Consider a commutative diagram of bounded linear operators between Banach spaces

    \[ \begin{CD} \qquad\qquad\qquad\qquad 0 @>>> X @>{J}>> Y @>{Q}>> Z @>>> 0\\ @. @V{A}VV @V{B}VV @V{C}VV @. \qquad\qquad\qquad \qquad \\ \qquad\qquad\qquad\qquad 0 @>>> X @>>{J}> Y @>>{Q}> Z @>>> 0 \end{CD} \]

with exact rows.
In what ways are the spectral and local spectral properties of B related to those of the pairs of operators A and C?
In this paper, we give our answers to this general question using tools from local spectral theory.

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Zeng, Qingping