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

2025/02/11 by Arup Majumdar, Majumdar, Arup
Computer Science · Mathematics · #Matrix Theory and Algorithms #Advanced Topics in Algebra #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2502.07361

Abstract

In this paper, we characterize the essential spectra and the resolvent set of the off-diagonal block linear relation \beginbmatrix 0 amp; A
B amp; 0 \endbmatrix 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 A = \beginbmatrix 0 amp; T^†
T amp; 0 \endbmatrix, 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.

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