Short five lemmas in Banach spaces

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.