The angle of an operator and range and kernel complementarity

Abstract

We show that if the angle of a bounded linear operator on a Banach space, with closed range and closed sum of its range and kernel, is less than $\pi$, then its range and kernel are complementary. In finite dimensions and up to rotations this simple geometric property characterizes operators for which the above complementarity holds. Applying our result we get simple proofs concerning eigenvalues lying in the boundary of the numerical range. Concluding, for an operator on a Hilbert space, we present a sufficient condition, involving the distance of the boundary of the numerical range from the origin, for its range and kernel to be complementary.