On polynomial connections between projections (Q750849)

A result of \textit{M. Tremon} [Linear Algebra Appl. 64, 115-132 (1985; Zbl 0617.46054)] is used 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. It is not known whether this result can be extended to Banach algebras. An affirmative answer would follow from a spectral conjecture arising in \textit{B. Aupetit}, \textit{T. J. Laffey} and \textit{J. Zemanek} [Linear Algebra Appl. 41, 131-135 (1981; Zbl 0471.46030)]. But the authors construct two similar projections P and Q, acting on a separable Hilbert space, such that 1 is an eigenvalue of \(A+B\) for all operators A and B with \(\| P-A\| <1\) and \(\| Q- A\| <1\), which disproves this conjecture.