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Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & & \vdots \\\\\ a_{2} & a_{1} & \ddots & \ddots & \ddots& \vdots \\\\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\\\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \\\\ \vdots \end{bmatrix} $$ where $a=(a_k)$, $k\in \mathbb{Z}$.

In the book by Hagen, Roch and Silbermann you can find the conditions when this operator is invertible in $\ell_2$, theorem 1.31, in page 48.

My question is whether there similar conditions for existence of invertibility in space $\ell_1$?

The condition in the book is the following. Denote $\mathbb{T}$ the unit circle in $\mathbb{C}$. Let $a\in L^{\infty}(\mathbb{T})$ and let

$$a_k=\frac{1}{2\pi}\int_0^{2\pi}a(e^{i\theta})e^{-ik\theta}d\theta$$

Theorem. 1.31 (Coburn's theorem) Let $a\in C(\mathbb{T})$. Operator $T(a)$ is invertible in $\ell_2$ if and only if $0\notin a(\mathbb{T})$ and if $\mathrm{wind}\text{ }a(\mathbb{T})=0$.

Here $\mathrm{wind}\text{ }a(\mathbb{T})$ is a winding number of the curve $a(\mathbb{T})$, provided with the orientation inherited by the usual counter-clockwise orientation of the unit circle, around the origin. (The definition taken from the book)

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