# Spectra and pseudo-spectra of tridiagonal $k$-Toeplitz matrices and the
topological origin of the non-Hermitian skin effect

Habib Ammari  Habib Ammari ETH Zürich, Department of Mathematics, Rämistrasse
101, 8092 Zürich, Switzerland<EMAIL_ADDRESS>,  Silvio Barandun
Silvio Barandun ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092
Zürich, Switzerland<EMAIL_ADDRESS>,  Yannick De Bruijn
Yannick De Bruijn ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092
Zürich, Switzerland<EMAIL_ADDRESS>,  Ping Liu  Ping Liu ETH
Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland.
<EMAIL_ADDRESS>and  Clemens Thalhammer  Clemens Thalhammer ETH
Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland.
<EMAIL_ADDRESS>
###### Abstract.

We establish new results on the spectra and pseudo-spectra of tridiagonal
$k$-Toeplitz operators and matrices. In particular, we prove the connection
between the winding number of the eigenvalues of the symbol function and the
exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our
results elucidate the topological origin of the non-Hermitian skin effect in
general one-dimensional polymer systems of subwavelength resonators with
imaginary gauge potentials, proving the observation and conjecture in [5]. We
also numerically verify our theory for these systems.

Keywords. Tridiagonal $k$-Topelitz operator, block-Toeplitz operator,
Tridiagonal $k$-Laurent operator, pseudospectra, Coburn’s lemma, non-Hermitian
skin effect, gauge capacitance matrix, eigenmode condensation.

AMS Subject classifications. 15A18, 15B05, 81Q12.

##  1\. Introduction

Motivated by the study of the non-Hermitian skin effect in polymer systems of
subwavelength resonators and its topological origin, in this paper we extend
the theory on the spectra and pseudo-spectra of Toeplitz operators and
matrices to tridiagonal $k$-Toeplitz operators and matrices. Our main goal is
to establish a connection between, on one hand, the winding number of the
eigenvalues of the symbol function of a tridiagonal $k$-Toeplitz operator and,
on the other hand, the exponential decay of the associated eigenvectors or
pseudo-eigenvectors. By doing so, we elucidate the topological origin of the
eigenmode condensation at one edge of a polymer system of subwavelength
resonators where an imaginary gauge potential is added inside the resonators
to break Hermiticity. We show that, due to a nonzero total winding of the
eigenvalues of the symbol of the $k$-Toeplitz operator corresponding to the
polymer system, the eigenmodes exhibit exponential decay and condensate at one
of the edges of the structure.

In addition to advancing our understanding of the non-Hermitian skin effect,
which can be seen as the classical analogue of the non-Hermitian Anderson
model [16], one of our key contributions in this paper is to optimally
characterize the spectra of tridiagonal $k$-Toeplitz operators. Notably, we
reveal the distinctive property that the eigenvectors of these operators
undergo exponential decay. Using the notion of pseudo-spectrum, we rigorously
justify the exponential decay of the pseudo-eigenvectors of the associated
$k$-Toeplitz matrices and significantly generalize the results presented in
[15, 18, 20, 21, 9].

Our paper is organized as follows. In Section 2, we first provide an optimal
characterization of the spectra of tridiagonal $k$-Toeplitz and Laurent
operators. Then we prove the existence of an exponentially decaying
eigenvector for Topelitz operators. In Section 3, we establish a connection
between the eigenvectors of a tridiagonal $k$-Toeplitz operator and the
associated tridiagonal $k$-Toeplitz matrix. In Section 4 we discuss the
topological origin of the non-Hermitian skin effect in polymer systems of
subwavelength resonators and numerically verify our findings on the spectra of
tridiagonal $k$-Toeplitz operators. In Section 5, we make some concluding
remarks and give some possible extensions of our present work.

##  2\. Spectra of tridiagonal $k$-Toeplitz operators

In this section, we characterize the spectrum of tridiagonal $k$-Toeplitz
operators and prove that their associated eigenvectors exhibit exponential
decay. This type of operators and their associated truncated operators, i.e.
matrices, are crucial in the study of the one-dimensional polymer systems of
subwavelength resonators [1, 6, 11, 5]. We define a _tridiagonal $k$-Toeplitz
operator_ by

$A=\begin{pmatrix}a_{1}&b_{1}&0&&&&\\\ c_{1}&a_{2}&b_{2}&\ddots&\\\
0&\ddots&\ddots&\ddots&\ddots\\\ &\ddots&c_{k-2}&a_{k-1}&b_{k-1}&\ddots\\\
&&\ddots&c_{k-1}&a_{k}&b_{k}&\ddots\\\ &&&\ddots&c_{k}&a_{1}&b_{1}&\ddots\\\
&&&&\ddots&\ddots&\ddots&\ddots\\\ \end{pmatrix},$ (2.1)

where $a_{i},b_{i},c_{i}\in\operatorname{\mathbb{C}}$ for $i\in\mathbb{N}$ and
$A_{ij}=0$ if $|i-j|>1$ for $i,j\in\mathbb{N}$. Its finite, truncated,
counterpart will be called a _tridiagonal $k$-Toeplitz matrix_.

To characterize the spectrum of tridiagonal $k$-Toeplitz operators, we
introduce the notions of Fredholm operator and Fredholm index. Let $X$ be a
Banach space and $\mathcal{B}(X)$ the set of bounded linear operators on $X$.
We say that an operator $A\in\mathcal{B}(X)$ is a _Fredholm operator_ if
$\operatorname{Im}A$ is a closed subspace in $X$ and

$\operatorname{dim}\mathrm{Ker}(A)<\infty,\quad\operatorname{dim}\mathrm{Coker}A<\infty,$

where $\mathrm{Coker}A:=X/\mathrm{Im}A$. The _index_ of the Fredholm operator
$A$ is defined as

$\operatorname{Ind}A\coloneqq\operatorname{dimKer}A-\operatorname{dimCoker}A.$
(2.2)

###  2.1. Tridiagonal $k$-Toeplitz operators

Observe that the tridiagonal $k$-Toeplitz operator $A$ presented in (2.1) can
be reformulated as a tridiagonal block Toeplitz operator, where the blocks
repeat in a $1$-periodic way:

$A=\begin{pmatrix}A_{0}&A_{-1}&\\\ A_{1}&A_{0}&\ddots\\\ &\ddots&\ddots\\\
\end{pmatrix},$ (2.3)

with

$A_{0}=\begin{pmatrix}a_{1}&b_{1}&0&\cdots&0\\\
c_{1}&a_{2}&b_{2}&\ddots&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&c_{k-2}&a_{k-1}&b_{k-1}\\\ 0&\cdots&0&c_{k-1}&a_{k}\\\
\end{pmatrix},A_{-1}=\begin{pmatrix}0&\cdots&\cdots&0\\\
\vdots&\ddots&&\vdots\\\ 0&&\ddots&\vdots\\\
b_{k}&0&\cdots&0\end{pmatrix},A_{1}=\begin{pmatrix}0&\cdots&0&c_{k}\\\
\vdots&\ddots&&0\\\ \vdots&&\ddots&\vdots\\\ 0&\cdots&\cdots&0\end{pmatrix}.$

The _symbol_ of the tridiagonal block Toeplitz operator (2.3) (and thus also
of the tridiagonal $k$-Toeplitz operator (2.1)) is defined by

$\displaystyle f:S^{1}$ $\displaystyle\to\operatorname{\mathbb{C}}^{k\times
k}$ $\displaystyle z$ $\displaystyle\mapsto A_{-1}z^{-1}+A_{0}+A_{1}z,$ (2.4)

or explicitly

$f(z)=\begin{pmatrix}a_{1}&b_{1}&0&\dots&0&c_{k}z\\\
c_{1}&a_{2}&b_{2}&0&\dots&0\\\ 0&c_{2}&a_{3}&b_{3}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\ 0&\dots&0&c_{k-2}&a_{k-1}&b_{k-1}\\\
b_{k}z^{-1}&0&\dots&0&c_{k-1}&a_{k}\\\ \end{pmatrix}.$ (2.5)

We will write $A=T(f)$. In the following sections, we will characterize the
spectrum of the operator $T(f)$ based on its symbol $f(z)$.

###  2.2. Hardy Spaces

To get a better understanding of the connection between the symbol $f$ and the
operator $T(f)$ itself, it is necessary to introduce the so-called _Hardy
Spaces_. Let $\mathbb{T}$ be the unit circle in $\mathbb{C}$.

######  Definition 2.1.

The Hardy spaces $H^{2}:=H^{2}(\mathbb{T})$ and
$H^{2}_{-}:=H^{2}_{-}(\mathbb{T})$ are defined by

$\displaystyle H^{2}:=\\{f\in L^{2}(\mathbb{T}):f_{n}=0\text{ for }n<0\\},$
$\displaystyle H^{2}_{-}:=\\{f\in L^{2}(\mathbb{T}):f_{n}=0\text{ for }n\geq
0\\},$

where $f_{n}$ denotes the Fourier coefficients of $f$.

The spaces $H^{2}$ and $H^{2}_{-}$ are closed, orthogonal subspaces of
$L^{2}\coloneqq L^{2}(\mathbb{T})$. Hence, one may write

$L^{2}=H^{2}\oplus H^{2}_{-}.$

Denote by $P$ the projection of $L^{2}$ onto $H^{2}$. The functions

$\left\\{\frac{1}{\sqrt{2\pi}}e^{in\theta}\right\\}_{n=0}^{\infty}$

form an orthonormal basis of $H^{2}$. One important property of functions in
$H^{2}$ is given by the following theorem [10, Theorem 6.13].

######  Theorem 2.2 (F. and M. Riesz).

If f is a nonzero function in $H^{2}$, then the set
$\\{e^{it}\in\mathbb{T}:f(e^{it})=0\\}$ has measure zero.

It is well-known [21] that for a Toeplitz operator $T(f)$ with symbol $f\in
L^{\infty}$, $T(f)$ is the matrix representation of the operator

$H^{2}\rightarrow H^{2},g\mapsto P(fg)$

in the above orthonormal basis. A similar notion holds for block Toeplitz
operators. We denote by $(H^{2})^{k}$ the $k$-dimensional vector space whose
entries are elements of $H^{2}$. If $\tilde{P}$ is the orthogonal projection
of $(L^{2})^{k}$ onto $(H^{2})^{k}$ and $f\in(L^{\infty})^{k\times k}$, then
$T(f)$ is the matrix representation of the operator

$(H^{2})^{k}\rightarrow(H^{2})^{k},\quad\bm{\mathbf{g}}\mapsto\tilde{P}(f\bm{\mathbf{g}}),$

where $f\bm{\mathbf{g}}$ is the usual matrix-vector multiplication. Note that
the $(H^{2})^{k}$ equivalent of Theorem 2.2 holds verbatim with $H^{2}$
replaced by $(H^{2})^{k}$. With this in mind, we are now able to prove a
generalization of Coburn’s lemma (for the Toeplitz case, see [10, Theorem
1.10]).

######  Theorem 2.3 (Coburn’s lemma; tridiagonal $k$-Toeplitz version).

Let $f\in\operatorname{\mathbb{C}}^{k\times k}(\mathbb{T})$ be the symbol of a
tridiagonal $k$-Toeplitz operator such that $\det(f(z))$ does not vanish
identically on $\mathbb{T}$. Then one of the following statements holds:

  1. (i)

$T(f)$ has a trivial kernel;

  2. (ii)

$T(f)$ has a dense range;

  3. (iii)

The leading $(k-1)\times(k-1)$ principal minor of $A_{0}$ has a nonzero
kernel. In particular, there exists some
$z_{0}\in\operatorname{\mathbb{C}}\cup\\{\infty\\}$ such that
$\ker(z_{0}^{-1}A_{-1}+A_{0})\cap\ker(A_{1})\neq\bm{\mathbf{0}}$.

###### Proof.

It is not hard to see that the adjoint of $T(f)$ is $T(\bar{f}^{\top})$, where
the superscript $\top$ denotes the transpose. Assume that $T(f)$ has a
nontrivial kernel and a nondense range, i.e., conditions (i) and (ii) do not
hold. This implies that $T(\bar{f}^{\top})$ has a nontrivial kernel as well.
Hence there exist nonzero functions
$\bm{\mathbf{g}}_{+},\bm{\mathbf{h}}_{+}\in(H^{2})^{k}$ such that

$f\bm{\mathbf{g}}_{+}=\bm{\mathbf{g}}_{-}\in(H^{2}_{-})^{k}\quad\text{and}\quad\bar{f}^{\top}\bm{\mathbf{h}}_{+}=\bm{\mathbf{h}}_{-}\in(H^{2}_{-})^{k},$
(2.6)

where $\bm{\mathbf{g}}_{\pm}=(g_{\pm,1},g_{\pm,2},\cdots,g_{\pm,k})^{\top}$
and $\bm{\mathbf{h}}_{\pm}=(h_{\pm,1},h_{\pm,2},\cdots,h_{\pm,k})^{\top}$.
Since $\bm{\mathbf{g}}_{+},\bm{\mathbf{h}}_{+}\in(H^{2})^{k}$, there exist two
sequences of vectors
$(\bm{\mathbf{v}}_{n})_{n},(\bm{\mathbf{w}}_{n})_{n}\in\operatorname{\mathbb{C}}^{k}$
such that

$\displaystyle\bm{\mathbf{g}}_{+}$
$\displaystyle=\sum_{n\in\mathbb{N}}\bm{\mathbf{v}}_{n}z^{n-1},$ (2.7)
$\displaystyle\bm{\mathbf{h}}_{+}$
$\displaystyle=\sum_{n\in\mathbb{N}}\bm{\mathbf{w}}_{n}z^{n-1},$

for $z=e^{i\theta}$, where we have absorbed the factor $\frac{1}{\sqrt{2\pi}}$
into $\bm{\mathbf{v}}_{n}$ and $\bm{\mathbf{w}}_{n}$. This allows us to write

$\displaystyle\bm{\mathbf{g}}_{-}$
$\displaystyle=z^{-1}A_{-1}\bm{\mathbf{v}}_{1}+(A_{0}\bm{\mathbf{v}}_{1}+A_{-1}\bm{\mathbf{v}}_{2})$
(2.8)
$\displaystyle\phantom{=}+\sum_{n\in\mathbb{N}}z^{n}(A_{-1}\bm{\mathbf{v}}_{n+2}+A_{0}\bm{\mathbf{v}}_{n+1}+A_{1}\bm{\mathbf{v}}_{n})$
$\displaystyle=z^{-1}A_{-1}\bm{\mathbf{v}}_{1},$

where we have used that $\bm{\mathbf{g}}_{-}\in(H^{2}_{-})^{k}$ in the last
equality. Similarly, using that $\bm{\mathbf{h}}_{-}\in(H^{2}_{-})^{k}$, we
obtain the following expression for $\bm{\mathbf{h}}_{-}$:

$\displaystyle\bm{\mathbf{h}}_{-}=$ $\displaystyle
z^{-1}\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{1}+(\bar{A_{0}}^{\top}\bm{\mathbf{w}}_{1}+\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{2})$
(2.9)
$\displaystyle+\sum_{n\in\mathbb{N}}z^{n}(\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{n+2}+\bar{A_{0}}^{\top}\bm{\mathbf{w}}_{n+1}+\bar{A_{-1}}^{\top}\bm{\mathbf{w}}_{n})$
$\displaystyle=$ $\displaystyle z^{-1}\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{1}.$

By Theorem 2.2, it follows that $\bm{\mathbf{g}}_{+},\bm{\mathbf{h}}_{+}\neq
0$ almost everywhere on $\mathbb{T}$. We define

$\varphi:=\bar{\bm{\mathbf{h}}}_{+}^{\top}\bm{\mathbf{g}}_{-}=\bar{\bm{\mathbf{h}}}_{+}^{\top}f\bm{\mathbf{g}}_{+}=\bar{(\bar{f}^{\top}{\bm{\mathbf{h}}}_{+})}^{\top}\bm{\mathbf{g}}_{+}=\bar{\bm{\mathbf{h}}}_{-}^{\top}\bm{\mathbf{g}}_{+}.$

It holds that $\varphi\in L^{1}$. Moreover,
$\varphi_{n}=(\bar{\bm{\mathbf{h}}}_{-}^{\top}\bm{\mathbf{g}}_{+})_{n}=0$ for
$n\leq 0$ and
$\varphi_{n}=(\bar{\bm{\mathbf{h}}}_{+}^{\top}\bm{\mathbf{g}}_{-})_{n}=0$ for
$n\geq 0$ since by (2.7), (2.8), and (2.9),

$\displaystyle\bar{\bm{\mathbf{h}}}_{+}^{\top}\bm{\mathbf{g}}_{-}$
$\displaystyle=\sum_{n\in\mathbb{N}}z^{-n}\bar{\bm{\mathbf{w}}}_{n}^{\top}A_{-1}\bm{\mathbf{v}}_{1},$
$\displaystyle\bar{\bm{\mathbf{h}}}_{-}^{\top}\bm{\mathbf{g}}_{+}$
$\displaystyle=\sum_{n\in\mathbb{N}}z^{n}(\bar{\bar{A_{1}}^{\top}\bm{\mathbf{w}}}_{1})^{\top}\bm{\mathbf{v}}_{n}.$

Hence $\varphi=0$. This implies
$\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{1}\perp\bm{\mathbf{v}}_{n}$ and
$A_{-1}\bm{\mathbf{v}}_{1}\perp\bm{\mathbf{w}}_{n}$ for $n\in\mathbb{N}$. Thus
either $\bm{\mathbf{v}}_{n}\perp\bm{\mathbf{e}}_{k},n=1,\cdots$, or
$\bm{\mathbf{w}}_{1}\in\ker\overline{A_{1}}^{\top}$. However, if
$\bm{\mathbf{w}}_{1}\in\ker\overline{A_{1}}^{\top}$, then by (2.9)

$\displaystyle\bar{f}^{\top}\bm{\mathbf{h}}_{+}=\bm{\mathbf{h}}_{-}=z^{-1}\bar{A_{1}}^{\top}\bm{\mathbf{w}}_{1}=\bm{\mathbf{0}},$

which contradicts the fact that $\det(\bar{f}^{\top})=\bar{\det(f)}$ does not
vanish identically on $\mathbb{T}$. Therefore,
$\bm{\mathbf{v}}_{n}\perp\bm{\mathbf{e}}_{k}$ for $n\in\mathbb{N}_{\geq 1}$.
By the above arguments, we conclude that if $T(f)$ has non-dense range and
nontrivial kernel, then $\bm{\mathbf{v}}_{n}\perp\bm{\mathbf{e}}_{k}$ for
$n\in\mathbb{N}_{\geq 1}$ and for any $\bm{\mathbf{v}}_{n}$ defined by (2.7).

Now we prove that if $T(f)$ has a non-dense range and a nontrivial kernel,
then the condition (iii) must hold. Since
$f\bm{\mathbf{g}}_{+}=\bm{\mathbf{g}}_{-}\in(H^{2}_{-})^{k}$, by expansion
(2.8) it holds that
$A_{-1}\bm{\mathbf{v}}_{n+2}+A_{0}\bm{\mathbf{v}}_{n+1}+A_{1}\bm{\mathbf{v}}_{n}=\bm{\mathbf{0}}$
for $n\in\mathbb{N}$. Since $\bm{\mathbf{v}}_{n}\perp\bm{\mathbf{e}}_{k}$ for
$n\in\mathbb{N}_{\geq 1}$ it follows that $\bm{\mathbf{v}}_{n}\in\ker(A_{1})$.
Hence it holds that
$A_{-1}\bm{\mathbf{v}}_{n+1}+A_{0}\bm{\mathbf{v}}_{n}=\bm{\mathbf{0}}$ for
$n\in\mathbb{N}_{\geq 2}$. Note that the expansion (2.8) also gives
$A_{-1}\bm{\mathbf{v}}_{2}+A_{0}\bm{\mathbf{v}}_{1}=0$. Therefore,
$A_{-1}\bm{\mathbf{v}}_{n+1}+A_{0}\bm{\mathbf{v}}_{n}=\bm{\mathbf{0}}$ for
$n\in\mathbb{N}_{\geq 1}$. Furthermore, since the image of $A_{-1}$ is
confined to the span of $\bm{\mathbf{e}}_{k}$ and
$\bm{\mathbf{v}}_{n}\perp\bm{\mathbf{e}}_{k}$ for $n\in\mathbb{N}_{\geq 1}$,
we have

$A_{0}|_{(k-1)\times(k-1)}\bm{\mathbf{v}}_{n}|_{1:k-1}=\bm{\mathbf{0}},$

where $A_{0}|_{(k-1)\times(k-1)}$ is the leading principal $(k-1)\times(k-1)$
submatrix of $A_{0}$. This also means that $A_{0}|_{(k-1)\times(k-1)}$ cannot
have full rank. On the other hand, one can compute that

$(z^{-1}A_{-1}+A_{0})\bm{\mathbf{v}}_{n}=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
c_{k-1}\bm{\mathbf{v}}_{n}^{(k-1)}+z^{-1}b_{k}\bm{\mathbf{v}}_{n}^{(1)}\end{pmatrix},\quad
n\geq 1$ (2.10)

where $\bm{\mathbf{v}}_{n}^{(j)}$ is the $j$-th element of
$\bm{\mathbf{v}}_{n}$. Considering the case when $n=1$, by (2.8), we know that
$\bm{\mathbf{v}}_{1}^{(1)}\neq 0$, otherwise,

$f\bm{\mathbf{g}}_{+}=\bm{\mathbf{g}}_{-}=z^{-1}A_{-1}\bm{\mathbf{v}}_{1}=\bm{\mathbf{0}},$

which contradicts $\det f$ does not vanish identically on $\mathbb{T}$.
Similarly, we can also show that $b_{k}\neq 0$. Thus by (2.10) and $b_{k}\neq
0$ there exists some $z_{0}\in\operatorname{\mathbb{C}}\cup\\{\infty\\}$ such
that $\bm{\mathbf{v}}_{1}$ lies in the kernel of $z_{0}^{-1}A_{-1}+A_{0}$.
Therefore it holds that
$\ker(z_{0}^{-1}A_{-1}+A_{0})\cap\ker(A_{1})\neq\bm{\mathbf{0}}$, which
concludes the proof. ∎

###  2.3. Spectra of tridiagonal $k$-Toeplitz operators

In this section, we will provide a full characterization of the spectrum of
tridiagonal $k$-Toeplitz operators. We begin by first recalling (and
generalizing) a few theorems on Toeplitz operators. The first important
theorem is a consequence of [15, Chapter 23, Theorem 4.3]. Its counterpart for
Toeplitz operators was already established in [15, Chapter 23, Corollary 4.4].

We denote by $\sigma(\cdot),\sigma_{ess}(\cdot)$ the spectrum and the
essential spectrum of the operator, respectively. We define

$\sigma_{\mathrm{det}}(f)=\\{\lambda\in\mathbb{C}:\det(f(z)-\lambda)=0,\
\exists z\in\mathbb{T}\\}.$ (2.11)

We first have the following theorem characterizing the essential spectrum of
$T(f)$.

######  Theorem 2.4.

The operator $T(f)$ in (2.3) is Fredholm if and only if $\mathrm{det}(f)$ has
no zeros on $\mathbb{T}$. Furthermore, the essential spectrum of $T(f)$ is
given by

$\sigma_{ess}(T(f))=\sigma_{\mathrm{det}}(f).$

###### Proof.

The first part has been proven in [15, Chapter 23, Theorem 4.3]. For the
second part, note that $\lambda\in\sigma_{\text{ess}}(T(f))$ if and only if
$T(f)-\lambda$ is not a Fredholm operator. By the first part, this happens
only if $\det(f(z_{0})-\lambda)=0$ for some $z_{0}\in\mathbb{T}$. That is,
$\lambda$ is an eigenvalue of $f(z_{0})$. On the other hand, if $\lambda$ is
an eigenvalue for some $z_{0}\in\mathbb{T}$, it holds that
$\det(f(z_{0})-\lambda)=0$, which concludes the proof. ∎

We now recall the following Theorem due to Gohberg [15, Chapter 23, Theorem
5.1].

######  Theorem 2.5.

Let $f:\mathbb{T}\to\operatorname{\mathbb{C}}^{k\times k}$ be such that $T(f)$
is a Toeplitz operator. Then, $T(f)$ is Fredholm on the space $\ell^{2}$ if
and only if $\mathrm{det}(f)$ has no zeros on $\mathbb{T}$, in which case

$\operatorname{Ind}T(f)=-\operatorname{wind}(\det(f(\mathbb{T})),0),$

with $\operatorname{wind}(\det(f(\mathbb{T})),0)$ being the winding number
about the origin of the determinant of $f(z)$ with $z\in\mathbb{T}$.

Utilizing this theorem together with Theorem 2.3, we are now able to establish
one of the main results of our paper, an optimal characterization of the
spectra of tridiagonal $k$-Toeplitz operators. Before stating the theorem, we
first define

$\sigma_{\mathrm{wind}}(f):=\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\operatorname{wind}(\det(f(\mathbb{T})-\lambda),0)\neq
0\\}.$ (2.12)

By Lemma A.1,

$\det(f(z)-\lambda)=(-1)^{k+1}\left(\prod_{i=1}^{k}c_{i}\right)z+(-1)^{k+1}\left(\prod_{i=1}^{k}b_{i}\right)z^{-1}+g(\lambda)$

with $g(\lambda)$ being given by (A.2), which yields

$\sigma_{\mathrm{wind}}(f)=\left\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\operatorname{wind}\left((-1)^{k+1}\left((\prod_{i=1}^{k}c_{i})z+(\prod_{i=1}^{k}b_{i})z^{-1}\right),-g(\lambda)\right)\neq
0\right\\}.$ (2.13)

Furthermore, since

$\mathrm{det}(f(z)-\lambda)=\prod_{j=1}^{k}(\lambda_{j}(z)-\lambda),$

where $\lambda_{j}(z),1\leq j\leq k,$ are the eigenvalues of the matrix
$f(z)$, we obtain

$\displaystyle\sigma_{\mathrm{wind}}(f)=$
$\displaystyle\left\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T})-\lambda,0)\neq
0\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\neq
0\right\\}.$ (2.14)

Note that the representations (2.12), (2.13) and (2.14) have their own pro and
contra. For relating our results to the conjecture in [5], we utilize
representation (2.14) in our main results. We now establish the following
theorem for the spectrum of the operator $T(f)$.

######  Theorem 2.6.

Let $f\in\operatorname{\mathbb{C}}^{k\times k}(\mathbb{T})$ be the symbol of a
tridiagonal $k$-Toeplitz operator $T(f)$. Denote by $B_{0}$ the leading
$(k-1)\times(k-1)$ principal minor of $A_{0}$. It holds that

$\sigma_{\mathrm{det}}(f)\cup\sigma_{\mathrm{wind}}(f)\subset\sigma(T(f))\subset\sigma_{\mathrm{det}}(f)\cup\sigma_{\mathrm{wind}}(f)\cup\sigma(B_{0}),$

where $\sigma_{\mathrm{det}}(f),\sigma_{\mathrm{wind}}(f)$ are given by
(2.11), (2.14), respectively.

###### Proof.

The first inclusion is an immediate consequence of Theorems 2.4 and 2.5. For
the second inclusion, note that if

$\lambda\in\sigma(T(f))\setminus\left(\sigma_{\mathrm{det}}(f)\cup\sigma_{\mathrm{wind}}(f)\right),$
(2.15)

then it must hold that $T(f)-\lambda I$ is Fredholm and
$\operatorname{Ind}(T(f)-\lambda I)=\operatorname{wind}(\det(f-\lambda),0)=0$.
Based on the definition (2.2), this implies

$\mathrm{dim}\ \mathrm{Ker}(T(f)-\lambda
I)=\mathrm{dim}(\ell^{2}\setminus\mathrm{Im}(T(f)-\lambda I)).$ (2.16)

Moreover, since $\lambda\in\sigma(T(f))$, $T(f)-\lambda I$ has a non-trivial
kernel and non-dense image. By Theorem 2.3, this implies that $\lambda$ must
be an eigenvalue of $B_{0}$. ∎

This characterization is optimal. It cannot be made more precise, as it cannot
be guaranteed that an eigenvalue $\lambda$ of $B_{0}$ with
$\operatorname{wind}(\det(f-\lambda),0)=0$ is also an eigenvalue of $T(f)$. To
illustrate this, consider

$f(z)=\begin{pmatrix}0&1+\frac{1}{2}z\\\ 1+\frac{1}{2}z^{-1}&1\\\
\end{pmatrix}.$

The determinant of this symbol is given by
$\det(f)=-(1+\frac{1}{2}z)(1+\frac{1}{2}z^{-1})$. We have
$\operatorname{wind}(\det(f(\mathbb{T})),0)=0$ and $\det f(z)\neq 0,\forall
z\in\mathbb{T}$. In this case, $B_{0}=0$ is just the top left entry of $f(z)$.
Furthermore, by (2.2), if $0\in\sigma(T(f))$, then the kernel space of $T(f)$
must be non-trivial. Since

$T(f)=\begin{pmatrix}0&1&0&0&\\\ 1&1&\frac{1}{2}&0&\\\ 0&\frac{1}{2}&0&1&\\\
0&0&1&1&\\\ &&&&\ddots\end{pmatrix},$

one can check that, after scaling, the only possible eigenvector of $T(f)$
corresponding to the eigenvalue $0$ is
$\bm{\mathbf{u}}=(1,0,-2,0,4,\dots)^{\top}$. However, $\bm{\mathbf{u}}$ does
not lie in $\ell^{2}$, which means that $0\not\in\sigma(T(f))$.

On the other hand, consider

$\tilde{f}(z)=\begin{pmatrix}0&1+2z\\\ 1+2z^{-1}&1\\\ \end{pmatrix}.$

The determinant is $\det(\tilde{f})=-(1+2z)(1+2z^{-1})$, hence
$0\notin\sigma_{\mathrm{det}}(f)$ and
$\operatorname{wind}(\det(\tilde{f}(\mathbb{T})),0)=0$. Therefore, by Theorem
2.3, the operator $T(\tilde{f})$ is either invertible or has a nontrivial
kernel. By explicitly writing down $T(\tilde{f})$

$T(\tilde{f})=\begin{pmatrix}0&1&0&0&\\\ 1&1&2&0&\\\ 0&2&0&1&\\\ 0&0&1&1&\\\
&&&&\ddots\end{pmatrix},$

we find that
$\tilde{\bm{\mathbf{u}}}=(1,0,-\frac{1}{2},0,\frac{1}{4},\dots)^{\top}$
satisfies $T(\tilde{f})\tilde{\bm{\mathbf{u}}}=\bm{\mathbf{0}}$. Then in this
case, $\tilde{\bm{\mathbf{u}}}\in\ell^{2}$ and $0\in\sigma(T(f))$.

######  Remark 2.7.

Note that (2.13) and (2.14) also provide ways to compute
$\sigma_{\mathrm{wind}}(f)$ explicitly. We will utilize (2.14) in Section 4
for numerical illustrations of the topological origin of the skin effect.

For the sake of completeness, we will devote the last part in this section to
tridiagonal $k$-Laurent operators that are defined as

$L(f)=\begin{pmatrix}\ddots&\ddots&\ddots&\\\ &A_{1}&A_{0}&A_{-1}&\\\
&&A_{1}&A_{0}&A_{-1}&\\\ &&&\ddots&\ddots&\ddots\\\ \end{pmatrix},$ (2.17)

with the generating symbol $f$ being (2.5). Using the identification
$L(f-\lambda)=L(f)-\lambda$, which we have used to characterize the spectrum
of tridiagonal $k$-Toeplitz operators, one can obtain an explicit
characterization of the spectrum of tridiagonal $k$-Laurent operators as a
consequence of [15, Chapter 23, Corollary 2.5]. The following theorem holds.

######  Theorem 2.8.

Let $f\in(\mathcal{L}(\mathbb{T})^{\infty})^{k\times k}$ and denote the
associated tridiagonal $k$-Laurent operator $L(f)$. Then, it holds that

$\sigma(L(f))=\sigma_{\text{ess}}(L(f))=\sigma_{\det}(f)$

with $\sigma_{\det}(f)$ being defined by (2.11).

###  2.4. Eigenvectors of tridiagonal $k$-Toeplitz operators

Now that we have characterized the spectra of tridiagonal $k$-Toeplitz
operators, we will venture onwards to explore their eigenvectors. The
following theorem is the second main result of this paper.

######  Theorem 2.9.

Suppose $\Pi_{j=1}^{k}c_{j}\neq 0$ and $\Pi_{j=1}^{k}b_{j}\neq 0$. Let
$f(z)\in\mathbb{C}^{k\times k}$ be the symbol (2.5) and let
$\lambda\in\operatorname{\mathbb{C}}\setminus\sigma_{ess}(T(f))$. If
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)<0$ for
$\lambda_{j}$ being the eigenvalues of $f(z)$, then there exists an
eigenvector $\bm{x}\in\ell^{2}$ of $T(f)$ associated to $\lambda$ and some
$\rho<1$ such that

$\frac{\lvert\bm{x}_{j}\rvert}{\max_{i}\lvert\bm{x}_{i}\rvert}\leq C\lceil
j/k\rceil\rho^{\lceil j/k\rceil-1},\quad j\geq 1,$ (2.18)

where $C>0$ is a constant depending only on $\lambda,a_{p},b_{p},c_{p}$ for
$1\leq p\leq k$.

###### Proof.

Let $\lambda\in\mathbb{C}$ be such that
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)=\operatorname{wind}(\det(f(\mathbb{T})-\lambda),0)<0$.
Since $\prod_{i=1}^{k}b_{i}\neq 0,\prod_{i=1}^{k}c_{i}\neq 0$, by Lemma A.1,
$\det(f(z)-\lambda)$ is a meromorphic function in $\mathbb{C}$, with poles at
$0$ and $\infty$. Then the argument principle implies that
$\det(f(z)-\lambda)=0$ for two values $z_{1},z_{2}$ with
$1<\frac{1}{\rho}\leq\left|{z_{j}}\right|,j=1,2$, counted with multiplicity.
Moreover, since $\det(f(z_{i})-\lambda)=0$, we find that $\lambda$ is an
eigenvalue to $f(z_{i})$ and hence there exists an associated eigenvector
$\mathbf{v}_{i}$. Assume for now that $z_{1}\neq z_{2}$. Consider the vectors
defined by

$\mathbf{u}_{i}=(\mathbf{v}_{i}^{\top},z_{i}^{-1}\mathbf{v}_{i}^{\top},z_{i}^{-2}\mathbf{v}_{i}^{\top},\dots)^{\top},\quad
i\in\\{1,2\\}.$ (2.19)

The vector $\mathbf{u}_{i}$ satisfies the eigenvalue equation
$T(f)\mathbf{u}_{i}=\lambda\mathbf{u}_{i}$ in all but the first row. Since
$\bm{\mathbf{u}}_{1},\bm{\mathbf{u}}_{2}$ are linearly independent, there
exists a linear combination of them that is indeed an eigenvector of $T(f)$.

Next, we consider the case where $z_{1}=z_{2}$ and first suppose that
$\dim\operatorname{Eig}(f(z_{1}),\lambda)>1$, where
$\operatorname{Eig}(f(z_{1}),\lambda)$ denotes the eigenspace of $f(z_{1})$
associated to the eigenvalue $\lambda$. Then there exist linearly independent
eigenvectors $\bm{\mathbf{v}}_{1},\bm{\mathbf{v}}_{2}$ of $f(z_{1})$
associated to $\lambda$ and we can use

$\mathbf{u}_{i}=(\mathbf{v}_{i}^{\top},z_{1}^{-1}\mathbf{v}_{i}^{\top},z_{1}^{-2}\mathbf{v}_{i}^{\top},\dots)^{\top}\quad
i\in\\{1,2\\},$ (2.20)

to construct an exponentially decaying eigenvector of $T(f)$. Finally, it
remains to treat the case when $z_{1}=z_{2}$ and
$\dim\operatorname{Eig}(f(z_{1}),\lambda)=1$. Denote $\bm{\mathbf{v}}_{1}$ the
eigenvector of $f(z_{1})$ associated to $\lambda$. We will construct a vector
$\bm{\mathbf{v}}_{2}$ such that

$\bm{\mathbf{u}}_{2}=(\bm{\mathbf{v}}_{2}^{\top},\
z_{1}^{-1}\bm{\mathbf{v}}_{2}^{\top}+\bm{\mathbf{v}}_{1}^{\top},\
z_{1}^{-2}\bm{\mathbf{v}}_{2}^{\top}+2z_{1}^{-1}\bm{\mathbf{v}}_{1}^{\top},\
\dots)^{\top}$ (2.21)

satisfies the eigenvalue problem in all but the first row. Similarly to the
previous arguments, this is enough to construct the eigenvector of $T(f)$. By
explicitly writing down $(T(f)-\lambda)\bm{\mathbf{u}}_{2}=0$ (except for the
first row), we find that $\bm{\mathbf{v}}_{2}$ must satisfy the condition

$(A_{1}+(A_{0}-\lambda)z_{1}^{-1}+A_{-1}z_{1}^{-2})\bm{\mathbf{v}}_{2}=-((A_{0}-\lambda)+2A_{-1}z_{1}^{-1})\bm{\mathbf{v}}_{1},$
(2.22)

which is also a sufficient condition. From Lemma A.1, it is not hard to see
that for all $\tilde{\lambda}$ in a small neighborhood of $\lambda$, the
equation $\det(f(z)-\tilde{\lambda})$ has two distinct roots. Let
$(\varepsilon_{j})_{j}$ be a sequence converging to zero. We define
$\lambda(\varepsilon_{j})=\lambda+\varepsilon_{j}$. Then by the previous
observation, we may assume that to each $\lambda(\varepsilon_{j})$ there exist
two distinct roots $z_{1}(\varepsilon_{j}),z_{2}(\varepsilon_{j})$ of the
equation $\det(f(z)-\lambda(\varepsilon_{j}))=0$. Let
$\bm{\mathbf{w}}_{1}(\varepsilon_{j}),\bm{\mathbf{w}}_{2}(\varepsilon_{j})$ be
the corresponding unit eigenvectors. We may assume that

$\bm{\mathbf{w}}_{1}(\varepsilon_{j})\rightarrow\bm{\mathbf{v}}_{0},\quad\bm{\mathbf{w}}_{2}(\varepsilon_{j})\rightarrow\bm{\mathbf{v}}_{0}.$
(2.23)

The limit of $\bm{\mathbf{w}}_{i}(\varepsilon_{j})$ (or a subsequence thereof)
exists by a standard compactness argument and it must be an eigenvector of
$f(z_{1})$ associated to $\lambda$. In particular, for each $j$ it holds that

$\displaystyle\left(A_{1}+(A_{0}-\lambda(\varepsilon_{j}))z_{1}(\varepsilon_{j})^{-1}+A_{-1}z_{1}(\varepsilon_{j})^{-2}\right)\bm{\mathbf{w_{1}}}(\varepsilon_{j})=0,$
$\displaystyle\left(A_{1}+(A_{0}-\lambda(\varepsilon_{j}))z_{2}(\varepsilon_{j})^{-1}+A_{-1}z_{2}(\varepsilon_{j})^{-2}\right)\bm{\mathbf{w_{2}}}(\varepsilon_{j})=0.$

We define

$\Delta
z(\varepsilon_{j})=z_{1}(\varepsilon_{j})^{-1}-z_{2}(\varepsilon_{j})^{-1}.$

Then we have

$\displaystyle
A_{1}\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}+(A_{0}-\lambda(\varepsilon_{j}))\frac{z_{1}(\varepsilon_{j})^{-1}\bm{\mathbf{w}}_{1}(\varepsilon_{j})-z_{2}(\varepsilon_{j})^{-1}\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}$
$\displaystyle\qquad+A_{-1}\frac{z_{1}(\varepsilon_{j})^{-2}\bm{\mathbf{w}}_{1}(\varepsilon_{j})-z_{2}(\varepsilon_{j})^{-2}\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}=0.$

Further expansion of this expression then yields

$\displaystyle
A_{1}\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}+(A_{0}-\lambda(\varepsilon_{j}))\frac{z_{1}(\varepsilon_{j})^{-1}(\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta
z(\varepsilon_{j})}+A_{-1}\frac{z_{1}(\varepsilon_{j})^{-2}(\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta
z(\varepsilon_{j})}$ (2.24) $\displaystyle=$
$\displaystyle-\left((A_{0}-\lambda(\varepsilon_{j}))\frac{(z_{1}(\varepsilon_{j})^{-1}-z_{2}(\varepsilon_{j})^{-1})\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta
z(\varepsilon_{j})}+A_{-1}\frac{(z_{1}(\varepsilon_{j})^{-2}-z_{2}(\varepsilon_{j})^{-2})\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}\right).$

Consider now the sequence
$\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}$. If a bounded subsequence exists, we may define
$\bm{\mathbf{v}}_{2}$ as the limit of this subsequence. In this case, it is
not hard to see that $\bm{\mathbf{v}}_{2}$ satisfies the condition (2.22).
Otherwise, there exists a subsequence of
$\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}$ whose norm tends to infinity. Define

$\Delta\bm{\mathbf{w}}(\varepsilon_{j})=\lVert\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})\rVert.$

Then (upon passing to said subsequence) it holds that

$\frac{\Delta
z(\varepsilon_{j})}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}\rightarrow 0.$
(2.25)

Multiplying (2.24) with this yields

$\displaystyle
A_{1}\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}+(A_{0}-\lambda(\varepsilon_{j}))\frac{z_{1}(\varepsilon_{j})^{-1}(\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}+A_{-1}\frac{z_{1}(\varepsilon_{j})^{-2}(\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}$
$\displaystyle=$
$\displaystyle-\left((A_{0}-\lambda(\varepsilon_{j}))\frac{(z_{1}(\varepsilon_{j})^{-1}-z_{2}(\varepsilon_{j})^{-1})\bm{\mathbf{w}}_{2}(\varepsilon_{j}))}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}+A_{-1}\frac{(z_{1}(\varepsilon_{j})^{-2}-z_{2}(\varepsilon_{j})^{-2})\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}\right).$

In the limit, the right-hand side tends to $0$. Thus
$\bm{\mathbf{v}}_{2}=\lim_{\varepsilon_{j}\to
0}\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta\bm{\mathbf{w}}(\varepsilon_{j})}$
satisfies

$\left(A_{1}+(A_{0}-\lambda)z_{1}^{-1}+A_{-1}z_{1}^{-2}\right)\bm{\mathbf{v}}_{2}=0.$

This indicates that $\bm{\mathbf{v}}_{2}$ is an eigenvector. But from the
construction of $\bm{\mathbf{v}}_{2}$, it becomes apparent that
$\bm{\mathbf{v}}_{2}$ must be orthogonal to $\bm{\mathbf{v}}_{1}$, which
contradicts the assumption that the eigenspace of $f(z_{1})$ associated with
$\lambda$ is one-dimensional. Therefore, the sequence
$\frac{\bm{\mathbf{w}}_{1}(\varepsilon_{j})-\bm{\mathbf{w}}_{2}(\varepsilon_{j})}{\Delta
z(\varepsilon_{j})}$ is uniformly bounded and arguments for the first case
already demonstrate the existence of such $\bm{\mathbf{v}}_{2}$ satisfying
(2.22).

The exponential bound in (2.18) is clear by the construction of the vectors
$\bm{\mathbf{u}}_{1}$ and $\bm{\mathbf{u}}_{2}$. ∎

######  Remark 2.10.

Note that the constructive approach used in the proof of [21, Theorem 7.2]
does not work here. We have used a new approximation approach to construct the
eigenvectors.

######  Remark 2.11.

For the case when
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)>0$,
$\lambda\in\sigma(T(f))$ is because $T(f)$ has a non-dense image. In
particular, one can construct the eigenvector $\bm{x}$ of
$\overline{T(f)}^{\top}-\overline{\lambda}$ in the same spirit of Theorem 2.9,
which implies $T(f)-\lambda$ has a non-dense image.

##  3\. Pseudo-spectra of tridiagonal $k$-Toeplitz matrices

In this section, we will apply the results of the previous section to
matrices. As already described by Trefethen in [20], the spectrum of non-
Hermitian matrices is very sensitive to small perturbations. For this reason,
it is often advisable to study their pseudo-spectra, rather than their
spectra. For the convenience of the reader, we recall the definition of
pseudoeigenvalues and pseudoeigenvectors.

######  Definition 3.1.

Let $\varepsilon>0$. Then, $\lambda\in\mathbb{C}$ is an
$\varepsilon$-pseudoeigenvalue of $\mathbf{A}\in\mathbb{C}^{N\times N}$ if one
of the following conditions is satisfied:

  1. (i)

$\lambda$ is a proper eigenvalue of $\mathbf{A}+\mathbf{E}$ for some
$\mathbf{E}\in\mathbb{C}^{N\times N}$ such that
$\lVert\mathbf{E}\rVert\leq\varepsilon$;

  2. (ii)

$\lVert(\mathbf{A}-\lambda I)\mathbf{u}\rVert<\varepsilon$ for some vector $u$
with $\lVert\bm{\mathbf{u}}\rVert=1$;

  3. (iii)

$\lVert(\mathbf{A}-\lambda I)^{-1}\rVert^{-1}\leq\varepsilon$.

The set of all $\varepsilon$-pseudoeigenvalues of $\bm{\mathbf{A}}$, the
$\varepsilon$-pseudospectrum, is denoted by $\sigma_{\varepsilon}(A)$. If some
nonzero $\bm{\mathbf{u}}$ satisfies
$\lVert(\lambda-\mathbf{A})\mathbf{u}\rVert<\varepsilon$, then we say that
$\bm{\mathbf{u}}$ is an $\varepsilon$-pseudoeigenvector of $\mathbf{A}$.

By the equivalence of norms in finite dimensions, any norm can be used. For a
general treatment of pseudospectra, see [21].

We consider the family of tridiagonal $k$-Toeplitz matrices
$\left\\{\mathbf{A}_{N}\right\\}$ of various dimensions obtained as $N\times
N$ finite sections of the infinite operator $T(f)$. Results on the
pseudospectra of the family of Toeplitz matrices can be found in [20]. On the
other hand, the spectrum of tridiagonal $k$-Toeplitz matrices is fully
understood and closed form characteristic polynomials have been found in [13].

Let us also introduce the sets

$\displaystyle\Omega_{r}$
$\displaystyle:=\\{\lambda\in\operatorname{\mathbb{C}}:\operatorname{wind}(\operatorname{det}(f(\mathbb{T}_{r})-\lambda),0)>0\\},$
(3.1) $\displaystyle\Omega^{R}$
$\displaystyle:=\\{\lambda\in\operatorname{\mathbb{C}}:\operatorname{wind}(\operatorname{det}(f(\mathbb{T}_{R})-\lambda),0)<0\\},$
(3.2)

where $\mathbb{T}_{r}=\\{z\in\operatorname{\mathbb{C}}:\lvert z\rvert=r\\}$
and let $\sigma_{\varepsilon}(\mathbf{A}_{N})$ be the set of
$\varepsilon$-pseudoeigenvalues of $\mathbf{A}_{N}$.

The connection between the eigenvectors of a tridiagonal $k$-Toeplitz operator
and those of the associated tridiagonal $k$-Toeplitz matrix can be made with
an argument similar to [21, Theorem 7.2]. The first $N$ components of an
eigenvector of the Toeplitz operator can be used as the
$\varepsilon$-pseudoeigenvector of the tridiagonal $k$-Toeplitz matrix
$\mathbf{A}_{N}$. Since we have proven the existence of an exponentially
decaying eigenvector in Theorem 2.9, we can conclude with the following
statement on tridiagonal $k$-Toeplitz matrices.

######  Theorem 3.2.

Let $\\{\mathbf{A}_{N}\\}$ be a family of tridiagonal $k$-Toeplitz matrices
such that $\prod_{i=1}^{k}b_{i}\neq 0$, $\prod_{i=1}^{k}c_{i}\neq 0$. Then,
for any $r<1$ and $\rho>r$, we have

$\Omega_{r}\cup\Omega^{1/r}\cup\left(\sigma(T(f))+\Delta_{\varepsilon}\right)\subseteq\sigma_{\varepsilon}\left(A_{N}\right)\quad\text{
for }\quad\varepsilon=\max(C_{1},\lceil N/k\rceil C_{2})\rho^{\lceil
N/k\rceil-1},$ (3.3)

where $\Delta_{\varepsilon}$ is a closed unit disk of radius $\varepsilon$ and
$C_{1},C_{2}$ are constants depending on $r,\rho$, and $a_{j},b_{j},c_{j}$ for
$1\leq j\leq k$ but independent of $N$. In particular, for the corresponding
$\lambda\in\Omega_{r}\cup\Omega^{1/r}$, there exist nonzero pseudoeigenvectors
$\mathbf{v}^{(N)}$ satisfying

$\frac{\lVert(\mathbf{A}_{N}-\lambda)\mathbf{v}^{(N)}\rVert}{\lVert\mathbf{v}^{(N)}\rVert}\leq\max(C_{3},\lceil
N/k\rceil C_{4})\rho^{\lceil N/k\rceil-1}$ (3.4)

such that

$\frac{|\bm{\mathbf{v}}_{j}^{(N)}|}{\max_{i}|\bm{\mathbf{v}}_{i}^{(N)}|}\leq\begin{cases}C_{5}\lceil
N/k\rceil\rho^{\lceil j/k\rceil-1},&\text{if }\lambda\in\Omega^{1/r},\\\
C_{5}\lceil N/k\rceil\rho^{\lceil(N-j)/k\rceil-1},&\text{if
}\lambda\in\Omega_{r},\end{cases}\quad 1\leq j\leq N,$ (3.5)

where $C_{3},C_{4},C_{5}$ are constants independent of $N$.

###### Proof.

Firstly, we note that the inclusion
$\sigma(T(f))+\Delta_{\varepsilon}\subseteq\sigma_{\varepsilon}\left(A_{N}\right)$
is triviality valid for any matrix or operator. Secondly, by symmetry,
$\Omega_{r}$ must satisfy an estimate of the type (3.3) if $\Omega^{1/r}$
does. Thus all that we have to prove is
$\Omega^{1/r}\subseteq\sigma_{\varepsilon}\left(\bm{\mathbf{A}}_{N}\right)$.

Given any $r<1$, let $\lambda\in\Omega^{1/r}$ be arbitrary. Since
$\operatorname{wind}(\det(f(\mathbb{T}_{\frac{1}{r}})-\lambda),0)<0$,
similarly to the arguments in the proof of Theorem 2.9, we can show that
$\det(f(z)-\lambda)=0$ for two values $z_{1},z_{2}$ with
$1<\frac{1}{\rho}\leq\left|{z_{j}}\right|,j=1,2$, counted with multiplicity.
Also, since $\det(f(z_{i})-\lambda)=0$, we find that $\lambda$ is an
eigenvalue to $f(z_{i})$ and hence there exists an associated eigenvector
$\mathbf{v}_{i}$. Assume for now that $z_{1}\neq z_{2}$. Consider the vectors
defined by

$\mathbf{u}_{i}=(\mathbf{v}_{i}^{\top},z_{i}^{-1}\mathbf{v}_{i}^{\top},z_{i}^{-2}\mathbf{v}_{i}^{\top},\dots,z_{i}^{-\lceil
N/k\rceil+1}\bm{\mathbf{v}}_{i}^{\top})^{\top}\quad i\in\\{1,2\\},$ (3.6)

where without loss of generality we assume that $N$ is divisible by $k$. This
vector satisfies the eigenvalue equation
$\bm{\mathbf{A}}_{N}\mathbf{u}_{i}=\lambda\mathbf{u}_{i}$ in all but the first
and last rows. Since $\bm{\mathbf{u}}_{1},\bm{\mathbf{u}}_{2}$ are linearly
independent, there exists a normalized linear combination
$\bm{\mathbf{v}}^{(N)}=a_{1}\bm{\mathbf{u}}_{1}+a_{2}\bm{\mathbf{u}}_{2}$ so
that $(\bm{\mathbf{A}}_{N}-\lambda)\bm{\mathbf{v}}^{(N)}=\bm{\mathbf{0}}$ in
all but the last row. It is not hard to see that

$\lVert(\bm{\mathbf{A}}_{N}-\lambda)\bm{\mathbf{v}}^{(N)}\rVert\leq
C_{3}\max\\{\left|{z_{1}}\right|^{-\lceil
N/k\rceil+1},\left|{z_{2}}\right|^{-\lceil N/k\rceil+1}\\},$ (3.7)

where $C_{3}$ depends on $a_{j},b_{j},c_{j},\lambda$ but not on $N$. Then
since $1<\frac{1}{\rho}\leq\left|{z_{j}}\right|,j=1,2,$ we obtain the bound

$\lVert(\bm{\mathbf{A}}_{N}-\lambda)\bm{\mathbf{v}}^{(N)}\rVert\leq
C_{3}\rho^{\lceil N/k\rceil-1},$ (3.8)

which proves the claim (3.4) if $z_{1}\neq z_{2}$. If the roots $z_{1},z_{2}$
are distinct for all $\lambda\in\overline{\Omega^{1/r}}$, then the function

$\sup_{N\geq 1}\frac{\sigma_{N}\left(\lambda
I-\bm{\mathbf{A}}_{N}\right)}{r^{\lceil N/k\rceil-1}}$

is a continuous function of $\lambda$ on the compact set
$\overline{\Omega^{1/r}}$; its maximum provides the constant $C_{1}$ in (3.3)
that independent of $\lambda,N$, and we can take $\rho=r$.

In the case where $z_{1}=z_{2}$, one can resort to a similar argument as the
one in Theorem 2.9 to utilize

$\displaystyle\mathbf{u}_{1}$ $\displaystyle=(\mathbf{v}_{1}^{\top},\
z_{1}^{-1}\mathbf{v}_{1}^{\top},\ z_{1}^{-2}\mathbf{v}_{1}^{\top},\ \dots,\
z_{1}^{-\lceil\frac{N}{k}\rceil+1}\bm{\mathbf{v}}_{1}^{\top})^{\top},$ (3.9)
$\displaystyle\bm{\mathbf{u}}_{2}$
$\displaystyle=(\bm{\mathbf{v}}_{2}^{\top},\
z_{1}^{-1}\bm{\mathbf{v}}_{2}^{\top}+\bm{\mathbf{v}}_{1}^{\top},\
z_{1}^{-2}\bm{\mathbf{v}}_{2}^{\top}+2z_{1}^{-1}\bm{\mathbf{v}}_{1}^{\top},\
\dots,\
z_{1}^{-\lceil\frac{N}{k}\rceil+1}\bm{\mathbf{v}}_{2}^{\top}+(\lceil\frac{N}{k}\rceil-1)z_{1}^{-\lceil\frac{N}{k}\rceil+2}\bm{\mathbf{v}}_{1}^{\top})^{\top}$

to construct the pseudoeigenvector. This yields a similar bound except with
$C_{3}\rho^{\lceil N/k\rceil-1}$ in (3.4) replaced by an algebraically growing
factor at worst $\lceil N/k\rceil C_{4}\rho^{\lceil N/k\rceil-1}$. Then since
the function

$\sup_{N\geq 1}\frac{\sigma_{N}\left(\lambda
I-\bm{\mathbf{A}}_{N}\right)}{\lceil N/k\rceil r^{\lceil N/k\rceil-1}}$

is a continuous function of $\lambda$ on the compact set
$\overline{\Omega^{1/r}}$, its maximum provides the constant $C_{2}$ in (3.3).
Finally, (3.5) is deduced from construction (3.6) and (3.9). ∎

##  4\. Topological origin of the skin effect in polymer systems of
subwavelength resonators

The non-Hermitian skin effect is the phenomenon whereby in the subwavelength
regime a large proportion of the bulk eigenmodes of a non-Hermitian open
system of subwavelength resonators are localised at one edge of the structure
[8, 19]. It has been realized experimentally in topological photonics,
phononics, and other condensed matter systems [14, 19, 17]. Its significance
lies in its substantial contribution to advancing the realm of active
metamaterials, paving the way for novel opportunities to guide and control
energy on subwavelength scales [3, 6, 7, 12].

Recently, the non-Hermitian skin effect has been demonstrated in one-
dimensional systems of subwavelength resonators using first-principle
mathematical analysis. In particular, [1] demonstrates the skin effect for the
monomer case when an imaginary gauge potential is introduced inside the
resonators to break Hermiticity and [5] generalizes the results in [1] to
dimer systems. We also refer to [2] for the skin effect in a three-dimensional
setting and to [4] for stability analysis.

By the results obtained in the previous sections, we provide the topological
origin of the skin effect in the one-dimensional polymer system, i.e., in
systems with periodically repeated cells of $k$ resonators. This also
elucidates the numerical findings in [5] for the topological origin of the
skin effect.

Let us first introduce the model setting. We consider a chain of $N$ disjoint
one-dimensional resonators $D:=(x_{i}^{\mathrm{L}},x_{i}^{\mathrm{R}}$), where
$(x_{i}^{\mathrm{L},\mathrm{R}})_{1\leq i\leq N}\subset\mathbb{R}$ are the
$2N$ extremities such that
$x_{i}^{\mathrm{L}}<x_{i}^{\mathrm{R}}<x_{i+1}^{\mathrm{L}}$ for any $1\leq
i\leq N$. The lengths of the $i$-th resonators will be denoted by
$\ell_{i}=x_{i}^{\mathrm{R}}-x_{i}^{\mathrm{L}}$ and the spacings between the
$i$-th and the $(i+1)$-th resonators will be denoted by
$s_{i}=x_{i+1}^{\mathrm{L}}-x_{i}^{\mathrm{R}}$. Furthermore, we assume that
$k$ resonators are periodically repeated, that is, $s_{i+k}=s_{i}$. The
resulting system is illustrated in Figure 4.1.

$x_{1}^{\mathrm{L}}$$x_{1}^{\mathrm{R}}$$\ell_{1}$$x_{2}^{\mathrm{L}}$$x_{2}^{\mathrm{R}}$$\ell_{2}$$x_{3}^{\mathrm{L}}$$x_{3}^{\mathrm{R}}$$\ell_{3}$…$x_{k}^{\mathrm{L}}$$x_{k}^{\mathrm{R}}$$\ell_{k}$$x_{k+1}^{\mathrm{L}}$$x_{k+1}^{\mathrm{R}}$$\ell_{1}$…$x_{N-2}^{\mathrm{L}}$$x_{N-2}^{\mathrm{R}}$$\ell_{k-2}$$x_{N-1}^{L}$$x_{N-1}^{R}$$\ell_{k-1}$$x_{N}^{\mathrm{L}}$$x_{N}^{\mathrm{R}}$$\ell_{k}$$s_{1}$$s_{2}$$s_{k}$$s_{k-2}$$s_{k-1}$

Figure 4.1. A chain of $N$ times periodically repeated $k$ subwavelength
resonators, with lengths $(\ell_{i})_{1\leq i\leq N}$ and spacings
$(s_{i})_{1\leq i\leq N-1}$.

Let $D$ be the domain consisting of the union of all resonators

$D=\bigcup_{i=1}^{N}(x_{i}^{\mathrm{L}},x_{i}^{\mathrm{R}})\subset\mathbb{R}.$
(4.1)

The underlying wave equation that governs the system is given by

$\begin{cases}u^{\prime\prime}(x)+\gamma
u^{\prime}(x)+\frac{\omega^{2}}{v^{2}}u=0,&x\in D,\\\
u^{\prime\prime}(x)+\frac{\omega^{2}}{v_{b}^{2}}u=0,&x\in\mathbb{R}\backslash
D,\\\
\left.u\right|_{\mathrm{R}}\left(x_{i}^{\mathrm{L},\mathrm{R}}\right)-\left.u\right|_{\mathrm{L}}\left(x_{i}^{\mathrm{L},\mathrm{R}}\right)=0,&\text{
for all }1\leq i\leq N,\\\
\left.\frac{\mathrm{d}u}{\mathrm{\leavevmode\nobreak\
d}x}\right|_{\mathrm{R}}\left(x_{i}^{\mathrm{L}}\right)=\left.\delta\frac{\mathrm{d}u}{\mathrm{\leavevmode\nobreak\
d}x}\right|_{\mathrm{L}}\left(x_{i}^{\mathrm{L}}\right),&\text{ for all }1\leq
i\leq N,\\\ \left.\frac{\mathrm{d}u}{\mathrm{\leavevmode\nobreak\
d}x}\right|_{\mathrm{L}}\left(x_{i}^{\mathrm{R}}\right)=\left.\delta\frac{\mathrm{d}u}{\mathrm{\leavevmode\nobreak\
d}x}\right|_{\mathrm{R}}\left(x_{i}^{\mathrm{R}}\right),&\text{ for all }1\leq
i\leq N,\\\ \frac{\mathrm{d}u}{\mathrm{\leavevmode\nobreak\
d}|x|}-\mathrm{i}\frac{\omega}{v}u=0,&\text{ at
}x=x_{1}^{\mathrm{L}},x_{N}^{\mathrm{R}},\text{ respectively},\end{cases}$
(4.2)

where $0<\delta\ll 1$, the wave speeds inside the resonator and in the
background are respectively $v_{b}$ and $v$ and are of order one, and $\gamma
u^{\prime}$ for $\gamma\in\mathbb{R}^{*}$ denotes the imaginary gauge
potential. Here, for a function $w$ we denote by

$\left.w\right|_{\mathrm{L}}(x):=\lim_{s\rightarrow 0,s>0}w(x-s)\text{ and
}\left.w\right|_{\mathrm{R}}(x):=\lim_{s\rightarrow 0,s>0}w(x+s)$

if the limits exist.

We are interested in nontrivial frequencies $\omega$ such that (4.2) is
satisfied and $\omega\rightarrow 0$ as $\delta\rightarrow 0$. Such $\omega$
are called subwavelength resonances.

In the high-contrast, low-frequency setting described above, gauge capacitance
matrices provide a discrete approximation of the subwavelength
eigenfrequencies induced in the structure.

######  Definition 4.1.

For $\gamma\in\mathbb{R}^{*}$, we define the _gauge capacitance matrix_
$C^{\gamma}\in\mathbb{R}^{N\times N}$ by

$C^{\gamma}_{i,j}=\begin{cases}\frac{\gamma}{s_{1}}\frac{\ell_{1}}{1-e^{-\gamma\ell_{1}}},&i=j=1,\\\
\frac{\gamma}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma\ell_{i}}}-\frac{\gamma}{s_{i-1}}\frac{\ell_{i}}{1-e^{\gamma\ell_{i}}},&1<i=j<N,\\\
-\frac{\gamma}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma\ell_{j}}},&1\leq i=j-1\leq
N-1,\\\ \frac{\gamma}{s_{j}}\frac{\ell_{i}}{1-e^{\gamma\ell_{j}}},&2<i=j+1\leq
N,\\\
-\frac{\gamma}{s_{N-1}}\frac{\ell_{N}}{1-e^{\gamma\ell_{N}}},&i=j=N,\end{cases},$
(4.3)

and all the other entries are zero.

In [1, Corollary 2.6], one can find a formal argumentation that the gauge
capacitance matrix gives indeed a valid approximation of the eigenmodes in a
subwavelength regime as $\delta\rightarrow 0$. For the remainder of this
paper, we assume that $\gamma=\ell_{i}=1$. Without loss of generality, we
consider a system of $nk$ resonators and obtain a gauge capacitance matrix of
the tridiagonal $k$-Toeplitz form (perturbed):

$\mathbf{A}^{(a,b)}_{nk}=\begin{pmatrix}a_{1}+a&b_{1}&0&\cdots&\cdots&0\\\
c_{1}&a_{2}&b_{2}&\ddots&&\vdots\\\ 0&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&c_{k-3}&a_{k-2}&b_{k-2}&0\\\
\vdots&&\ddots&c_{k-2}&a_{k-1}&b_{k-1}\\\
0&\cdots&\cdots&0&c_{k-1}&a_{k}+b\end{pmatrix}.$ (4.4)

For the semi-infinite systems we obtain a perturbed tridiagonal $k$-Toeplitz
operator

$T^{a}(f)=\begin{pmatrix}a_{1}+a&b_{1}&\\\ c_{1}&a_{2}&b_{2}&\\\
&\ddots&\ddots&\ddots\\\ &&c_{k-1}&a_{k}&b_{k}\\\ &&&c_{k}&a_{1}&b_{1}\\\
&&&&\ddots&\ddots&\ddots\end{pmatrix}$

with $f$ being the symbol (2.5).

It has been observed and conjectured [5] that the exponential decay of the
eigenvectors of $k$-Toeplitz matrices for $k\geq 2$ is due to the winding of
the eigenvalues of the symbol being nontrivial. Now, by the spectral theory
for the tridiagonal $k$-Toeplitz operators introduced in the previous
sections, we have the following results for the topological origin of the skin
effect in subwavelength resonator systems, validating the conjecture made in
[5].

######  Theorem 4.2.

Suppose $\Pi_{j=1}^{k}c_{j}\neq 0$ and $\Pi_{j=1}^{k}b_{j}\neq 0$. Let
$f(z)\in\mathbb{C}^{k\times k}$ be the symbol (2.5) and let
$\lambda\in\operatorname{\mathbb{C}}\setminus\sigma_{ess}(T^{a}(f))$. If
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)<0$, then
there exists an eigenvector $\bm{x}$ of $T^{a}(f)$ associated to $\lambda$ and
some $\rho<1$ such that

$\frac{\lvert\bm{x}_{j}\rvert}{\max_{i}\lvert\bm{x}_{i}\rvert}\leq C\lceil
j/k\rceil\rho^{\lceil j/k\rceil-1},$ (4.5)

where $C>0$ is a constant depending only on
$\lambda,a_{j},b_{j},c_{j},j=1,\cdots,k$. If
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)>0$, then
the above results hold for the left eigenvectors.

###### Proof.

Although with a perturbation $a$ on the first element, the proof of Theorem
2.9 can still be applied. ∎

Theorem 4.2 elucidates the topological origin of the skin effect in the
polymer system of subwavelength resonators. In particular, the skin effect
holds for all $\lambda$ in the region

$G:=\left\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\neq
0\right\\}.$ (4.6)

This is a generalization for the topological origin of the skin effect in the
Toeplitz operator case given in [1].

The last part of the section is devoted to illustrating numerically the skin
effect and its topological origin in chains of $2$ and $3$ periodically
repeated resonators. We start by illustrating in Figure 4.2 the results of a
system of $2$ periodically repeated resonators as in [5]. In Figure 4.2A, we
show the spectrum and pseudospectrum of the gauge capacitance matrix of a
system of $25$ dimers together with the winding of the two eigenvalues of the
symbol of the corresponding 2-Toeplitz operator. Figure 4.2B shows that all
the eigenvectors (black eigenvectors) associated with eigenvalues inside the
region $G$ in (4.6) are localised and the only non-decreasing eigenvector
(gray eigenvector) corresponds to the eigenvalue $0$ in the boundary of the
region $G$. On the other hand, in [5] it is observed and conjectured that the
non-trivial winding of the eigenvalues $\lambda_{j}(z)$ predicts the
exponential decay of the eigenmodes. This is due to
$\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\leq 0,j=1,2$ in the
example, which yields

$G=\left\\{\lambda\in\mathbb{C}\setminus\sigma_{\mathrm{det}}(f):\bigcup_{j=1}^{2}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\neq
0\right\\}.$

(a) The region of non-trivial winding of the
eigenvalues and the pseudo-spectrum.

(b) Eigenmodes superimposed on one another to portray the skin effect.

Figure 4.2. The region of $\lambda$ so that
$\sum_{j=1}^{k}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\neq 0$ and
the localization of the eigenvectors. Computation performed for
$s_{1}=1,s_{2}=2,$ and $N=50$.

(a) Computation performed for $s_{1}=1,s_{2}=2,s_{3}=3,$ and $N=50$.

(b) Simulation performed with $s_{1}=1,s_{2}=2,s_{3}=3,$ and $N=50$.

(c) Computation performed for $s_{1}=2,s_{2}=3,s_{3}=4,$ and $N=50$.

(d) Simulation performed with $s_{1}=2,s_{2}=3,s_{3}=4,$ and $N=50$.

Figure 4.3. Figures A and C show the spectrum of the operator. The green
regions consist of all the eigenvalues $\lambda$ that satisfy
$\sum_{j=1}^{3}\operatorname{wind}(\lambda_{j}(\mathbb{T}),\lambda)\neq 0$.
The black dots along the real line denote the spectrum of the gauge
capacitance matrix $C^{\gamma}$ and the solid blue and orange lines around the
spectrum are the $\varepsilon$-pseudospectra for $\varepsilon=10^{k}$ and
$k=-5,-2$. Figures B and D show the eigenvectors of $C^{\gamma}$.

Figure 4.3 illustrates the results for $3$ periodically repeated resonators.
We numerically verify that indeed all the eigenvectors, except the one
associated with eigenvalue $0$ on the boundary of $G$, are localised at the
left edge of the structure.

######  Remark 4.3.

We remark that the green regions for $k=1,2,3,4$ can be computed analytically
through the formula (2.14). For the case when $k\geq 5$, one can utilize
(2.14) to compute numerically the green region.

##  5\. Concluding remarks

We have developed new theories of spectra and pseudo-spectra for tridiagonal
$k$-Toeplitz operators and matrices. Specifically, we have established the
relationship between the winding number of the symbol function’s eigenvalues
and the exponential decay property of the corresponding eigenvectors (or
pseudo-eigenvectors). This discovery sheds light on the topological origin of
the non-Hermitian skin effect observed in one-dimensional polymer systems of
subwavelength resonators. This paper also opens the door to the study of the
spectral theory of block Toeplitz operators.

## Acknowledgements

This work was partially supported by Swiss National Science Foundation grant
number 200021-200307.

## Code Availability

The data that support the findings of this work are openly available at
https://doi.org/10.5281/zenodo.10438679.

## Conflict of interest

The authors have no competing interests to declare that are relevant to the
content of this article.

##  Appendix A Computation of the determinant

######  Lemma A.1.

For $\lambda\in\operatorname{\mathbb{C}}\setminus\sigma_{ess}(T(f))$, the
determinant $\det(f(z)-\lambda)$, where $f$ is the symbol of the tridiagonal
Toeplitz operator $T$ given in (2.5), has the form

$\det(f(z)-\lambda)=(-1)^{k+1}(\prod_{i=1}^{k}c_{i})z+(-1)^{k+1}(\prod_{i=1}^{k}b_{i})z^{-1}+g(\lambda),$
(A.1)

where $g$ is a polynomial in $\lambda$ of degree $k$, defined by (A.2). In
particular, there are at most $2k$ $\lambda\in\mathbb{C}$ so that
$\det(f(z)-\lambda)=0$ admits a double root.

###### Proof.

The case when $k=1,2$ is easy to verify. For the case when $k\geq 3$, by
repeated Laplace expansion, one can show that

$\displaystyle\det(f(z)-\lambda)$
$\displaystyle=(-1)^{k+1}b_{1}b_{k}\begin{vmatrix}b_{2}&0&\dots&\dots&0\\\
a_{3}-\lambda&\ddots&\ddots&&\vdots\\\ c_{3}&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&0\\\ 0&\dots&c_{k-2}&a_{k-1}-\lambda&b_{k-1}\\\
\end{vmatrix}z^{-1}$
$\displaystyle\qquad\qquad+(-1)^{k+1}c_{k}c_{k-1}\begin{vmatrix}c_{1}&a_{2}-\lambda&b_{2}&0&\dots&0\\\
0&c_{2}&a_{3}-\lambda&b_{3}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\
\vdots&&\ddots&\ddots&\ddots&b_{k-3}\\\
\vdots&&&\ddots&c_{k-3}&a_{k-2}-\lambda\\\ 0&\dots&\dots&\dots&0&c_{k-2}\\\
\end{vmatrix}z$
$\displaystyle+(a_{k}-\lambda)\begin{vmatrix}a_{1}-\lambda&b_{1}&0&\dots&0\\\
c_{1}&a_{2}-\lambda&b_{2}&\ddots&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{k-1}\\\ 0&\dots&0&c_{k-2}&a_{k-1}-\lambda\\\
\end{vmatrix}$ $\displaystyle\qquad\qquad-
b_{k-1}c_{k-1}\begin{vmatrix}a_{1}-\lambda&b_{1}&0&\dots&0\\\
c_{1}&a_{2}-\lambda&b_{2}&\ddots&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{k-3}\\\ 0&\dots&0&c_{k-3}&a_{k-2}-\lambda\\\
\end{vmatrix}$ $\displaystyle\qquad\qquad\qquad\qquad-
b_{k}c_{k}\begin{vmatrix}a_{2}-\lambda&b_{2}&0&\dots&0\\\
c_{3}&a_{3}-\lambda&b_{3}&\ddots&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{k-2}\\\ 0&\dots&0&c_{k-1}&a_{k-1}-\lambda\\\
\end{vmatrix}.$

Hence (A.1) holds for

$g(\lambda)=\det(A_{0}-\lambda)-b_{k}c_{k}p(\lambda),$ (A.2)

where

$p(\lambda)=\begin{cases}0,&k=1,\\\ 1,&k=2,\\\
\begin{vmatrix}a_{2}-\lambda&b_{2}&0&\dots&0\\\
c_{3}&a_{3}-\lambda&b_{3}&\ddots&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{k-2}\\\ 0&\dots&0&c_{k-1}&a_{k-1}-\lambda\\\
\end{vmatrix},&k\geq 3.\end{cases}$

To demonstrate the last claim, we write

$\det(f(z)-\lambda)=(Az+Bz^{-1})+g(\lambda)$ (A.3)

with $A=(-1)^{k+1}(\prod_{i=1}^{k}c_{i}),B=(-1)^{k+1}(\prod_{i=1}^{k}b_{i})$.
By multiplication with $z$, we obtain that

$z\det(f(z)-\lambda)=Az^{2}+B+g(\lambda)z.$

Thus only when

$g(\lambda)^{2}-4AB=0,$

we have a double root. Since $g(\lambda)$ is a polynomial of order $k$, we
have at most $2k$ solutions $\lambda$ of the above equation so that the root
is a double root. ∎

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