Invertibility of Toeplitz operator in $\ell_1$ Suppose we have a Toeplitz operator
$$
T(a) = \begin{bmatrix} a_{0} &amp; a_{-1} &amp; a_{-2} &amp; \ldots &amp; \ldots &amp;a_{-n+1} &amp;\dots \\\\ 
a_{1} &amp; a_0 &amp; a_{-1} &amp; \ddots &amp; &amp; \vdots \\\\\ a_{2} &amp; a_{1} &amp; \ddots &amp; \ddots &amp; \ddots&amp; \vdots \\\\
\vdots &amp; \ddots &amp; \ddots &amp; \ddots &amp; a_{-1} &amp; a_{-2}\\\\
\vdots &amp; &amp; \ddots &amp; a_{1} &amp; a_{0}&amp; a_{-1} \\\\ a_{n-1} &amp; \ldots &amp; \ldots &amp; a_{2} &amp; a_{1} &amp; a_{0} \\\\
\vdots
\end{bmatrix}
$$
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)
the theory is pretty much complete in all the lp spaces. relevant sources include:

Analysis of Toeplitz Operators By Albrecht Böttcher, Bernd Silbermann (Ch 6.)

an algebraic formula for the inverse: gohberg-semencul

books by gohberg-feldman, gohberg-kaashoek, gohberg-goldberg.

A general framework is the theory of Fredholm operators on Banach
spaces. For the toeplitz operator, the winding number is just 
the "fredholm index". A fredholm operator, when invertible, has zero index.
but the converse statement does not hold.

if the function is lp and only piecewise continuous on T, one of the sources
(i think, gohberg-feldman) provides a way to fill the gaps of a(T) by circles.
now, coburn's condition generalizes as follows: this modified curve should 
avoid the origin.
the theory is pretty much complete in all the lp spaces. relevant sources include:

Analysis of Toeplitz Operators By Albrecht Böttcher, Bernd Silbermann (Ch 6.)

an algebraic formula for the inverse: gohberg-semencul

books by gohberg-feldman, gohberg-kaashoek, gohberg-goldberg.

A general framework is the theory of Fredholm operators on Banach
spaces. For the toeplitz operator, the winding number is just 
the "fredholm index". A fredholm operator, when invertible, has zero index.
but the converse statement does not hold.

if the function is lp and only piecewise continuous on T, one of the sources
(i think, gohberg-feldman) provides a way to fill the gaps of a(T) by circles.
now, coburn's condition generalizes as follows: this modified curve should 
avoid the origin.