We define the angle of a bounded linear operator A along a curve emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if \(\sigma (A)\) does not separate 0 from \(\infty \), then \(X=R(A)\oplus N(A)\) if and only if R(A) is closed and some angle of A is less than \(\pi \). We first apply this result to invertible operators that have a spectral set that does not separate 0 from \(\infty \). Next we extend the notion of angle along a curve to Banach algebras and use it to prove two characterizations of elements in a semisimple and in a \(C^*\) commutative algebra respectively, whose spectrum does not separate 0 from \(\infty \).

我们沿从原点发出的曲线定义有界线性算子A的角度，并用它来表征范围核互补性。特别地，我们证明如果\(\sigma (A)\)没有将 0 与\(\infty \)分开，那么\(X=R(A)\oplus N(A)\)当且仅当R ( A ) 是闭合的并且A 的某个角度小于\(\pi \)。我们首先将此结果应用于具有不将 0 与\(\infty \)分开的谱集的可逆算子。接下来，我们将角度的概念沿曲线扩展到 Banach 代数，并用它来证明半简单和\(C^*\)分别为交换代数，其谱不将 0 与\(\infty \)分开。