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

TL;DR

This paper explores the spectral characteristics of off-diagonal block linear relations in Hilbert spaces, linking their spectra to those of related product operators and extending analysis via Moore--Penrose inverses.

Contribution

It provides a detailed spectral characterization of off-diagonal block linear relations and extends the analysis using Moore--Penrose inverses in Hilbert spaces.

Findings

Characterization of essential spectra and resolvent sets of block linear relations.

Establishment of spectral relationships between block relations and product operators.

Extension of spectral analysis using Moore--Penrose inverses for closed linear relations.

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.