Given a bounded positive linear operator A on a Hilbert space \(\mathcal {X}\) , this operator induces a semi-Hilbertian structure on \(\mathcal {X}\) . For a bounded linear operator T defined on the corresponding semi-Hilbertian space, we introduce a new definition of a Fredholm operator that is compatible with the semi-Hilbertian structure. Based on this definition, we proceed to define the essential spectra of T and establish their fundamental properties. Within this framework, we consider the bounded off-diagonal operator matrix \(\mathbb {T} = \begin{pmatrix} 0 & M\\ N & 0 \end{pmatrix}\) acting on the semi-Hilbertian space and demonstrate that the essential spectra of \(\mathbb {T}\) are entirely determined by the essential spectra of the products MN and NM. Finally, an illustrative example is provided to substantiate the theoretical conclusions.
