  Spectral sections of families of self-adjoint Fredholm operators were
introduced by Melrose and Piazza for the needs of index theory. The basic
result about spectral sections is a theorem of Melrose and Piazza to the effect
that a family admits a spectral section if and only if its analytic index
vanishes. The present paper is devoted to two proofs of this theorem. These
proofs allow to generalize this theorem and to clarify some subtle aspects
related to the definition of the analytic index and trivializations of Hilbert
bundles. It is based on ideas of author's paper arXiv:2111.15081, but is
largely independent from it.
