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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A Toeplitz matrix has lower-shift displacement rank at most two

Statement

Let $T = (t_{i - j})_{0 \leq i, j < n}$ be any square Toeplitz matrix, and let $Z = Z_{n}$ be the lower shift of Lower-shift displacement rank and generator pairs. With $e_{0} = (1, 0, \dots, 0)^{T}$ and $t_{0} = T_{00}$ , $T - Z T Z^{*} = e_{0} (e_{0}^{*} T - t_{0} e_{0}^{*}) + (T e_{0}) e_{0}^{*} .$ Consequently, its displacement rank is at most two. A width-two generator is $G = [e_{0} T e_{0}], H^{*} = [\begin{array}{c} e_{0}^{*} T - t_{0} e_{0}^{*} \\ e_{0}^{*} \end{array}] .$

Facts & Assumptions

Given: A square Toeplitz matrix $T = (t_{i - j})$ , its first diagonal entry $t_{0}$ , and the lower shift $Z$ .

[F1]

The lower-shift displacement is $\nabla_{Z} (T) = T - Z T Z^{*}$ , and a generator is a factorization $\nabla_{Z} (T) = G H^{*}$ . Lower-shift displacement rank and generator pairs

[F2]

Toeplitz entries satisfy $T_{i j} = t_{i - j}$ for zero-based indices. Toeplitz, circulant, and finite-index conventions for structured solvers

[F3]

Matrix multiplication is defined by the finite entry sums. Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes

Proof

1.1F1F2F3algebra

For $i, j \geq 1$ , $(Z T Z^{*})_{i j} = T_{i - 1, j - 1} = t_{(i - 1) - (j - 1)} = t_{i - j} = T_{i j}$ by [F2]. If $i = 0$ or $j = 0$ , the corresponding entry of $Z T Z^{*}$ is zero because the first row of $Z$ and the first column of $Z^{*}$ are zero. Thus $T - Z T Z^{*}$ agrees with $T$ on its first row and first column and is zero in the interior. The displayed expression has exactly these entries: its first row is $e_{0}^{*} T$ , its first column is $T e_{0}$ , and its $(0, 0)$ entry is $t_{0}$ .

2.1F1step 1.1algebra∎

The factorization in the statement multiplies to the displayed expression from step 1.1. It has two columns, so the rank of $T - Z T Z^{*}$ is at most two by the definition of rank and a product through $C^{2}$ .

Depends on

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Sources

Daniel Kressner and Robert Luce, Fast computation of the matrix exponential for a Toeplitz matrix, §2.1 (standard reference, not scraped)
Ralf Hiptmair, Numerical Methods for Computational Science and Engineering, Lecture Document, §4.5.1 (standard reference, not scraped)