We have the Toeplitz operator $T:l^{\infty}(Z, R^2) \to l^{\infty}(Z, R^2)$. We computed spectrum of $T$ on $l^2$ using its symbol (symbol is continuous function $\varphi(z)$ and eigenvalues of $\varphi(z)$ are real for every $z$, $|z|=1$). What can be said about spectrum of $T$ on $l^{\infty}$? We suspect that it coincides with $l^2$ spectrum.

In our case elements of matrix $T$ are $2 \times 2$ real matrices and $\varphi(z)$ is complex matrix $2 \times 2$.

We have the Toeplitz operator $T:l^{\infty}(Z, R^2) \to l^{\infty}(Z, R^2)$. We computed spectrum of $T$ on $l^2$ using its symbol (symbol is continuous function $\varphi(z)$ and eigenvalues of $\varphi(z)$ are real for every $z$, $|z|=1$). What can be said about spectrum of $T$ on $l^{\infty}$? We suspect that it coincides with $l^2$ spectrum.

In our case elements of matrix $T$ are $2 \times 2$ real matrices and $\varphi(z)$ is complex matrix $2 \times 2$.