Spectral Properties of Off Diagonal Block Linear Relations via Moore Penrose Inverses in Hilbert Spaces

In this paper, we characterize the essential spectra and the resolvent set of the off-diagonal block linear relation $$\left[\begin{array}{cc}0& A\\ B& 0\end{array}\right]$$ in terms of the essential spectra and resolvent sets of the products $AB$ and $BA$. Our approach establishes precise spectral relationships that connect the structural properties of the block linear relation with those of the associated compositions. Furthermore, we investigate the Moore–Penrose inverses of closed linear relations in Hilbert spaces and employ these results to extend the spectral analysis to the off-diagonal block linear relation $$\mathcal{A}=\left[\begin{array}{cc}0& T^{†}\\ T& 0\end{array}\right],$$ where $T$ is a closed, continuous linear relation with closed range from a Hilbert space $H$ to a Hilbert space $K$, and $T^{†}$ denotes its Moore–Penrose inverse.