# The numerical range of periodic banded Toeplitz operators

Benjamín A. Itzá-Ortiz, Rubén A. Martínez-Avendaño and Hiroshi Nakazato
Universidad Autónoma del Estado de Hidalgo<EMAIL_ADDRESS>Instituto
Tecnológico Autónomo de México<EMAIL_ADDRESS>Hirosaki
University<EMAIL_ADDRESS>
###### Abstract.

We prove that the closure of the numerical range of a $(n+1)$-periodic and
$(2m+1)$-banded Toeplitz operator can be expressed as the closure of the
convex hull of the uncountable union of numerical ranges of certain symbol
matrices. In contrast to the periodic $3$-banded (or tridiagonal) case, we
show an example of a $2$-periodic and $5$-banded Toeplitz operator such that
the closure of its numerical range is not equal to the numerical range of a
single finite matrix.

###### Key words and phrases:

Numerical range, Toeplitz operators, periodic operators, banded operators

###### 2020 Mathematics Subject Classification:

47A12, 15A60

The second author’s research is partially supported by the Asociación Mexicana
de Cultura A.C

##  1\. Introduction

Tridiagonal operators have long been a subject of interest not only for their
remarkable mathematical properties but more recently for their applications to
mathematical physics. In [11] a tridiagonal operator known as the “hopping
sign model” was introduced and subsequently studied by many researchers [8, 7,
12, 10, 14]. By following the work in [2] (see also [3]), the first two
authors showed in [13] that the numerical range of the $(n+1)$-periodic
tridiagonal Toeplitz operator on $\ell^{2}({\mathbb{N}}_{0})$, given by

$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}b_{0}&c_{0}&&&&&&&&\\\
a_{1}&b_{1}&c_{1}&&&&&&&\\\ &a_{2}&b_{2}&c_{2}&&&&&&\\\
&&\ddots&\ddots&\ddots&&&&&\\\ &&&a_{n}&b_{n}&c_{n}&&&&\\\
&&&&a_{0}&b_{0}&c_{0}&&&\\\ &&&&&\ddots&\ddots&\ddots&&\\\
&&&&&&a_{n-1}&b_{n-1}&c_{n-1}&\\\ &&&&&&&a_{n}&b_{n}&c_{n}&\\\
&&&&&&&&\ddots&\ddots&\ddots\end{pmatrix},$

is equal to the closure of the convex hull of the uncountable union of
numerical ranges of certain symbol matrices. In this paper, we extend this
result to periodic banded Toepliz operators of arbitrary band. While the
results in [2] already include the case for banded Toeplitz operators of
arbitrary band, they are constrained to biperiodic Toeplitz operators. Hence,
our results may also be regarded as a generalization of those in [2]. Later,
in [14], the authors were able to show that the closure of the numerical range
of a $(n+1)$-periodic tridiagonal operator is equal to the numerical range of
a single $2(n+1)\times 2(n+1)$ complex matrix. It is therefore natural to ask
if a similar result would also apply for general periodic banded operators. We
answer this question in the negative by exhibiting a biperiodic $5$-banded
Toeplitz operator for which the closure of its numerical range cannot be the
numerical range of a single complex matrix.

Also, as a consequence of the above representation of the numerical range of a
periodic banded Toeplitz operator as the closure of the convex hull of the
union of numerical ranges of finite matrices, we obtain the result that there
exists a real binary polynomial $g(x,y)$ such that the boundary of the
numerical range of a periodic banded Toeplitz operator is a subset of the set
of zeroes of $g(x,y)$.

We divide this work in three sections. In Section 2 we establish the
definitions needed and state and prove the main result of this paper. In
Section 3 we prove a consequence of the main result; namely the existence of a
real binary form such that the boundary of the numerical range of a periodic
banded Toeplitz operator is contained in the zero set of such form. Finally,
in Section 4 we give an example of a 2-periodic and 5-banded Toeplitz operator
for which the closure of the numerical range is not the numerical range of a
single matrix.

As usual, we denote by ${\mathbb{N}}_{0}$, ${\mathbb{N}}$, ${\mathbb{Z}}$,
${\mathbb{R}}$ and ${\mathbb{C}}$, the sets of nonnegative integers, the set
of positive integers, the set of integers, the set of real numbers, and the
set of complex numbers, respectively.

##  2\. The numerical range of a periodic banded Toeplitz operator

For $m\in{\mathbb{N}}_{0}$, we consider $2m+1$ complex-valued bi-infinite
sequences denoted by

$a^{(r)}=\left(a_{j}^{(r)}\right)_{j\in{\mathbb{Z}}},$

for $r=-m,\dots,0,\dots,m$. For $n\in{\mathbb{N}}_{0}$, we assume that each
sequence is $(n+1)$-periodic; i.e., $a_{j}^{(r)}=a_{j+n+1}^{(r)}$ for each
$r=-m,\dots,m$ and each $j\in{\mathbb{Z}}$. If $|r|>m$, then we define
$a_{j}^{(r)}=0$ for all $j\in{\mathbb{Z}}$.

We define the operator $T:\ell^{2}({\mathbb{Z}})\to\ell^{2}({\mathbb{Z}})$ by
the bi-infinite matrix with $(j,k)$ entry the number $a_{j}^{(j-k)}$; i.e., by
the bi-infinite matrix

$T=\left(a_{j}^{(j-k)}\right)_{j,k\in{\mathbb{Z}}}.$

In other words, the operator $T$ is $(2m+1)$-banded and the sequence $a^{(r)}$
appears in the $r$-th diagonal, with $r=-m,\dots,0,\dots,m$. The operator $T$
is a $(n+1)$-periodic and $(2m+1)$-banded Toeplitz operator on
$\ell^{2}({\mathbb{Z}})$.

We define the operator
$T_{+}:\ell^{2}({\mathbb{N}}_{0})\to\ell^{2}({\mathbb{N}}_{0})$ as
$T_{+}=PT_{|\ell^{2}({\mathbb{N}}_{0})}$, where $P$ is the orthogonal
projection of $\ell^{2}({\mathbb{Z}})$ onto $\ell^{2}({\mathbb{N}}_{0})$,
thought of as a subspace of $\ell^{2}({\mathbb{Z}})$. The matrix of the
operator $T_{+}$ is given by

$T_{+}=\left(a_{j}^{(j-k)}\right)_{j,k\in{\mathbb{N}}_{0}},$

that is, the $(j,k)$ entry of the matrix of $T_{+}$ is

$\langle T_{+}e_{k},e_{j}\rangle=a_{j}^{(j-k)}$ (1)

for all $j,k\in{\mathbb{N}}_{0}$. In expanded form,

$T_{+}=\begin{pmatrix}a_{0}^{(0)}&a_{0}^{(1)}&\dots&a_{0}^{(m)}&0&\cdots\\\\[5.0pt]
a_{1}^{(-1)}&a_{1}^{(0)}&a_{1}^{(1)}&\cdots&a_{1}^{(m)}&0&\cdots&\\\\[5.0pt]
\vdots&&\ddots&&&\ddots\\\\[5.0pt]
a_{m}^{(-m)}&\cdots&a_{m}^{(-1)}&a_{m}^{(0)}&a_{m}^{(1)}&\cdots&a_{m}^{(m)}&\cdots\\\\[5.0pt]
0&a_{m+1}^{(-m)}&\cdots&a_{m+1}^{(-1)}&a_{m+1}^{(0)}&a_{m+1}^{(1)}&\cdots&\cdots\\\\[5.0pt]
\vdots&0&a_{m+2}^{(-m)}&\cdots&a_{m+2}^{(-1)}&a_{m+2}^{(0)}&a_{m+2}^{(1)}&\cdots\\\\[5.0pt]
\vdots&\vdots&0&a_{m+3}^{(-m)}&\cdots&a_{m+3}^{(-1)}&a_{m+3}^{(0)}&\cdots\\\\[5.0pt]
\vdots&&&\ddots&\ddots&&&\ddots\end{pmatrix}.$

We also will refer to $T_{+}$ as an $(n+1)$-periodic and $(2m+1)$-banded
Toeplitz operator, acting on $\ell^{2}({\mathbb{N}}_{0})$.

The operators $T$ and $T_{+}$ defined by the matrices above are bounded, since
the sum of the moduli of each row and each column is uniformly bounded (see,
e.g., [15, Example 2.3])

As in [13], we introduce the symbol $\Phi(\theta)$ of the $(n+1)$-periodic and
$(2m+1)$-banded Toeplitz operators $T$ and $T_{+}$. We shall show that the
matrix $\Phi(\theta)$ with trigonometric polynomial entries in $\theta$ plays
a crucial role in expressing the closure of the numerical range $W(T_{+})$ of
the operator $T_{+}$.

######  Definition 2.1.

We define the symbol $\Phi$ of a $(n+1)$-periodic and $(2m+1)$-banded Toeplitz
operator $T$ on $\ell^{2}({\mathbb{Z}})$ as the function $\Phi:[0,2\pi)\to
M_{n+1}({\mathbb{C}})$ given as follows. For each $j,k=0,1,2,\dots,n$, the
$(j,k)$ entry of $\Phi(\theta)$ is

$\sum_{u\in{\mathbb{Z}}}e^{iu\theta}\langle Te_{k+u(n+1)},e_{j}\rangle.$

We define the symbol $\Phi$ of a $(n+1)$-periodic and $(2m+1)$-banded Toeplitz
operator $T_{+}$ on $\ell^{2}({\mathbb{N}}_{0})$ as the symbol of $T$.

Remark. We should point out that this definition of the symbol does not
coincide with the symbol $T_{\theta}$ defined in [2] (cf. [3]) for the case
$n+1=2$, but it is simple to see that both matrices, in this case, are
unitarily equivalent: $\Phi(\theta)={\rm diag}(1,e^{-i\theta})T_{\theta}\ {\rm
diag}(1,e^{i\theta})$. On the other hand, our definition of symbol, when
$m=1$, coincides with that of [13].

In [13], one can see the explicit form for the symbol when $m=1$. If $m=2$,
the explicit form of the symbols $\Phi(\theta)$ will be shown next. For this,
consider the $(n+1)$-periodic and $5$-banded Toeplitz operator $T$ and we set
$p_{j}:=a_{j}^{(-2)}$, $q_{j}:=a_{j}^{(-1)}$, $r_{j}=a_{j}^{(0)}$,
$s_{j}:=a_{j}^{(1)}$, and $t_{j}:=a_{j}^{(2)}$, with $j=0,1,2,\ldots,n$.

If $n+1=2$, then the symbol of $T$ is the $2\times 2$ matrix

$\Phi(\theta)=\begin{pmatrix}p_{0}e^{-i\theta}+r_{0}+t_{0}e^{i\theta}&q_{0}e^{-i\theta}+s_{0}\\\
q_{1}+s_{1}e^{i\theta}&p_{1}e^{-i\theta}+r_{1}+t_{1}e^{i\theta}\end{pmatrix}.$

If $n+1=3$, then the symbol of $T$ is the $3\times 3$ matrix

$\Phi(\theta)=\begin{pmatrix}r_{0}&p_{0}e^{-i\theta}+s_{0}&q_{0}e^{-i\theta}+t_{0}\\\
q_{1}+t_{1}e^{i\theta}&r_{1}&p_{1}e^{-i\theta}+s_{1}\\\
s_{2}e^{i\theta}+p_{2}&q_{2}+t_{2}e^{i\theta}&r_{2}\end{pmatrix}.$

If $n+1=4$, then the symbol of $T$ is the $4\times 4$ matrix

$\Phi(\theta)=\begin{pmatrix}r_{0}&s_{0}&p_{0}e^{-i\theta}+t_{0}&q_{0}e^{-i\theta}\\\
q_{1}&r_{1}&s_{1}&a_{1}e^{-i\theta}+e_{1}\\\
p_{2}+t_{2}e^{i\theta}&q_{2}&r_{2}&s_{2}\\\
s_{3}e^{i\theta}&p_{3}+t_{3}e^{i\theta}&q_{3}&r_{3}\end{pmatrix}.$

If $n+1\geq 5$, then the symbol of $T$ is the $(n+1)\times(n+1)$ matrix

$\Phi(\theta)=\begin{pmatrix}r_{0}&s_{0}&t_{0}&&&p_{0}e^{-i\theta}&q_{0}e^{-i\theta}\\\
q_{1}&r_{1}&s_{1}&t_{1}&&&p_{1}e^{-i\theta}\\\
p_{2}&q_{2}&r_{2}&s_{2}&t_{2}&&\\\ &\ddots&\ddots&\ddots&\ddots&\ddots&\\\
&&p_{n-2}&q_{n-2}&r_{n-2}&s_{n-2}&t_{n-2}\\\
t_{n-1}e^{i\theta}&&&p_{n-1}&q_{n-1}&r_{n-1}&s_{n-1}\\\
s_{n}e^{i\theta}&t_{n}e^{i\theta}&&&p_{n}&q_{n}&r_{n}\end{pmatrix}.$

Observe that, for arbitrary $n$ and $m$ the sum defining the symbols is
finite: indeed, for each $(j,k)$ we are summing over the $(j,k+u(n+1))$
entries of $T$ as $u$ ranges over the integers; that is, we are looking at the
$j$-th row of $T$ and there are only finitely many nonzero entries in this row
for if $|j-(k+u(n+1)|>m$, then $\langle Te_{k+u(n+1)},e_{j}\rangle=0$, since
$T$ is $(2m+1)$-banded.

Furthermore, notice that for $|j-(k+u(n+1)|\leq m$ to hold we need
$j-(k+u(n+1))$ to be in the set $\\{-m,-(m-1),\dots,-1,0,1,\dots(m-1),m\\}$;
since, as $u$ ranges over ${\mathbb{Z}}$, the different values of
$j-(k+u(n+1))$ vary by multiples of $n+1$, there are at most
$\left\lceil\frac{2m+1}{n+1}\right\rceil$ values of $u$ for which $-m\leq
j-(k+u(n+1))\leq m$.

In the case $n+1\geq 2m+1$; i.e., the period is greater or equal than the
length of the band, then there is at most one integer value of $u$ for which
$|j-(k+u(n+1)|\leq m$ and hence we have that $\Phi(\theta)$ equals

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}a_{0}^{(0)}&a_{0}^{(1)}&\cdots&\cdots&a_{0}^{(m)}&0&\cdots&0&a_{0}^{(-m)}e^{-i\theta}&\cdots&a_{0}^{(-2)}e^{-i\theta}&a_{0}^{(-1)}e^{-i\theta}\\\\[5.0pt]
a_{1}^{(-1)}&a_{1}^{(0)}&\ddots&&\cdots&a_{1}^{(m)}&0&\cdots&0&\ddots&&a_{1}^{(-2)}e^{-i\theta}\\\\[5.0pt]
\vdots&\ddots&\ddots&&&&&&&&\ddots&\vdots\\\\[5.0pt]
\vdots&&&\ddots&&&&&&&0&a_{m-1}^{(-m)}e^{-i\theta}\\\\[5.0pt]
a_{m}^{(-m)}&&&&a_{m}^{(0)}&&&&&&&0\\\\[5.0pt]
0&a_{m+1}^{(-m)}&&&&a_{m+1}^{(0)}&&&&&&\vdots\\\\[5.0pt]
\vdots&0&&&&&\ddots\\\\[5.0pt]
\vdots&&&&&&&\ddots&&&a_{n-m-1}^{(m)}&0\\\\[5.0pt]
0&\vdots&&&&&&&&&&a_{n-m}^{(m)}\\\\[5.0pt]
a_{n-m+1}^{(m)}e^{i\theta}&0\\\\[5.0pt] \vdots&\ddots&&&&&&&&\ddots\\\\[5.0pt]
a_{n-1}^{(2)}e^{i\theta}&&&\ddots&0&\cdots&0&a_{n-1}^{(-m)}&&&a_{n-1}^{(0)}&a_{n-1}^{(1)}\\\\[5.0pt]
a_{n}^{(1)}e^{i\theta}&a_{n}^{(2)}e^{i\theta}&\cdots&&a_{n}^{(m)}e^{i\theta}&0&\cdots&0&a_{n}^{(-m)}&\cdots&a_{n}^{(-1)}&a_{n}^{(0)}\end{pmatrix}$
(2)

Now we define a matrix which will be useful in what follows.

######  Definition 2.2.

Fix $s\in{\mathbb{N}}$, $s\geq 2$, such that $s(n+1)\geq 2m+1$ and let
$\mu:=s(n+1)$. We then define a $\mu\times\mu$ matrix $C_{\mu}$ as follows.
For each $j,k=0,1,\dots,\mu-1$, the $(j,k)$ entry of $C_{\mu}$ is

$\sum_{u\in{\mathbb{Z}}}\langle Te_{k+u\mu},e_{j}\rangle.$

Observe that all terms in the above sum are zero, except at most one. Indeed,
the above sum is formed by adding some of the terms in the $j$-th row of $T$;
namely, the entries $(j,k+u\mu)$ for $u\in{\mathbb{Z}}$. But assume there are
two entries in the $j$-th row inside the band, the entries $(j,k+u_{1}\mu)$
and $(j,k+u_{2}\mu)$. Then, we must have

$-m\leq j-(k+u_{1}\mu)\leq m\quad\text{ and }\quad-m\leq j-(k+u_{2}\mu)\leq m$

but this implies

$-2m\leq(u_{2}-u_{1})\mu\leq 2m,$

which, since $\mu\geq 2m+1$ this is impossible unless $u_{1}=u_{2}$.

Notice that if the sequences $a^{(r)}$, for $r=-m,\dots,0,\dots,m$, are
regarded as $\mu$-periodic, instead of $n+1$ periodic, then $C_{\mu}$ is just
the symbol $\Phi$ evaluated at $0$, and hence it has the same form as in
expression (2) (but it is of size $\mu\times\mu$ instead of
$(n+1)\times(n+1)$). Hence if we erase the last $m$ columns and $m$ rows of
$C_{\mu}$, we obtain $T_{\mu-m}$ a compression of the Toeplitz operator
$T_{+}$. Therefore, we have

$W(T_{s(n+1)-m})\subseteq W(C_{\mu})$ (3)

under the given condition $s(n+1)\geq 2m+1$.

As in [13], for $0\leq p\leq n$ and $0\leq q<s$ we define the unit vectors in
${\mathbb{C}}^{\mu}$

$\displaystyle f_{p,q}$
$\displaystyle=\frac{1}{\sqrt{s}}\sum_{u=0}^{s-1}\rho^{uq}e_{p+u(n+1)}$
$\displaystyle=\frac{1}{\sqrt{s}}(\underbrace{0,\dots,0}_{p},1,\underbrace{0,\dots,0}_{n},\rho^{q},\underbrace{0,\dots,0}_{n},\rho^{2q},\underbrace{0,\dots,0}_{n},\dots,\rho^{(s-1)q},\underbrace{0,\dots,0}_{n-p})^{\mathsf{T}},$

where $\rho=\exp(i2\pi/s)$, and we define the $\mu\times\mu$ unitary matrix
$U$ by

$U=\setcounter{MaxMatrixCols}{14}\begin{pmatrix}\vrule&\vrule&&\vrule&\vrule&\vrule&&\vrule&&&\vrule&\vrule&&\vrule\\\
f_{0,0}&f_{1,0}&\dots&f_{n,0}&f_{0,1}&f_{1,1}&\dots&f_{n,1}&\dots&\dots&f_{0,s-1}&f_{1,s-1}&\dots&f_{n,s-1}\\\
\vrule&\vrule&&\vrule&\vrule&\vrule&&\vrule&&&\vrule&\vrule&&\vrule\end{pmatrix}.$

The matrix $U$ is useful in the next proposition.

######  Proposition 2.3.

Let $n$, $s$ and $\mu$ be as above. Then $C_{\mu}$ is unitarily equivalent to
a block diagonal matrix. More precisely,

$U^{*}C_{\mu}U=\Phi(0)\oplus\Phi(2\pi/s)\oplus\Phi(4\pi/s)\oplus\cdots\oplus\Phi(2(s-1)\pi/s).$

###### Proof.

Let $p_{1},p_{2},q_{1},q_{2}$ be integers such that $0\leq q_{1},q_{2}\leq
s-1$ and $0\leq p_{1},p_{2}\leq n$. Observe that

$\displaystyle\langle C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle$
$\displaystyle=\frac{1}{s}\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\langle
C_{\mu}\rho^{uq_{1}}e_{p_{1}+u(n+1)},\rho^{vq_{2}}e_{p_{2}+v(n+1)}\rangle$
$\displaystyle=\frac{1}{s}\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\rho^{uq_{1}}\rho^{-vq_{2}}\langle
C_{\mu}e_{p_{1}+u(n+1)},e_{p_{2}+v(n+1)}\rangle$
$\displaystyle=\frac{1}{s}\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\rho^{uq_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+u(n+1)+w\mu},e_{p_{2}+v(n+1)}\rangle$
$\displaystyle=\frac{1}{s}\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\rho^{uq_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+(u-v)(n+1)+w\mu},e_{p_{2}}\rangle,$

where the last equality follows from the $(n+1)$-periodicity of $T$.

Making the change of variables $u=v+r$, we obtain

$\displaystyle\langle C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle$
$\displaystyle=\frac{1}{s}\sum_{v=0}^{s-1}\sum_{r=-v}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle.$

For each $v=0,1,2,\dots,s-1$, we consider the inner sum of the above
expression and rewrite it as

$\displaystyle\sum_{r=-v}^{-v+s-1}$
$\displaystyle\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{r=-v}^{-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle+\sum_{r=0}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle.$

Making another change of variables in the first summand above, we obtain

$\displaystyle\sum_{r=-v}^{-1}$
$\displaystyle\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle+\sum_{r=0}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{r=s-v}^{s-1}\rho^{(v+r-s)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+(r-s)(n+1)+w\mu},e_{p_{2}}\rangle+\sum_{r=0}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{r=s-v}^{s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+(w-1)\mu},e_{p_{2}}\rangle+\sum_{r=0}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{r=s-v}^{s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle+\sum_{r=0}^{-v+s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+w\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{r=0}^{s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+(w+1)\mu},e_{p_{2}}\rangle.$

Hence

$\begin{split}\langle
C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle&=\frac{1}{s}\sum_{v=0}^{s-1}\sum_{r=0}^{s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\sum_{w\in{\mathbb{Z}}}\langle
Te_{p_{1}+r(n+1)+(w+1)\mu},e_{p_{2}}\rangle\\\
&=\frac{1}{s}\sum_{w\in{\mathbb{Z}}}\sum_{r=0}^{s-1}\sum_{v=0}^{s-1}\rho^{(v+r)q_{1}}\rho^{-vq_{2}}\langle
Te_{p_{1}+r(n+1)+(w+1)\mu},e_{p_{2}}\rangle\\\
&=\frac{1}{s}\sum_{w\in{\mathbb{Z}}}\sum_{r=0}^{s-1}\rho^{rq_{1}}\langle
Te_{p_{1}+r(n+1)+(w+1)\mu},e_{p_{2}}\rangle\sum_{v=0}^{s-1}\rho^{v(q_{1}-q_{2})}.\end{split}$
(4)

Now, assume $q_{1}\neq q_{2}$. Since this implies that
$\sum_{v=0}^{s-1}\rho^{v(q_{1}-q_{2})}=0$, equation (4) gives

$\langle C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle=0.$

If $q_{1}=q_{2}$, then $\sum_{v=0}^{s-1}\rho^{v(q_{1}-q_{2})}=s$ and hence
equation (4) gives

$\displaystyle\langle C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle$
$\displaystyle=\sum_{w\in{\mathbb{Z}}}\sum_{r=0}^{s-1}\rho^{rq_{1}}\langle
Te_{p_{1}+r(n+1)+(w+1)\mu},e_{p_{2}}\rangle$
$\displaystyle=\sum_{w\in{\mathbb{Z}}}\sum_{r=0}^{s-1}\rho^{rq_{1}}\langle
Te_{p_{1}+(r+(w+1)s)(n+1)},e_{p_{2}}\rangle$
$\displaystyle=\sum_{w\in{\mathbb{Z}}}\sum_{r=0}^{s-1}\rho^{(r+(w+1)s)q_{1}}\langle
Te_{p_{1}+(r+(w+1)s)(n+1)},e_{p_{2}}\rangle,$

since $\rho^{s}=1$.

Observe that when $r$ ranges from $0$ to $s-1$ and $w$ ranges in
${\mathbb{Z}}$, the number $r+(w+1)s$ ranges over all the integers, and hence
the previous expression can be written as

$\langle
C_{\mu}f_{p_{1},q_{1}},f_{p_{2},q_{2}}\rangle=\sum_{u\in{\mathbb{Z}}}\rho^{uq_{1}}\langle
Te_{p_{1}+u(n+1)},e_{p_{2}}\rangle,$

which equals the $(p_{2},p_{1})$ entry of the matrix $\Phi\left(\frac{2\pi
q_{1}}{s}\right)$.

The above computations show that the matrix of $C_{\mu}$ with respect to the
orthogonal basis $\\{f_{p,q}\,:\,0\leq p\leq n,\ 0\leq q\leq s-1\\}$ is block-
diagonal and each of the $s$ blocks on the diagonal is of the form
$\Phi\left(\frac{2\pi q}{s}\right)$, for $q=0,1,\dots,s-1$. This proves the
desired result. ∎

The proof of the next proposition follows the same reasoning as that of
Corollary 2.4 in [13]. We include the details here for completeness.

######  Proposition 2.4.

Let $T_{+}$ be a $(n+1)$-periodic and $(2m+1)$-banded Toeplitz operator on
$\ell^{2}({\mathbb{N}}_{0})$ and let $\Phi$ be its symbol. Then

$\overline{W(T_{+})}\subseteq\overline{\mathrm{Conv}\left(\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))\right)}.$

###### Proof.

For every positive integer $s$ satisfying $(n+1)s>m$, we define the matrix
$C_{(n+1)s}$ as in Definition 2.2. By Proposition 2.3, we get

$W(C_{(n+1)s})=\mathrm{Conv}\left(\bigcup_{k=0}^{s-1}W\left(\Phi\left(\tfrac{2k\pi}{s}\right)\right)\right).$

By removing the last $m$ rows and the last $m$ columns in $C_{(n+1)s}$, we
obtain the matrix $T_{(n+1)s-m}$ and therefore, by the inclusion (3), we have

$W(T_{(n+1)s-m})\subseteq\mathrm{Conv}\left(\bigcup_{k=0}^{s-1}W\left(\Phi\left(\tfrac{2k\pi}{s}\right)\right)\right).$

Now, clearly

$\bigcup_{k=0}^{s-1}W\left(\Phi\left(\tfrac{2k\pi}{s}\right)\right)\subseteq\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))$

and thus it follows that

$W(T_{(n+1)s-m})\subseteq\mathrm{Conv}\left(\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))\right).$

Hence, since

$W(T_{1})\subseteq W(T_{2})\subseteq W(T_{3})\subseteq\dots,$

we obtain

$\bigcup_{k=1}^{\infty}W(T_{k})\subseteq\mathrm{Conv}\left(\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))\right).$

Applying now Proposition 2.3 in [13], we obtain the desired result. ∎

For the next theorem, we first establish the following lemma.

######  Lemma 2.5.

Let $n,s\in{\mathbb{N}}$, $s>1$, and let $\mu=s(n+1)$. For each
$r=0,1,2,\dots,s-1$, if $\lambda$ is an eigenvalue of $\Phi(\tfrac{2r\pi}{s})$
with eigenvector $\vec{v}=(v_{0},v_{1},\ldots,v_{n})\in{\mathbb{C}}^{n+1}$,
then $\lambda$ is an eigenvalue of the matrix $C_{\mu}$ with eigenvector

$\overrightarrow{V_{r}}=(v_{0},\dots,v_{n},v_{0}\rho^{r},\dots,v_{n}\rho^{r},\dots,v_{0}\rho^{(s-1)r},\dots,v_{n}\rho^{(s-1)r})\in{\mathbb{C}}^{\mu},$

where $\rho=\exp(i2\pi/s)$. Conversely, if $\lambda$ is an eigenvalue of
$C_{\mu}$, there is $r=0,1,2,\dots,s-1$, such that $\lambda$ is an eigenvalue
of $\Phi(\tfrac{2r\pi}{s})$ with eigenvector $\vec{v}$ and the corresponding
$\overrightarrow{V_{r}}$ is an eigenvector for $C_{\mu}$ with eigenvalue
$\lambda$.

###### Proof.

A straightforward computation shows that $\overrightarrow{V_{r}}$ is indeed an
eigenvector for the eigenvalue $\lambda$ of $C_{\mu}$ if $\vec{v}$ is
eigenvector for the eigenvalue $\lambda$ of $\Phi(\tfrac{2r\pi}{s})$.

For the last assertion, as a consequence of Proposition 2.3, we have that

$\sigma(C_{\mu})=\bigcup_{r=0}^{s-1}\sigma\left(\Phi\left(\tfrac{2r\pi}{s}\right)\right).$

Therefore, if $\lambda\in\sigma(C_{\mu})$ is given, there is $r$ such that
$\lambda\in\sigma\left(\Phi(\tfrac{2r\pi}{s})\right)$. Let $\vec{v}$ be an
eigenvector for $\Phi(\tfrac{2r\pi}{s})$ corresponding to the eigenvalue
$\lambda$. Then the corresponding vector $\overrightarrow{V_{r}}$ does the
job. ∎

The following generalizes Theorem 2.6 in [13].

######  Theorem 2.6.

Let $n,m\in{\mathbb{N}}$ and let $T_{+}$ be an $(n+1)$-periodic and
$(2m+1)$-banded selfadjoint Toeplitz operator on $\ell^{2}({\mathbb{N}}_{0})$
and let $\Phi$ be its symbol (which is a Hermitian matrix). Let
$\lambda^{-}(\theta)$ and $\lambda^{+}(\theta)$ denote the smallest and
largest eigenvalues of $\Phi(\theta)$ and set

$a=\min_{\theta\in[0,2\pi)}\lambda^{-}(\theta)\quad\text{ and }\quad
b=\max_{\theta\in[0,2\pi)}\lambda^{+}(\theta).$

Then the closure of $W(T_{+})$ is the closed interval

$\overline{W(T_{+})}=[a,b].$

###### Proof.

The proof follows the same argument as that of Theorem 2.6 in [13]. We include
the first part since some details are different and for the sake of
completeness. We first show that $[a,b]\subseteq\overline{W(T_{+})}$. Let
$s\in{\mathbb{N}}$ such that $\mu:=s(n+1)\geq 2m+1$. Let
$\lambda_{s}\in\sigma(C_{s(n+1)})$. Lemma 2.5 then implies that there exists
$r=0,1,2,\dots,s-1$ such that
$\lambda_{s}\in\sigma\left(\Phi(\frac{2r\pi}{s})\right)$. We can choose an
eigenvector $\vec{v}=(v_{0},v_{1},\ldots,v_{n})$ of $\Phi(\frac{2r\pi}{s})$ of
norm $1/\sqrt{s}$. In the manner of Lemma 2.5, the vector
$\overrightarrow{V_{r}}$ is an eigenvector of norm $1$ for the matrix
$C_{s(n+1)}$.

Recall that $T_{s(n+1)}$ denotes the compression of the operator $T_{+}$ to
the subspace of $\ell^{2}({\mathbb{N}}_{0})$ spanned by the first $\mu=s(n+1)$
vectors in the canonical basis: $\\{e_{0},e_{1},e_{2},\dots,e_{\mu-1}\\}$. We
then have

$\displaystyle\left\langle
T_{s(n+1)}\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle$
$\displaystyle=\left\langle
C_{s(n+1)}\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle+\left\langle\left(T_{s(n+1)}-C_{s(n+1)}\right)\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle$
$\displaystyle=\lambda_{s}+\left\langle\left(T_{s(n+1)}-C_{s(n+1)}\right)\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle.$

Since the real number $\left\langle
T_{s(n+1)}\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle$ is
contained in $W(T_{s(n+1)})\subseteq W(T_{+})$, we see that

$\lambda_{s}+\left\langle\left(T_{s(n+1)}-C_{s(n+1)}\right)\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle\in
W(T_{+}).$ (5)

Observe that the entries of $C_{s(n+1)}$ and $T_{s(n+1)}$ coincide everywhere
except at the “triangular corners” on the top-right and down-left positions.
Hence $b_{j,k}$, the $(j,k)$ entry of $C_{s(n+1)}-T_{s(n+1)}$, is zero if
$|j-k|<s(n+1)-m$, and therefore at most $m(m+1)$ entries of the matrix
$C_{s(n+1)}-T_{s(n+1)}$ are nonzero. If we set
$\overrightarrow{V_{r}}=(w_{0},w_{1},w_{2},\dots,w_{\mu-1})$, and we recall
that the modulus of each coordinate of $\overrightarrow{V_{r}}$ is equal to
the modulus of some coordinate of $\vec{v}$, we have
$|w_{j}|\leq\|\vec{v}\,\|$, and hence

$\displaystyle\left|\left\langle\left(C_{s(n+1)}-T_{s(n+1)}\right)\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle\right|$
$\displaystyle\leq\sum_{j=0}^{\mu-1}\sum_{k=0}^{\mu-1}|b_{j,k}|\ |w_{j}|\
|w_{k}|$ $\displaystyle\leq\sum_{j=0}^{\mu-1}\sum_{k=0}^{\mu-1}|b_{j,k}|\
\|\vec{v}\,\|^{2}$
$\displaystyle\leq\max\left\\{|b_{j,k}|\,:\,j,k=0,1,2,\dots,\mu-1\right\\}\
m(m+1)\ \|\vec{v}\,\|^{2}.$

Since each nonzero $b_{j,k}$ is multiple, of modulus one, of an element of the
finite set $\left\\{a_{j}^{(k)}\right\\}$, and since
$\|\vec{v}\|^{2}=\frac{1}{s}$ it follows that

$\left|\left\langle\left(C_{s(n+1)}-T_{s(n+1)}\right)\overrightarrow{V_{r}},\overrightarrow{V_{r}}\right\rangle\right|\leq\max\left\\{\left|a_{j}^{(k)}\right|\,:\,j=0,1,\dots,n,\
k=-m,\dots,0,\dots,m\right\\}\ \frac{m(m+1)}{s}.$

Hence, the previous estimate ensures that if we can find a sequence
$(\lambda_{s})$ which converges as $s\to\infty$, then, by expression (5), we
would have

$\lim_{s\to\infty}\lambda_{s}\in\overline{W(T_{+})}.$ (6)

We now propose two suitable sequences of eigenvalues $(\lambda_{s})$. For each
$s$, let $\lambda_{s}^{-}$ and $\lambda_{s}^{+}$ denote the smallest and
largest eigenvalues of $C_{s(n+1)}$, respectively. We will prove that

$\lim_{s\to\infty}\lambda_{s}^{-}=\min_{\theta\in[0,2\pi)}\lambda^{-}(\theta)$
(7)

where $\lambda^{-}(\theta)$ is the smallest eigenvalue of $\Phi(\theta)$. It
will follow then, by taking the limit as $s\to\infty$, that
$a=\min_{\theta\in[0,2\pi)}\lambda^{-}(\theta)\in\overline{W(T_{+})}$.

By Proposition 2.3 and since $\lambda_{s}^{-}$ is the smallest eigenvalue of
$C_{s(n+1)}$, we have

$\sigma(C_{s(n+1)})=\bigcup_{r=0}^{s-1}\sigma\left(\Phi\left(\frac{2r\pi}{s}\right)\right).$

Bu then $\lambda_{s}^{-}$ is not only an eigenvalue of $\Phi(\frac{2\pi
r}{s})$ for some $r$ but it is, in fact, the smallest among the eigenvalues of
all symbols $\Phi(\frac{2r\pi}{s})$ for $r=0,1,\dots,s-1$; i.e.,
$\lambda^{-}_{s}=\lambda^{-}\left(\frac{2r\pi}{s}\right)$ for some
$r=0,1,2,\dots,s-1$.

Let

$\lambda^{-}(\theta^{*})=\min_{\theta\in[0,2\pi)}\lambda^{-}(\theta)$

where $\theta^{*}$ is the point where the minimum is reached. Therefore
$\lambda^{-}(\theta^{*})\leq\lambda^{-}\left(\frac{2r\pi}{s}\right)=\lambda_{s}^{-}$.
Using the continuity of $\lambda^{-}(\cdot)$, for all $\epsilon>0$ there is
$\delta>0$ such that $|\theta-\theta^{*}|<\delta$ implies
$\lambda^{-}(\theta^{*})\leq\lambda^{-}(\theta)<\lambda^{-}(\theta^{*})+\epsilon$.
Also, there exists $N\in{\mathbb{N}}$ such that for $s\geq N$ we may choose
$0\leq j\leq s-1$ such that $\left|\frac{2j\pi}{s}-\theta^{*}\right|<\delta$.
Thus for all $s\geq N$,

$\lambda^{-}(\theta^{*})\leq\lambda_{s}^{-}=\lambda^{-}\left(\frac{2r\pi}{s}\right)\leq\lambda^{-}\left(\frac{2j\pi}{s}\right)<\lambda^{-}(\theta^{*})+\epsilon.$

This proves Equation (7). Analogously, one can show that

$\lim_{s\to\infty}\lambda_{s}^{+}=\max_{\theta\in[0,2\pi)}\lambda^{+}(\theta),$

where $\lambda_{s}^{+}$ is the largest eigenvalue of $C_{s(n+1)}$ and
$\lambda^{+}(\theta)$ is the largest eigenvalue of $\Phi(\theta)$.

Hence, Expression (6) proves that

$a=\min_{\theta\in[0,2\pi)}\lambda^{-}(\theta)=\lim_{s\to\infty}\lambda_{s}^{-}\in\overline{W(T_{+})}$

and

$b=\max_{\theta\in[0,2\pi)}\lambda^{+}(\theta)=\lim_{s\to\infty}\lambda_{s}^{+}\in\overline{W(T_{+})}.$

Convexity then shows that $[a,b]\subseteq\overline{W(T_{+})}$.

The proof of the inclusion $\overline{W(T_{+})}\subseteq[a,b]$ is now exactly
the same as that of Theorem 2.6 in [13] and we omit the proof. ∎

Acting as in the proof of Theorem 2.8 in [13], the above Theorem 2.6 implies
the following result.

######  Theorem 2.7.

Let $n,m\in{\mathbb{N}}$ and let $T_{+}$ be an $(n+1)$-periodic and
$(2m+1)$-banded Toeplitz operator acting on $\ell^{2}({\mathbb{N}}_{0})$. Let
$\Phi(\theta)$ be the symbol of $T_{+}$. Then

$\overline{W(T_{+})}=\overline{\mathrm{Conv}\left(\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))\right)}.$

Also, imitating the proof of Proposition 1.1 in [13] we obtain that
$\overline{W(T)}=\overline{W(T_{+})}$ and hence we have the following result.

######  Corollary 2.8.

Let $n,m\in{\mathbb{N}}$ and let $T$ be an $(n+1)$-periodic and
$(2m+1)$-banded Toeplitz operator acting on $\ell^{2}({\mathbb{Z}})$. Let
$\Phi(\theta)$ be the symbol of $T$. Then

$\overline{W(T)}=\overline{\mathrm{Conv}\left(\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))\right)}.$

##  3\. The boundary of the numerical range of a periodic banded Toeplitz
operator

In this section we shall consider the boundary of the numerical range
$W(T_{+})$ of the $(n+1)$-periodic and $(2m+1)$-banded Toeplitz operator
$T_{+}$. By Theorem 2.7, every boundary point of $W(T_{+})$ lies on the
boundary of the set

${\mathcal{O}}:=\bigcup_{\theta\in[0,2\pi)}W(\Phi(\theta))$

or on one of the support lines of $\mathcal{O}$. Each symbol matrix
$\Phi(\theta)$ is an $(n+1)\times(n+1)$ complex matrix whose entries are
trigonometric polynomials in $\theta$. So we can express $\Phi(\theta)$ as a
finite sum of the form

$\Phi(\theta)=\sum_{r=-L}^{L}A(r)e^{ir\theta},$

for some $L\in{\mathbb{N}}$ and where each $(n+1)\times(n+1)$ matrix $A(r)$
does not depend on $\theta$. Hence, for each
$\xi=(\xi_{0},\xi_{1},\dots,\xi_{n})\in{\mathbb{C}}^{n+1}$, we have

$\langle\Phi(\theta)\xi,\xi\rangle=\sum_{r=-L}^{L}\langle
A(r)e^{ir\theta}\xi,\xi\rangle=\sum_{r=-L}^{L}e^{ir\theta}\sum_{j=0}^{n}\sum_{k=0}^{n}a_{j,k}(r)\xi_{j}\overline{\xi_{k}},$

where $a_{j,k}(r)$ is the $(j,k)$ entry of the matrix $A(r)$. Hence

${\mathcal{O}}=\left\\{\sum_{r=-L}^{L}e^{ir\theta}\sum_{j=0}^{n}\sum_{j=0}^{n}a_{j,k}(k)\xi_{j}\overline{\xi_{k}}:\xi=(\xi_{0},\xi_{1},\dots,\xi_{n})\in{\mathbb{C}}^{n+1},\|\xi\|=1,0\leq\theta<2\pi\right\\}.$

Under the identification of ${\mathbb{C}}$ with ${\mathbb{R}}^{2}$, the set

$\\{\xi\in{\mathbb{C}}^{n+1}\ :\ \|\xi\|=1\\}\times S^{1}$

can be written as

$\\{(x_{0},y_{0},x_{1},y_{1},\dots,x_{n}y_{n})\in{\mathbb{R}}^{2n+2}\ :\
x_{0}^{2}+y_{0}^{2}+x_{1}^{2}+y_{1}^{2}+\dots
x_{n}^{2}+y_{n}^{2}=1\\}\times\\{(x,y)\in{\mathbb{R}}^{2}:x^{2}+y^{2}=1\\},$

which is an algebraic set (see, for example, [4]). But then, ${\mathcal{O}}$
is the image of a real algebraic set under a polynomial map from
${\mathbb{R}}^{2n+4}$ to ${\mathbb{R}}^{2}$. Then, by the Tarski-Seidenberg
Theorem (cf. [4], page 60, Theorem 2.3.4), ${\mathcal{O}}$ is a semi-algebraic
set. Hence, the boundary of $\mathcal{O}$ lies on some real algebraic curve;
that is, there is a non-zero real polynomial $f(x,y)$ for which every boundary
point $(x,y)$ of ${\mathcal{O}}$ satisfies

$f(x,y)=0.$

Since the set $\\{\xi\in{\mathbb{C}}^{n+1}\,:\,\|\xi\|=1\\}\times S^{1}$ is
compact and connected, its image ${\mathcal{O}}$ under a continuous map is
also compact and connected. Since the number of connected components of a
semialgebraic set is finite [5, Theorem 2.4.5], this implies that the boundary
of $\mathcal{O}$ has a finite number of connected components.

Consider the set of polynomials $f(x,y)\in{\mathbb{R}}[x,y]$ such that

$\partial{\mathcal{O}}\subseteq\\{(x,y)\in{\mathbb{R}}^{2}\,:\,f(x,y)=0\\}.$

By the comment above, this set is nonempty. We choose
$f_{0}(x,y)\in{\mathbb{R}}[x,y]$, of smallest degree, satisfying

$\partial{\mathcal{O}}\subseteq\\{(x,y)\in{\mathbb{R}}^{2}:f_{0}(x,y)=0\\}.$

Clearly, since ${\mathbb{R}}[x,y]$ is a unique factorization domain, we can
write $f_{0}(x,y)$ as

$f_{0}=f_{1}^{m_{1}}f_{2}^{m_{2}}\cdots f_{k}^{m_{k}}$

where $f_{1},f_{2},\ldots,f_{k}$ are the mutually inequivalent irreducible
factors of $f_{0}$ and the numbers $m_{j}$ are positive integers. Since
$f_{0}(x,y)$ is of minimum degree, clearly we must have that $m_{j}=1$ for all
$j$.

The curve $\\{(x,y)\in{\mathbb{R}}^{2}:f_{0}(x,y)=0\\}$ may have
multitangents. Such a multitangent corresponds to some singular point of the
dual curve of the multiplicity free curve $f_{0}(x,y)=0$; therefore, the
number of such lines is at most finite. We denote the union of such
multitangents as $g_{0}(x,y)=0$ using some polynomial
$g_{0}(x,y)\in{\mathbb{R}}[x,y]$. By letting $g(x,y)=f_{0}(x,y)g_{0}(x,y)$, we
have the following inclusion

$\partial\mathrm{Conv}\left({\mathcal{O}}\right)\subseteq\\{(x,y):g(x,y)=0\\}.$

We have obtained the following theorem.

######  Theorem 3.1.

Let $T_{+}$ be an $(n+1)$-periodic and $(2m+1)$-banded Toeplitz operator
acting on $\ell^{2}({\mathbb{N}_{0}})$. Then there exists a nonzero real
binary polynomial $g(x,y)$ such that if $(x,y)\in\partial W(T_{+})$, then
$g(x,y)=0$.

##  4\. A Toeplitz operator for which the closure of the numerical range is
not the numerical range of a matrix

We examine the following example of a $2$-periodic and $5$-banded Toeplitz
operator $T_{+}$. We set the sequences $a^{(0)}$, $a^{(-1)}$ and $a^{(-2)}$ to
be the constant sequences of zeroes, we set the sequence
$a^{(1)}=\left(a_{j}^{(1)}\right)_{j\in{\mathbb{Z}}}$ with $a_{j}^{(1)}=-1$ if
$j$ is an even integer and $a_{j}^{(1)}=2$ if $j$ is an odd integer; and we
set the sequence $a^{(2)}$ to be the constant sequence of ones. That is, the
matrix of $T_{+}$ has the upper triangular form

$\begin{pmatrix}0&-1&1&0&\cdots\\\ 0&0&2&1&0&\cdots\\\ 0&0&0&-1&1&0&\cdots\\\
0&0&0&0&2&1&0&\cdots\\\ \vdots&0&0&0&0&-1&1&0&\cdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots\end{pmatrix}.$ (8)

By Definition 2.1, the symbol of $T_{+}$ is the $2\times 2$ matrix function

$\Phi(\theta)=\begin{pmatrix}e^{i\theta}&-1\\\
2e^{i\theta}&e^{i\theta}\end{pmatrix}.$

The numerical range of $\Phi(\theta)$ is the convex set bounded by the ellipse
with focii the points

$e^{i\theta}+i\sqrt{2}e^{i\theta/2}\quad\text{ and }\quad
e^{i\theta}-i\sqrt{2}e^{i\theta/2},$

major axis of length $3$, and minor axis of length $1$.

By Theorem 2.7, the closure of $W(T)$ is given as the closure of the convex
hull of the union of the above ellipses.

Each ellipse can be parametrized as

$\displaystyle X(t)$
$\displaystyle=\cos(\theta)+\frac{1}{2}\cos(\theta/2)\cos(t)-\frac{3}{2}\sin(\theta/2)\sin(t),$
$\displaystyle Y(t)$
$\displaystyle=\sin(\theta)+\frac{1}{2}\sin(\theta/2)\cos(t)+\frac{3}{2}\cos(\theta/2)\sin(t),$

where $t\in[0,2\pi)$. The Cartesian equation of each ellipse is

$(20+16\cos\theta)(X-\cos\theta)^{2}+(20-16\cos\theta)(Y-\sin\theta)^{2}+32\sin\theta(X-\cos\theta)(Y-\sin\theta)=9.$

or equivalently,

$(16X^{2}-16Y^{2}-40X+16)\cos\theta+(32XY-40Y)\sin\theta+(20X^{2}+20Y^{2}-32X+11)=0.$

We now consider the function

$\displaystyle H$ $\displaystyle(X,Y;\theta)$ $\displaystyle=$
$\displaystyle(16X^{2}-16Y^{2}-40X+16)\cos\theta+(32XY-40Y)\sin\theta+(20X^{2}+20Y^{2}-32X+11).$

Observe that, for each $\theta\in[0,2\pi)$, the boundary of the numerical
range of $\Phi(\theta)$ is given by the ellipse $H(X,Y;\theta)=0$. To find the
envelope of this family of ellipses, we write

$H(X,Y;\theta)=\alpha\cos\theta+\beta\sin\theta+\gamma$

where

$\alpha=16X^{2}-16Y^{2}-40X+16,\quad\beta=32XY-40Y,\quad\text{ and
}\quad\gamma=20X^{2}+20Y^{2}-32X+11.$

As is well-known [6, p. 76], the envelope consists of those points $(X,Y)$
such that there exists $\theta\in[0,2\pi)$ such that

$H(X,Y;\theta)=0\quad\text{ and
}\quad\frac{\partial}{\partial\theta}H(X,Y;\theta)=0.$

In other words, to find the points $(X,Y)$ in the envelope, we need to check
if there exists $\theta\in[0,2\pi)$ such that the system

$\alpha\cos\theta+\beta\sin\theta+\gamma=0,\quad\text{ and
}\quad\beta\cos\theta-\alpha\sin\theta=0,$

has a solution. A sufficient and necessary condition for this system to have a
solution is that $\alpha^{2}+\beta^{2}=\gamma^{2}$; that is

$(16X^{2}-16Y^{2}-40X+16)^{2}+(32XY-40Y)^{2}=(20X^{2}+20Y^{2}-32X+11)^{2}.$

Equivalently,

$16X^{4}+32X^{2}Y^{2}+16Y^{4}-72X^{2}-72Y^{2}+64X-15=0.$ (9)

Solving for $Y$ and using the second derivative test for convexity, it is
straightforward to check that the points satisfying the above equation are the
boundary of a convex set, except for the isolated point $(\frac{1}{2},0)$
which lies in the interior of said set. Hence, the union of the ellipses
$H(X,Y;\theta)=0$ and their interiors is a convex set. Also it is clear that
the union of the family of the ellipses $H(X,Y;\theta)=0$ and their interiors
is a closed set. Therefore, by Theorem 2.7, we conclude that a point $X+iY$ is
in the boundary of the numerical range of the operator $T_{+}$ if and only if
it satisfies the equation

$16X^{4}+32X^{2}Y^{2}+16Y^{4}-72X^{2}-72Y^{2}+64X-15=0,$

and $(X,Y)\neq(\frac{1}{2},0)$.

Hence, the boundary of $W(T_{+})$ consists of the points described by the set

$\left\\{(X,Y)\in{\mathbb{R}}^{2}:L(1,X,Y)=0\right\\}\setminus\left\\{(\tfrac{1}{2},0)\right\\}$

where

$L(U,X,Y)=16X^{4}+32X^{2}Y^{2}+16Y^{4}-72U^{2}X^{2}-72U^{2}Y^{2}+64U^{3}X-15U^{4},$
(10)

see Figure 1.

Figure 1. The red solid curve represents the points such that $L(1,X,Y)=0$ and
the black dotted curves represent the ellipses $H(X,Y;\theta)=0$ for six
different values of $\theta$.

The complex projective curve

$\left\\{[U;X;Y]\in{\mathbb{C}P}^{2}:L(U,X,Y)=0\right\\}$

has an ordinary double point at $(U,X,Y)=(2,1,0)$ and a pair of $(2,3)$-cusps
at $(U,X,Y)=(0,1,\pm i)$. This curve has no other singular points. These
properties imply that the quartic polynomial $L(U,X,Y)$ is irreducible in the
polynomial ring ${\mathbb{C}}[U,X,Y]$ (see [9]).

Also, we consider the dual curve $\ell(t,x,y)=0$ of the curve $L(U,X,Y)=0$
under the pairing

$tU+xX+yY=0.$

The curve $\ell(t,x,y)=0$ is also an irreducible quartic algebraic curve and a
computation shows that $\ell$ is given by

$\ell(t,x,y)=16t^{4}+32t^{3}x-72t^{2}(x^{2}+y^{2})-216t(x^{3}+xy^{2})-135x^{4}-162x^{2}y^{2}-27y^{4}.$
(11)

The irreducibility of $L$ and $\ell$ will be useful when answering the main
question of this section: Is there some $k\times k$ complex matrix $B$ for
which

$\overline{W(T_{+})}=W(B)?$

The following theorem shows that the answer is negative.

######  Theorem 4.1.

Let $a^{(0)}$, $a^{(-1)}$ and $a^{(-2)}$ be the constant sequences of zeroes,
let $a^{(1)}=\left(a_{j}^{(1)}\right)_{j\in{\mathbb{Z}}}$ be the sequence
defined by $a_{j}^{(1)}=-1$ if $j$ is an even integer and $a_{j}^{(1)}=2$ if
$j$ is an odd integer; and let $a^{(2)}$ be the constant sequence of ones.
Suppose that $T_{+}$ is the biperiodic and $5$-banded Toeplitz operator acting
on $\ell^{2}({\mathbb{N}}_{0})$ defined by these sequences, which has a matrix
representation given in (8). For every $k\in{\mathbb{N}}$, there is no
$k\times k$ matrix $B$ such that

$\overline{W(T_{+})}=W(B).$

Before proving this theorem, we introduce a notion which was originally
formulated for the principal symbol of a partial differential operator.

######  Definition 4.2.

Suppose that $F(t,x,y)$ is a real homogeneous polynomial of degree $k\geq 1$.
The form $F$ is said to be hyperbolic with respect to $(1,0,0)$ if for every
nonzero $(x_{0},y_{0})\in{\mathbb{R}}^{2}$, the equation

$F(t,x_{0},y_{0})=0$

as a polynomial in the variable $t$ has $k$ real solutions counting
multiplicities.

Observe that if $F_{B}$ is the Kippenhahn polynomial associated with an
$k\times k$ matrix $B$; i.e.,

$F_{B}(t,x,y)={\rm det}(tI_{m}+x\mathrm{Re}(B)+y\mathrm{Im}(B)),$

then the equation $F_{B}(t,x,y)=0$ is the characteristic equation of the
Hermitian matrix $-x\mathrm{Re}(B)-y\mathrm{Im}(B)$ and hence, for each
$(x,y)\in{\mathbb{R}}^{2}$, it has $k$ real roots counting multiplicites. So
the form $F_{B}$ is hyperbolic with respect to $(1,0,0)$ (cf. [16]).

###### Proof of Theorem 4.1.

By contradiction, we assume that for some $k\in{\mathbb{N}}$, there exists a
$k\times k$ matrix $B$ satisfying $W(B)=\overline{W(T_{+})}$.

By Kippenhahn’s theorem, the numerical range of a matrix $B$ equals the convex
hull of the real affine part of the dual curve of $F_{B}(t,x,y)=0$. Let us
denote by $f_{B}(U,X,Y)$ the dual curve of $F_{B}(t,x,y)$. As we have seen, a
point $(X_{0},Y_{0})$ on the boundary of the numerical range of $T_{+}$
satisfies the equation $L(1,X_{0},Y_{0})=0$, where $L(U,X,Y)$ is given in
(10). Then the assumption implies that $f_{B}(1,X_{0},Y_{0})=0$. By Bézout’s
Theorem, $L(U,X,Y)$ and $f_{B}(U,X,Y)$ must have a common component. Since
$L(U,X,Y)$ is irreducible, we conclude that $L(U,X,Y)$ is a factor of
$f_{B}(U,X,Y)$. It follows that its dual curve, $\ell(t,x,y)$, described in
(11), must be a factor of $F_{B}(t,x,y)$.

By the relation (3.12) of [1, p. 130], the ternary form $\ell(t,x,y)$ as a
factor of $F_{B}(t,x,y)$ has to be hyperbolic with respect to $(1,0,0)$, that
is, the quartic equation $\ell(t,-\cos\theta,-\sin\theta)=0$ in $t$ has $4$
real solutions counting its multiplicities for any angle $0\leq\theta<2\pi$.
But the equation for $\theta=\pi/2$ is expressed as

$16t^{4}-72t^{2}-27=\left(4t^{2}-(6\sqrt{3}+9)\right)\,\left(4t^{2}+(6\sqrt{3}-9)\right)=0.$

The equation $4t^{2}-(6\sqrt{3}+9)=0$ has two real solutions, but since
$6\sqrt{3}-9>0$, the equation $4t^{2}+(6\sqrt{3}-9)$ has no real solutions.
Hence, the quartic equation $16t^{4}-72t^{2}-27=0$ in $t$ does not have only
real solutions. So the ternary form $\ell(t,x,y)$ is not hyperbolic with
respect to $(1,0,0)$. This contradiction implies that our assumption on the
existence of a $k\times k$ matrix $B$ with $W(B)=\overline{W(T_{+})}$ is
false. ∎

## References

  * [1] M. F. Atiyah, R. Bott, and L. Gårding.  Lacunas for hyperbolic differential operators with constant coefficients i.  Acta Mathematica, 124(0):109–189, 1970.
  * [2] N. Bebiano, J. da Providência, and A. Nata.  The numerical range of banded biperiodic Toeplitz operators.  J. Math. Anal. Appl., 398(1):189–197, 2013.
  * [3] N. Bebiano and I. M. Spitkovsky.  Numerical ranges of Toeplitz operators with matrix symbols.  Linear Algebra Appl., 436(6):1721–1726, 2012.
  * [4] R. Benedetti and J.-J. Risler.  Real algebraic and semi-algebraic sets.  Actualités Mathématiques. [Current Mathematical Topics]. Hermann, Paris, 1990.
  * [5] J. Bochnak, M. Coste, and M.-F. Roy.  Real algebraic geometry, volume 36 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)].  Springer-Verlag, Berlin, 1998.  Translated from the 1987 French original, Revised by the authors.
  * [6] J. W. Bruce and P. J. Giblin.  Curves and singularities.  Cambridge University Press, Cambridge, second edition, 1992.  A geometrical introduction to singularity theory.
  * [7] S. Chandler-Wilde, R Chonchaiya, and M Lindner.  Eigenvalue problem meets sierpinski triangle: computing the spectrum of a non-self-adjoint random operator.  Oper. Matrices, 5:633–648, 2011.
  * [8] M.-T. Chien and H. Nakazato.  The numerical range of a tridiagonal operator.  J. Math. Anal. Appl., 373:297–304, 2011.
  * [9] M.-T. Chien and H. Nakazato.  Singular points of the ternary polynomials associated with 4-by-4 matrices.  Electron. J. Linear Algebra, 23:755–769, 2012.
  * [10] R.T Chien and I.M. Spitkovsky.  On the numerical ranges of some tridiagonal matrices.  Linear Algebra Appl., 470:228–240, 2015.
  * [11] J. Feinberg and A. Zee.  Spectral curves of non-hermitian hamiltonians.  Nuclear Physics B, 552(3):599–623, 1999.
  * [12] R. Hagger.  The eigenvalues of tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators.  J. Funct. Anal, 269:1563–1570, 2015.
  * [13] B. A. Itzá-Ortiz and R. A. Martínez-Avendaño.  The numerical range of a class of periodic tridiagonal operators.  Linear and Multilinear Algebra, 69(5):786–806, 2020.
  * [14] B.A. Itzá-Ortiz, R.A. Martínez-Avendaõ, and H.’ Nakazato.  The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix.  J. Math. Anal. Appl., 506:Paper No. 125713, 14, 2022.
  * [15] T. Kato.  Perturbation theory for linear operators.  Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Reprint of the 1980 edition.
  * [16] R. Kippenhahn.  Über den Wertevorrat einer Matrix.  Math. Nachr., 6:193–228, 1951.


# SerPyTor: A distributed context-aware computational graph execution
framework for durable execution

Anuran Roy
School of Computer Science and Engineering
Vellore Institute of Technology
Vellore, India
<EMAIL_ADDRESS>
Sridhar Raj S
School of Computer Science and Engineering
Vellore Institute of Technology
Vellore, India
<EMAIL_ADDRESS>

###### Abstract

Distributed computation is always a tricky topic to deal with, especially in
context of various requirements in various scenarios. A popular solution is to
use Apache Spark with a setup of multiple systems forming a cluster. However,
the prerequisite setup for a Spark cluster often induces an additional
overhead, often limiting usage in constrained scenarios, especially in
scenarios requiring context propagation. In this paper, we explore a
relatively lightweight computational graph execution framework requiring
little setup and fast speeds, coupled with context awareness.

_K_ eywords Computational Graphs $\cdot$ Distributed Computing $\cdot$ Durable
Execution $\cdot$ Python

##  1 Introduction

Distributed computing is a crucial aspect of services distributed across the
internet. Most operations of services that we consume on an everyday basis,
from ML recommendation algorithms on deciding the user’s next choice at scale,
to developer platforms helping developers deploy code faster. As such, setting
up distributed compute clusters is a basic requirement in almost all facets
across web services with varying degrees of use, but targeting a similar
objective - reducing computation latency for large-scale computations. Tools
like Apache Spark are available, which are language-agnostic in nature, but
require proper setup and tuning.

In this paper, we introduce a more compact framework for performant execution
of distributed computational graphs.

We discuss the physical hardware layer abstraction architecture and the
logical execution flow in separate forthcoming sections bearing the same name,
that are sections 3 and 4 respectively.

##  2 Literature Review

According to Brezillion et. al.[1], context is defined as the collection of
relevant conditions and surrounding influences that make a situation unique
and comprehensible. Further understanding and definition of context can be
found on the same paper. In a subsequent paper, Pomero and Brezillon et. al[2]
showed that there is a strong correlation between context and knowledge, and
knowledge has been divided into three types of categories in their 1999
paper[1]: external knowledge, contextual knowledge and proceduralized context.

The culmination of context-based policy works resulted in the formulation of
the conceptual context-aware framework by Brezillon et. al[3]. Graphs are
mentioned to be one of the data structures that fit the context usage
scenarios really well, according to Baldan, Paolo, et. al[4]. Similarly,
graph-based systems can act as vertex-centric models[5], adopting for more
context usage and relevance, while passing data to its child node(s) in a
discrete, well-defined (not necessarily well-structured) manner.

##  3 Physical Layer Abstraction Architecture

Figure 1: Hardware Abstraction Architecture

The Physical Layer Abstraction in SerPyTor comprises of primarily 3 objects:
HeartbeatServer, Server, and Gateway, as shown in Figure 1

The latter two can be customized to be fully generic as per the requirements,
while the first object (that is, the HeartbeatServer) has a few limitations,
primarily because it is a component of the generic Server class, and has a few
core functionalities to perform.

###  3.1 HeartbeatServer

The HeartbeatServer object basically functions as the resource monitor for a
server, and a successful response from it indicates that the server hardware
is up and running. It reports in the form of a JSON response that reports the
different types of resource usage for the server resource - for example, CPU
usage, disk usage, (possible) GPU usage and memory usage, which indicates the
status of the server to take the next task.

###  3.2 Server

The Server is the abstract representation of a connected server in a network
which is assigned to do computational tasks that require computational power /
storage that the client’s PC cannot perform (thus, the main motivation of the
framework). The Server object exposes one or more ports from the physical
server resource and responds to the request by either performing the
computation or refusing it. The task to ascertain whether the server can take
the task is delegated to the Gateway object, so that the task to determine the
optimal computational resource is delegated to a central authoritative entity
to reduce conflicts at high concurrency. As such, the task of the gateway to
determine optimal resources should be successfully executed as fast as
possible.

The Server object is generic and weakly opinionated in nature. Every component
in it can be modified as per the user’s requirements - including mappings,
execution mechanism, etc. Users can extend it with various user-defined or
pluggable mechanisms - security check pipelines, authentication and
authorization mechanisms, etc. Minimal number of assumptions have been made to
impose minimal constraints on the user and to provide maximum programming
flexibility.

The basic assumptions made in case of constructing a Server object are:

  1. 1.

The Heartbeat Server is a separate process than the Application Server, and
thus, requires a separate port.

  2. 2.

Each mapping is a function that gets all its dependencies through Dependency
Injection mechanism, so that the function itself forms an atomic task for
durable execution (mentioned in Section 4.2).

Separating the HeartbeatServer and Server processes, though might seem to be
uncalled for, actually would help users, and additionally programs to
troubleshoot and distinguish between system-level or application-level errors.

  * •

If there is a System-level error, then the corresponding Server Application
would fail automatically.

  * •

If there is an Application-level error, then the HeartbeatServer object would
respond with a successful response. On the other hand, the Server object would
respond with a failure response (or no response at all till timeout).

  * •

This would enable the user to distinguish between System-level and
Application-level errors, which can save lots of troubleshooting time in cases
of numerous servers.

Provided these assumptions are met, the Server is ready to be setup and
integrated into the code-base with just an import from the SerPyTor Package.

###  3.3 Gateway

The Gateway object stores the context required for the associated Server
objects, and stores the task routing information in them at regular intervals,
or after the next task arrives - whichever comes first. Tasks in a gateway are
queued in either a single-level queue or a queue silo, depending on the task
sorting and context determination algorithms. Appropriate sorting algorithms
along with fallback mechanisms are provided, in case of failures to reduce the
probability of a single point of failure, and to increase the probability of
graceful degradation. This ensures that there are least surprises during real
world, high-concurrency scenarios.

Similar to the Server Object, the Gateway object is weakly opinionated in
nature to ensure maximum developer freedom while not compromising on overall
setup integrity. It has a few minimal assumptions, which are summarized as
below:

  1. 1.

A predefined task to execute (weak requirement).

  2. 2.

A list of servers setup along with their Heartbeat and Application Processes
up.

  3. 3.

A predefined allocation algorithm, or a user-defined allocation algorithm that
can handle all possible edge cases that may arise (please note that
performance may take hit depending on implementation.)

Ideally, the gateway object should possess only one task so to ensure the
ideal scenario of durable execution (i.e., deterministic inputs and outputs).
However, for more flexibility, users can configure them as custom tasks with a
limited guarantee of durable execution. They can do it by mutating the task
definition as and when required, subject to failure risks, and hence, not
recommended unless there is no other way.

##  4 Logical Execution Flow

Figure 2: An example of logical execution flow in the distributed graph

###  4.1 Context Definition and Inheritance Mechanism

Since the context is an integral part of the Graph Execution, it is imperative
to define a robust system of mechanisms that involve context and its
transference.

The fundamental unit of a computational graph is a node. In the case of
context-aware graph, we cannot use the regular nodes of a graph. A context-
aware graph can only be fulfilled if the nodes itself are context-aware, i.e.,
they possess context of some sort, which is relevant to their computation and
output. To start defining the aforementioned context transference mechanism,
we need to understand what can be the cases for each node.

A node’s inherited context can be any of the following cases:

  1. 1.

Context from a Single Independent Origin

  2. 2.

Context from a Single Dependent Origin

  3. 3.

Context from Multiple Independent Origins

  4. 4.

Context from Multiple Dependent Origins

We can ensure a robust mechanism of context transference using a set of simple
rules:

  1. 1.

For the root node $\mathbb{R}$, it’s context is defined as:

$\xi(\mathbb{R})=\xi(\mathbb{0})\cup\Psi(\mathbb{R})$

, where:

     * •

$\Psi(\mathbb{R})$ means the data of Node $\mathbb{R}$,

     * •

$\xi(\mathbb{0})$ means the origin context of the root node

The origin context of the root node is represented as $\xi(\mathbb{0})$, which
can have the value of $\Phi$ or if there is no context prior to computation at
the root node, i.e., no environment variables or similar data of the sort,
supplied prior to the start of computation.

  2. 2.

For a node with either Single or Multiple independent origins, its context is
the union of the context of each origin node. In Figure 2, nodes G and F are
respective examples.

  3. 3.

For a node with either Single or Multiple dependent origins, its context is
the union of the context of their parents and each node that is codependent on
their parents, and other dependent nodes. In Figure 2, nodes A, B are
dependent on each other and can be thought to have formed some kind of a
"union node" A’ such that:

$\xi(A^{\prime})=\xi(A)\cup\xi(B)\cup\Psi(A)\cup\Psi(B)$

. As seen in the figure, all children of A and/or B are transferred the
origins of A’ instead of A or B.

####  4.1.1 Possible Pitfalls

A common possible pitfall of the aforementioned approach is that of the
existence of possible loops in a graph. It is to be remembered here that the
graph should be a Directed Acyclic Graph, commonly known as a DAG. This is a
barebones necessity as even in code-bases that run on a single hardware
resource (that is, a single system), dependencies cannot be circular in
nature, and failure to do so gives rise to the Circular Import Problem, and
Mohammed, Mawal, et. al., provide a search-based approach[6] to
minimize/eliminate circular dependencies . In extreme cases though, contextual
circular dependency resolution techniques similar to what is mentioned by
Athan, Tara, et. al.,[7] can be applied.

###  4.2 Durable Execution

Durable Programming is a recent paradigm of context-aware programming that
focuses on ensuring that tasks execute atomically. Instead of trying to
maximize chances of successful execution by trying out edge cases, it aims to
optimize for chances of failure by employing various methods to reduce
failures. It does so by breaking down a callable entity into atomic units of
computation that can be handled safely, and tractably - hence the name Durable
Execution.

Platforms such as Temporal (or Azure Durable Functions) that closely follow
the paradigm of durable execution, need deterministic outputs from workflows,
which can be thought of as akin to deterministic outputs from the nodes of a
graph. As such, context-aware nodes make a good use case of the durable
programming paradigm according to the Vertex-centric Model described by Liu,
et. al,[5], with a small but significant change - the context needs to be
propagated from origin during every call, as an individual context is
deterministic in nature with a deterministic output, but collectively, they
are non-deterministic in nature.

As mentioned in Section 3.2, Dependency Injection at run-time is the key to
deterministic input. Making context-aware policies for dependency injection
has been explored in other relevant literature, such as the one proposed by
Ekstrand, Michael et. al[8].

##  5 Conclusion and Future Works

In this paper, we have introduced examined the potential of a new programming
framework for distributed computations, employing context, context-aware
policies and a modified form of durable programming, which is a radical shift
from the traditional approach of programming. Scope for future work involves
improving benchmarks by optimizing internal implementations of various
algorithms, making troubleshooting easier through setting up guards at
different execution checkpoints (that is., adapting a hybrid approach to
maximize program durability as well as developer/user experience). In
particular, implementation of the Gateway object and server response timings
require a hefty amount of optimization, to extract maximum performance
possible, so that task queuing and delegation bottlenecks are minimized.
Otherwise, the bottlenecks will get magnified as soon as multiple Gateways are
involved. Thus while scaling up, small bottlenecks might become an issue.
Possible solutions involve rewriting IO and processing heavy tasks into
compiled and memory-safe languages, such as Rust or D. Another alternative
with possibly memory tradeoffs (insignificant in large-scale scenarios) would
be resorting to functional languages like Scala and Erlang to speed up
processes through real-time concurrency, along with Natively Implemented
Functions (NIFs) written in compiled Object-Oriented Languages plugged in
wherever necessary.

## References

  * [1] Patrick Brézillon.  Context in problem solving: A survey.  The Knowledge Engineering Review, 14(1):47–80, 1999.
  * [2] J C Pomerol and Patrick Brezillon.  About some relationships between knowledge and context.  In Modeling and Using Context: Third International and Interdisciplinary Conference, CONTEXT 2001 Dundee, UK, July 27–30 , 2001 Proceedings 3, pages 461–464. Springer, 2001.
  * [3] Patrick Brézillon.  Elaboration of the contextual graphs representation: From a conceptual framework to an operational software.  ISTE OpenScience, pages 1–26, 2017.
  * [4] Paolo Baldan.  Modelling concurrent computations: from contextual Petri nets to graph grammars.  PhD thesis, PhD thesis, Department of Computer Science, University of Pisa, 2000 …, 2000.
  * [5] Ning Liu, Dong-sheng Li, Yi-ming Zhang, and Xiong-lve Li.  Large-scale graph processing systems: a survey.  Frontiers of Information Technology & Electronic Engineering, 21(3):384–404, 2020.
  * [6] Mawal A Mohammed, Mohammad Alshayeb, and Jameleddine Hassine.  A search-based approach for detecting circular dependency bad smell in goal-oriented models.  Software and Systems Modeling, 21(5):2007–2037, 2022.
  * [7] Tara Athan.  Importation closure that is robust to circular dependencies.  RuleML (2), 1004, 2013.
  * [8] Michael D Ekstrand and Michael Ludwig.  Dependency injection with static analysis and context-aware policy.  J. Object Technol., 15(1):1–1, 2016.