Three-space theorem for semi-Fredholmness (Q1936514)

The authors prove that, for a given exact sequence \[ 0\longrightarrow X \mathop{\longrightarrow}^{J} Y \mathop{\longrightarrow}^{Q} Z\longrightarrow 0 \] of linear maps acting between Banach spaces and linear bounded maps \(A:X\to X\), \(B:Y\to Y\), \(C:Z\to Z\) commuting with \(J\) and \(Q\), respectively, \(B\) is upper (lower) semi-Fredholm provided that so are \(A\) and \(C\).