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arXiv · 2502.07361

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

Arup Majumdar
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Abstract

In this paper, we characterize the essential spectra and the resolvent set of the off-diagonal block linear relation \[ \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \] 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 \[ \mathcal{A} = \begin{bmatrix} 0 & T^{\dagger} \\ T & 0 \end{bmatrix}, \] where $T$ is a closed, continuous linear relation with closed range from a Hilbert space $H$ to a Hilbert space $K$, and $T^{\dagger}$ denotes its Moore--Penrose inverse.

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Arup Majumdar. 2025-02-11. Spectral Properties of Off Diagonal Block Linear Relations via Moore Penrose Inverses in Hilbert Spaces. https://arxiv.org/abs/2502.07361

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