On polynomial connections between projections

Abstract

We apply a result of Tremon to show that any two Banach-space projections of the same finite rank can be connected by a projection-valued polynomial path of degree not exceeding 3. Then we construct two similar infinite projections P and Q on a Hilbert space such that 1 is an eigenvalue of P′+Q′ for all projections P′ and Q′ with ‖ P− P′‖<1 and ‖ Q− Q′‖<1; this disproves a conjecture studied in [1].