Let \(\{e_{n}\}_{n\in\mathbb{Z}}\) be the standard basis of the Hilbert space \(\ell^{2}(\mathbb{Z})\) and let \(d:\ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})\) be the difference operator defined by \(de_{n}:=e_{n}-e_{n+1}\), \(n\in\mathbb{Z} .\) The free discrete Dirac operator \(D_0\) is a self-adjoint bounded operator in the Hilbert space \(\ell^2(\mathbb{Z})\oplus\ell^2(\mathbb{Z})\) given by the block operator matrix \[ D_0 := \begin{pmatrix}m & d \\ d^* & -m\end{pmatrix},\] where \(m\) is a non-negative constant. The authors investigate the operator \(D_0\) perturbed by a complex potential \(V\in \ell^p(\mathbb{Z},\mathbb{C}^{2\times 2})\) for \(1\leq p\leq\infty\). If \(p=1\), it is proved that the point spectrum satisfies \[\sigma_{\mathrm{p}}(D_V)\subset \left\{\lambda\in\mathbb{C} : |\lambda^2-m^2||\lambda^2-m^2-4| \leq \bigl(|\lambda+m|+|\lambda-m|\bigr)^2\|V\|^2_1 \right\}.\] As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small \(\ell^1\)-potential.
Several results provide a spectral estimates for \(V\in \ell^p\), \(p > 1\). Further possible improvements and sharpness of the obtained spectral bounds are also discussed.