On a comparison principle and the uniqueness of spectral flow

The spectral flow is a well-known quantity in theory that measures the variation of spectra about 0 along paths selfadjoint Fredholm operators. aim this work twofold. Firstly, we consider homotopy invariance properties and establish simple formula which comprises its classical yields comparison theorem for under compact perturbations. We apply our result to existence non-trivial solutions boundary value problems Hamiltonian systems. Secondly, was axiomatically characterised by Lesch, Ciriza, Fitzpatrick Pejsachowicz assumption endpoints operators are invertible. propose different approach uniqueness lifts additional assumption. As application latter result, discuss relation between Maslov index symplectic Hilbert spaces.