The Angle Along a Curve and Range-Kernel Complementarity

Abstract

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 $\infty$.