The angle of an operator and range-kernel complementarity
Dimosthenis Drivaliaris, Nikos Yannakakis · Journal of Operator Theory · 2016
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 π, then its range and kernel are complementary.In finite dimensions and up to scalar multiples this simple geometric property characterizes operators with complementary range and kernel.Applying our result we get simple proofs of two known facts concerning eigenvalues lying in the boundary of the numerical range.For an operator on a Hilbert space we present a sufficient condition for range-kernel complementarity, involving the distance of the boundary of the numerical range from the origin.Finally, we discuss some properties of operators whose spectrum does not intersect all rays emanating from the origin and show that in a Banach space which is uniformly convex and has uniformly convex dual such operators are surjective if and only if they are injective.