The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix

Consider $\mathcal{A}$ to be a finite set of complex numbers and let $a=(a_{i})_{i\in{\mathbb{Z}}}$ be a biinfinite sequence in the total shift space $\mathcal{A}^{\mathbb{Z}}$. In [13], the tridiagonal operator $A_{a}\colon\ell^{2}({\mathbb{Z}})\to\ell^{2}({\mathbb{Z}})$ associated to $a$ is defined as

where the square marks the matrix entry at $(0,0)$. In the particular case of the alphabet $\mathcal{A}=\{-1,1\}$, the corresponding operator $A_{a}$ is related to the so called “hopping sign model” introduced in [7] and subsequently studied in many other works, such as [1, 2, 3, 4, 5, 6, 9, 10, 13], just to name a few. On the other hand, when the alphabet is $\mathcal{A}=\{0,1\}$ some results for computing the numerical range of $A_{a}$ are presented in [13, 14]. In particular, work in [14] addresses the case when $a$ is an $n+1$-periodic sequence. Relying on the fact that the closure of the numerical range of $A_{a}$ may be written as the closure of the convex hull of an uncountable union of numerical ranges of certain matrices, in [14] the closure of the numerical range of the 2-periodic case is computed by substituting such uncountable union of numerical ranges by the convex hull of the union of the numerical ranges of just two $2\times 2$ matrices. In this work, we further contribute to the study of the numerical range of $A_{a}$ when $a$ is an $n+1$ periodic biinfinite sequence.

Instead of working with the operators $A_{a}$, we work with the more general tridiagonal operators $T=T(a,b,c)$ defined in Section 2, since, as can be seen in [14], the computation of the closure of the numerical range of $A_{a}$ is a particular case of that of $T$. Using a result of Plaumann and Vinzant [20], we show that the closure of the numerical range of the $n+1$ periodic tridiagonal operator $T$ is the numerical range of a $2(n+1)\times 2(n+1)$ matrix (cf. Theorem 2.6).

We divide this work in two sections. In Section 1 we briefly introduce the notation and terminologies needed in the rest of the paper. In Section 2 we develop the required machinery, first by computing the Kippenhahn polynomial of the symbol of $n+1$ periodc tridiagonal operatos $T$ on $\ell^{2}({\mathbb{N}}_{0})$ and then by combining our computations with results of Plaumann and Vizant. We will conclude that the closure of the numerical range of $T$ is equal to the numerical range of a $2(n+1)\times 2(n+1)$ matrix $A$. Furthermore, we provide some examples where $A$ can be explicitly computed and we show that the size of $A$ is optimal.

In this section we introduce the notation required which will be needed in the following sections. As usual, the symbols ${\mathbb{N}}$, ${\mathbb{N}}_{0}$, ${\mathbb{Z}}$, ${\mathbb{R}}$ and ${\mathbb{C}}$ will denote the set of positive integers, the sets of nonnegative integers, the set of integers, the set of real numbers and the set of complex numbers, respectively.

For a given $n\in{\mathbb{N}}$, let $a$, $b$ and $c$ be $(n+1)$-periodic infinite sequences in $\mathcal{A}^{{\mathbb{N}}_{0}}$. We will denote by $T=T(a,b,c)$ the $(n+1)$-periodic tridiagonal operator on $\ell^{2}({\mathbb{N}}_{0})$ given by

We should observe that $T$ is a bounded operator since the sum of the moduli of the entries in each column (and in each row) is uniformly bounded (see, e.g., [16, Example 2.3]). The biinfinite matrix $A_{a}$ is also a bounded operator, as long as the biinfinite sequence $a$ arises from a finite alphabet.

If $n>1$, for each $\phi\in[0,2\pi)$, following [1, 14] we define the symbol of $T$, as the following $(n+1)\times(n+1)$ matrix

while the symbol of $T$ for $n=1$ is the $2\times 2$ matrix

Recall that given a Hilbert space $\mathcal{H}$ and a bounded operator $A$ on it, the numerical range is defined as the set

The Toeplitz-Haussdorf Theorem establishes that $W(A)$ is a bounded convex subset of ${\mathbb{C}}$ (closed, if the Hilbert space is finite dimensional) and hence the closure of the numerical range can be seen as the intersection of the closed half-spaces containing the numerical range.

Kippenhahn [17] (see also [18]) characterized two vertical support lines of $W(A)$ for a given $n\times n$ matrix as $\mathrm{Re}(z)=\lambda_{1}(A)$ and $\mathrm{Re}(z)=\lambda_{n}(A)$, where $\lambda_{1}(A)$ and $\lambda_{n}(A)$ are the respective largest and least eigenvalues of $\mathrm{Re}(A)$ (recall that $\mathrm{Re}(A):=\frac{1}{2}(A+A^{*})$ and $\mathrm{Im}(A):=\frac{1}{2i}(A-A^{*})$). In fact, if $\alpha\in W(A)$ then $\lambda_{n}(A)\leq\mathrm{Re}(\alpha)\leq\lambda_{1}(A)$ (and the equalities hold for some points $\alpha_{1},\alpha_{2}\in W(A)$). Since $e^{i\theta}W(A)=W(e^{i\theta}A)$ for each $\theta\in[0,2\pi)$, it follows that if $\alpha\in W(A)$, then $e^{-i\theta}\alpha\in W(e^{-i\theta}A)$ and hence $\mathrm{Re}(e^{-i\theta}\alpha)\leq\lambda_{1}(e^{-i\theta}A)$. It follows that the lines $\mathrm{Re}(e^{-i\theta}z)=\lambda_{1}(e^{-i\theta}A)$ are support lines of $W(A)$. Hence the convex set $W(A)$ is uniquely determined by the numbers $\lambda_{1}(e^{-i\theta}A)$, as $\theta$ varies on the interval $[0,2\pi)$; i.e. $W(A)$ is determined by the largest eigenvalue of $\mathrm{Re}(e^{-i\theta}A)$, which equals $\cos(\theta)\mathrm{Re}(A)+\sin(\theta)\mathrm{Im}(A)$. Thus the numerical range is determined by the largest roots of the family of characteristic polynomials

The homogeneous polynomial $F_{A}(t,x,y)=\det(tI_{n}+x\mathrm{Re}(A)+y\mathrm{Im}(A))$ is called the Kippenhahn polynomial of the matrix $A$. It clearly follows that two matrices have the same numerical range if their Kippenhahn polynomials coincide. Furthermore,

for each $\theta\in[0,2\pi)$.

In this section, after some preliminary work, we show that the closure of the numerical range of a $n+1$-periodic tridiagonal operator $T$ is the numerical range of a $2(n+1)\times 2(n+1)$ matrix.

We will need the following lemma.

Let us set the following notation for the rest of this paper. For $0\leq j<n$ we define

and

We now find an expression for the Kippenhahn polynomial $F_{T_{\phi}}$ of the symbol matrix $T_{\phi}$ of an arbitrary $n+1$-periodic tridiagonal matrix $T$ acting on $\ell^{2}({\mathbb{N}}_{0})$, involving the determinants of some tridiagonal matrices. This expression will be useful in what follows.

For every $n\in{\mathbb{N}}$ and for a fixed point $(x,y)\in{\mathbb{R}}^{2}$, the angle $\phi\in[0,2\pi)$ is involved only in the constant term (with respect to the variable $t$) of the polynomial $F_{T_{\phi}}(t,x,y)$. Furthermore, for every $(x,y)\in{\mathbb{R}}^{2}$ and for every $\phi\in[0,2\pi)$, the polynomial $F_{T_{\phi}}(t,x,y)$, seen as a polynomial in $t$, has $n+1$ real roots, counting multiplicities, as it is the characteristic polynomial of the Hermitian matrix $-x\mathrm{Re}(T_{\phi})-y\mathrm{Im}(T_{\phi})$. The following lemma will be useful later when applied to the polynomial $F_{T_{\phi}}$.

In Theorem 2.7, we will show that the closure of the numerical range of $T$ is the numerical range of a single matrix. One of the key steps in the proof of said theorem will be to use the following proposition, which computes the closure of the numerical range of $T$ by using a single homogeneous polynomial, instead of the uncountable number of Kippenhahn polynomials of the symbols $T_{\phi}$, which Theorem 2.8 in [14] would suggest: this is achieved by getting rid of the parameter $\phi$ in the expression of the Kippenhahn polynomial of the symbol $T_{\phi}$ in Proposition 2.2.