{2}$. The resulting $b_{1}$ and $N_{0}$
coefficients are summarized in Table 1. The boundary condition leading to the
definition of $t_{min}$ (50 cases) was varied by 20% both up and down, which
leads to alternative results. These alternative results were found to
correspond with the nominal results within one standard deviation for 90% of
countries.

Country | $N_{0}$ |  $b_{1}$ [day-1] | $p_{KS}$
---|---|---|---
Italy | 6.61 $\pm$ 1.02 | 0.39 $\pm$ 0.01 | 1.00
Germany | 0.57 $\pm$ 0.13 | 0.35 $\pm$ 0.01 | 1.00
Portugal | 0.01 $\pm$ 0.00 | 0.34 $\pm$ 0.01 | 1.00
Sweden | 0.23 $\pm$ 0.06 | 0.31 $\pm$ 0.01 | 1.00
Spain | 0.92 $\pm$ 0.20 | 0.30 $\pm$ 0.01 | 1.00
France | 1.38 $\pm$ 0.39 | 0.30 $\pm$ 0.02 | 1.00
Austria | 0.13 $\pm$ 0.03 | 0.30 $\pm$ 0.01 | 1.00
Norway | 0.37 $\pm$ 0.14 | 0.29 $\pm$ 0.02 | 1.00
Denmark | 0.27 $\pm$ 0.05 | 0.29 $\pm$ 0.01 | 0.94
Czechia | 0.10 $\pm$ 0.03 | 0.27 $\pm$ 0.01 | 1.00
Switzerland | 0.80 $\pm$ 0.11 | 0.27 $\pm$ 0.01 | 1.00
Netherlands | 0.83 $\pm$ 0.15 | 0.26 $\pm$ 0.01 | 1.00
Poland | 0.08 $\pm$ 0.03 | 0.25 $\pm$ 0.01 | 1.00
Belgium | 0.94 $\pm$ 0.12 | 0.24 $\pm$ 0.00 | 1.00
United Kingdom | 1.85 $\pm$ 0.37 | 0.22 $\pm$ 0.01 | 1.00
Greece | 0.97 $\pm$ 0.19 | 0.19 $\pm$ 0.01 | 1.00
Romania | 0.84 $\pm$ 0.16 | 0.18 $\pm$ 0.01 | 1.00
Hungary | 0.26 $\pm$ 0.06 | 0.17 $\pm$ 0.01 | 1.00
Bulgaria | 2.96 $\pm$ 0.37 | 0.11 $\pm$ 0.00 | 1.00
Slovakia | 4.28 $\pm$ 0.07 | 0.10 $\pm$ 0.00 | 0.96
Table 1:  Parameters of fit ($N_{0}$, $b_{1}$) of the total number of cases in
European countries during the first period of the expansion (for the
definition see the text). Fit function is given by Equation (2) with $b_{i}=0$
for $i>1$. The $p$-values for Kolmogorov-Smirnov test. Countries are ordered
by a decreasing value of $b_{1}$ coefficient.

Table 1 contains also results of Kolmogorov-Smirnov (KS) test. Based on
$p$-values, one can accept the hypothesis that the exponential function
provides a suitable description of the spread of the disease in the early
stage for all the countries.

The average value of $b_{1}$ is $0.25\pm 0.04$ day-1. The value of $b_{1}$ for
11 out of 20 countries lies within one standard deviation from the average
value. The value of $N_{0}$ (which reflects the number of infected individuals
at $t=0$) oscillates between zero and one for the majority of countries, but
it is excessively high for the case of Italy (7 standard deviations from the
average value calculated excluding Italy). This may be interpreted as evidence
for the validity of a hypothesis that the spread in Italy has started sooner
than when the first cases were reported and systematically traced (see e.g.
Ref. [10]).

To further check the universality of the initial velocity of the expansion
quantified by $b_{1}$ coefficient, we studied magnitude of Pearson’s
correlation coefficient quantifying the correlation between $b_{1}$ and other
quantity $x$, $\rho(b_{1},x)$. For $x$ we tested: 1) $N_{0}$; 2) number of
days between 15th of February (first day when cases start to be collected
European-wide [7]) and the day when the number of identified cases exceeds 50
in a given country ($d_{50}$); 3) the number of inhabitants in a given country
$n_{inhab}$. The absolute value of the correlation coefficient was found to be
smaller than 0.2 for all the quantities, implying that $b_{1}$ is not
correlated with any of $N_{0}$, $d_{50}$, and $n_{inhab}$.

The value of $b_{1}$ can be further used as an input to specific
epidemiological studies. In particular, it can be used to estimate the basic
reproduction number, $R_{0}$, [11] using Laplace transform as
$R_{0}=1/[\int_{0}^{\infty}\exp(-b_{1}\kappa)h(\kappa)\mathrm{d}\kappa]$ [12,
13]. Here $\kappa$ and $h(\kappa)$ represent the generation interval (i.e. the
average time interval from onset of one individual to the onset of another
individual) and its probability distribution, respectively.

Figure 1:  The $N_{0}$ (left), $b_{2}$ (middle), and number of all cases
(right) as a function of time for Italy (green) and Czechia (red). The
published data on the number of registered cases (full markers) are shown
together with one-week predictions (open markers) (for details see the text).

###  3.2 Period of exponential expansion with damping

As shown in the previous section, the growth of the number of cases during the
first period of spread of the disease can be characterized by an exponential
function. The parameter $b_{1}$ can be interpreted as a parameter quantifying
the uncontrolled spread of the disease. This first period ends when the fit
starts to prefer a non-zero $b_{2}$. Function from $r.h.s.$ of Equation (2)
with $b_{i}=0$ for $i>2$ is fitted to the number of all registered cases in a
given country in a time window of 7 days. This time window moves in steps of
one day to quantify the evolution of $N_{0}$ and $b_{2}$ with time.

While the complete description of the spread can be done by the function
$a(t)$ from Equation (2), here we take only the first two coefficients of the
Taylor expansion of that function. We have tested that this provides a good
description of the data ($p>0.995$ from KS tests) in 7-day time window during
the full second period. Allowing for non-zero $b_{3}$ or higher-order terms
does not bring an improvement in the description. Therefore, one parameter,
$b_{2}$, quantifies the degree of deviation from a purely exponential
expansion in a given time. The more negative $b_{2}$ implies more significant
“damping” of the original exponential growth. The units of $b_{2}$ used here
are day-2.

The example of time evolution of $N_{0}$ and $b_{2}$ for one large country
(Italy) and one small country (Czechia) is shown in the left and middle panel
of Figure 1. The parameter $b_{2}$ starts at zero (purely exponential
expansion) and then rapidly decreases. The largest negative values are
achieved for Italy. The $b_{2}$ distributions are qualitatively similar: first
$b_{2}$ decreases fast, later the decrease is slower, ending at a minimum
followed by a shallow increase where the $b_{2}$ achieves similar values for
both countries. While the interpretation of $b_{2}$ parameter is
straigthforward, one can call it a “push-back” parameter, the interpration of
time-dependent $N_{0}$ is not straightforward anymore and it is just a
nuisance parameter.

In the case that $b_{2}$ turns to a constant in time, the evolution is
predictable by simple analytic calculation. In particular the time
$t^{\prime}$ when no new cases should be registered, is given by a derivative
of Equation (2) with respect to time, leading to $t^{\prime}=-b_{1}/(2b_{2})$.
Further, if $N_{0}$ is constant in time, number of all cases can be calculated
by evaluating (2) at time $t^{\prime}$. While $b_{2}$ and $N_{0}$ exhibit only
a mild evolution during certain periods, these parameters do not converge to
constant values. Prediction of the evolution of the value of registered cases
cannot therefore be done analytically in general. However, it can be done by
extrapolating the evolution of parameters to the future. Extrapolations based
on constant, linear, polynomial, and exponential functions were tested.
Prediction from extrapolation was calculated for a value of registered cases
in 7 days after a given time. These one-week predictions are shown along with
the data in the right panel of Figure 1. For the period when $b_{2}$ is
decreasing, it was found that an extrapolation using constant values of
parameters provides the best matching between estimates of the number of
registered cases based on these extrapolations and the actual data. For the
period when $b_{2}$ starts to increase, a linear extrapolation and exponential
extrapolation of $b_{2}$ and $N_{0}$, respectively, provide predictions that
are closest to the data.

For the case of Italy, the predictions differ from the data on average by
$265\%\pm 160\%$ during the first 14 days after the time when data start to
prefer the parameterization with non-zero $b_{2}$. This substantial
disagreement is given by a fast decrease of $b_{2}$ during the first two weeks
which we did not succeed to describe universally by the extrapolation. For the
next 50 days, the predictions differ from the data on average by $29\%\pm
19\%$ and $18\%\pm 10\%$ in the case of Italy and Czechia, respectively. These
findings imply that the extrapolation of trends in the parameter space used in
this study does not allow to make realistic predictions in the period of two
weeks after the time when $b_{2}$ starts to be non-zero. However, for the
later times, the method allows to make estimates with reasonable precision.

The method based on extrapolating the trends in the parametric space may
provide an elementary method to quantify the impact of restrictive measures on
the number of registered cases. This may be done by evaluating a difference
between the extrapolation and the data for a time period when a given
restrictive measure is applied.

The fact that the method based on extrapolations provide a reasonable estimate
of the future evolution for a period of one week is simple evidence for a
significant inertia in the evolution.

###  3.3 Period of linear evolution

The functional form of Equation (2) stops describing the data well at some
point. This reflects the fact that the spread of the disease turns from the
damped exponential expansion to a linear evolution. This is quantified in
Figure 2 for the case of Italy in terms of the sum of residuals for the damped
exponential fit and the linear fit. At around day 55, the linear function
starts to characterize the data significantly better. The transition period
between the damped exponential and linear growth takes about two weeks.

It is interesting to compare the value of $b_{2}$ for which the transition to
linear evolution appears with the value of $b_{1}$. The transition occurs for
$\sqrt{|b_{2}|}$ of $0.047-0.051$ day-1 and $0.047-0.049$ day-1 for Italy and
Czechia, respectively. These values are roughly five times smaller than the
average value of $b_{1}$ determined in Section 3.1, $b_{1}=0.25\pm 0.04$
day-1.

It might be interesting to further explore if the transition to the linear
evolution can be connected with some specific values of $b_{2}$ universally.
If so, then the transition point may be predictable within the proposed
parametric description of the spread. This is left open for further studies.

Figure 2:  Sum of fit residuals, $\sigma|\hat{r}|$, for the damped exponential
and a linear fit of the number of all registered case evaluated as a function
of time for Italy.

##  4 Discussion

Compartmental models represent an established way to model the spread of an
infectious disease. As the COVID-19 pandemic evolves, they are being tuned and
improved. In general, adding complexity into the description of the problem
and adding more free parameters do not automatically guarantee a better
precision of the description – see e.g. discussion in Refs. [14, 15, 16]. Work
presented in this paper represents an alternative to complex compartmental
modeling, which may provide new insight into the dynamics of the spread by its
parametric analysis.

The spread of the disease is also often described by growth-law models, see
e.g. Refs. [12, 17, 18, 19]. Compared to the growth-law models, the parametric
analysis presented here starts from generic analytic functions, a form of
which is constrained by the data via statistical tests. Parameters are then
let free to evolve. Trends in this evolution can be studied and extrapolated
to future. This allows to make predictions and compare with the data which may
provide an elementary method to judge the impact of restrictive measures. For
example, if the data show a larger decrease compared to the extrapolation,
then one gets an evidence that the restrictive measures applied earlier are
efficient. This allows to attempt to compare the impact of different
restrictive measures in a uniform way. This approach may be useful since the
restrictive measures are complex and therefore hard to model. The undemanding
approach used here may also allow to access larger datasets.

The cumulative number of registered COVID-19 cases from datasets used in this
analysis represents a complex quantity that may be influenced by various
external factors. In particular, these are the number of performed diagnostic
tests and their sensitivity, number of asymptomatic cases, or simply a lack of
accuracy (see e.g. discussion in Refs. [3, 20, 15]). This needs to be taken
into account when interpreting results such as those presented in Table 1.

##  5 Conclusions

The analysis presented in this paper aims to add new information helping to
understand the behavoir of the COVID-19 expansion in European countries. Data
collected from 20 European countries on number of registered COVID-19 cases
was analyzed for a period of 15th of February to 1st of June 2020. Three
regimes of the expansion are identified and quantified – early exponential
expansion, damped exponential, and linear expansion. Coefficient quantifying
the velocity of the expansion in the early period ($b_{1}$) and coefficient
quantifying the number of cases at time zero ($N_{0}$) were evaluated and
compared for 20 European countries. The $b_{1}$ coefficient was found to lie
within one standard deviation from the average value for 11 out of 20
countries. The $N_{0}$ for Italy was found to be excessively high, 7 standard
deviations from the average value. The transition from the damped exponential
regime to the linear regime is discussed and quantified in terms of the “push-
back” parameter ($b_{2}$).

A method to predict the evolution of the value of registered cases in the
period of damped exponential expansion which is based on extrapolations in
parametric space was put forward. It was shown that it has an average
precision of 18% and 29% in the period of $15-55$ days after the start of the
damped exponential expansion for the case of Italy and Czechia, respectively.
This method may provide an elementary method to quantify the impact of
restrictive measures on the number of registered cases.

## Acknowledgment

The author would like to thank Jiří Dolejší, Dalibor Nosek, and Alfredo Iorio
for useful discussions and careful reading of the manuscript.

## Disclosure statement

The author declares no conflict of interest.

## Data availability statement

The data that support the findings of this study are openly available in Ref.
[7].

## References

  * [1] World Health Organization - Coronavirus disease (COVID-2019) situation reports.
URL https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-
reports

  * [2] C. T. Bauch, J. O. Lloyd-Smith, M. P. Coffee, A. P. Galvani, Dynamically modeling SARS and other newly emerging respiratory illnesses, Epidemiology 16 (6) (2005) 791–801.  doi:10.1097/01.ede.0000181633.80269.4c.
URL https://doi.org/10.1097/01.ede.0000181633.80269.4c

  * [3] E. Bertuzzo, L. Mari, D. Pasetto, S. Miccoli, R. Casagrandi, M. Gatto, A. Rinaldo, The geography of COVID-19 spread in italy and implications for the relaxation of confinement measures, Nature Communications 11 (1).  doi:10.1038/s41467-020-18050-2.
URL https://doi.org/10.1038/s41467-020-18050-2

  * [4] C. Hou, J. Chen, Y. Zhou, L. Hua, J. Yuan, S. He, Y. Guo, S. Zhang, Q. Jia, C. Zhao, J. Zhang, G. Xu, E. Jia, The effectiveness of quarantine of wuhan city against the corona virus disease 2019 (COVID-19): A well-mixed SEIR model analysis, Journal of Medical Virology 92 (7) (2020) 841–848.  doi:10.1002/jmv.25827.
URL https://doi.org/10.1002/jmv.25827

  * [5] K. Prem, Y. Liu, T. W. Russell, A. J. Kucharski, R. M. Eggo, N. Davies, M. Jit, P. Klepac, S. Flasche, S. Clifford, C. A. B. Pearson, J. D. Munday, S. Abbott, H. Gibbs, A. Rosello, B. J. Quilty, T. Jombart, F. Sun, C. Diamond, A. Gimma, K. van Zandvoort, S. Funk, C. I. Jarvis, W. J. Edmunds, N. I. Bosse, J. Hellewell, The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in wuhan, china: a modelling study, The Lancet Public Health 5 (5) (2020) e261–e270.  doi:10.1016/s2468-2667(20)30073-6.
URL https://doi.org/10.1016/s2468-2667(20)30073-6

  * [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (4) (2000) 599–653.  doi:10.1137/s0036144500371907.
URL https://doi.org/10.1137/s0036144500371907

  * [7] Worldometer - collection of online world statistics.
URL https://www.worldometers.info/coronavirus/

  * [8] E. Dong, H. Du, L. Gardner, An interactive web-based dashboard to track COVID-19 in real time, The Lancet Infectious Diseasesdoi:10.1016/s1473-3099(20)30120-1.
URL https://doi.org/10.1016/s1473-3099(20)30120-1

  * [9] R. Brun, F. Rademakers, ROOT: An object oriented data analysis framework, Nucl. Instrum. Meth. A 389 (1997) 81–86.  doi:10.1016/S0168-9002(97)00048-X.
  * [10] A. R. Tuite, V. Ng, E. Rees, D. Fisman, Estimation of COVID-19 outbreak size in italy, The Lancet Infectious Diseases 20 (5) (2020) 537.  doi:10.1016/s1473-3099(20)30227-9.
URL https://doi.org/10.1016/s1473-3099(20)30227-9

  * [11] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 (1-2) (2002) 29–48.  doi:10.1016/s0025-5564(02)00108-6.
URL https://doi.org/10.1016/s0025-5564(02)00108-6

  * [12] S. Zhao, S. S. Musa, H. Fu, D. He, J. Qin, Simple framework for real-time forecast in a data-limited situation: the zika virus (ZIKV) outbreaks in brazil from 2015 to 2016 as an example, Parasites & Vectors 12 (1).  doi:10.1186/s13071-019-3602-9.
URL https://doi.org/10.1186/s13071-019-3602-9

  * [13] S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang, Y. Lou, D. Gao, L. Yang, D. He, M. H. Wang, Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in china, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak, International Journal of Infectious Diseases 92 (2020) 214–217.  doi:10.1016/j.ijid.2020.01.050.
URL https://doi.org/10.1016/j.ijid.2020.01.050

  * [14] N. P. Jewell, J. A. Lewnard, B. L. Jewell, Predictive mathematical models of the COVID-19 pandemic, JAMA 323 (19) (2020) 1893.  doi:10.1001/jama.2020.6585.
URL https://doi.org/10.1001/jama.2020.6585

  * [15] D. Adam, Special report: The simulations driving the world’s response to COVID-19, Nature 580 (7803) (2020) 316–318.  doi:10.1038/d41586-020-01003-6.
URL https://doi.org/10.1038/d41586-020-01003-6

  * [16] W. C. Roda, M. B. Varughese, D. Han, M. Y. Li, Why is it difficult to accurately predict the COVID-19 epidemic?, Infectious Disease Modelling 5 (2020) 271–281.  doi:10.1016/j.idm.2020.03.001.
URL https://doi.org/10.1016/j.idm.2020.03.001

  * [17] Y.-H. Hsieh, S. Ma, Intervention measures, turning point, and reproduction number for dengue, singapore, 2005, The American Journal of Tropical Medicine and Hygiene 80 (1) (2009) 66–71.  doi:10.4269/ajtmh.2009.80.66.
URL https://doi.org/10.4269/ajtmh.2009.80.66

  * [18] Y.-H. Hsieh, Temporal patterns and geographic heterogeneity of zika virus (ZIKV) outbreaks in french polynesia and central america, PeerJ 5 (2017) e3015.  doi:10.7717/peerj.3015.
URL https://doi.org/10.7717/peerj.3015

  * [19] P. Castorina, A. Iorio, D. Lanteri, Data analysis on coronavirus spreading by macroscopic growth laws, International Journal of Modern Physics C 31 (07) (2020) 2050103.  doi:10.1142/s012918312050103x.
URL https://doi.org/10.1142/s012918312050103x

  * [20] Y. Bai, L. Yao, T. Wei, F. Tian, D.-Y. Jin, L. Chen, M. Wang, Presumed asymptomatic carrier transmission of COVID-19, JAMA 323 (14) (2020) 1406.  doi:10.1001/jama.2020.2565.
URL https://doi.org/10.1001/jama.2020.2565