Let \(\Omega \subset {\mathbb{R}}^ N\) be a bounded domain, and let \(L:=\partial /\partial t+A(x,t,D)\) be a uniformly parabolic differential expression which is T-periodic in t. The authors study the eigenvalue problem (*) \(Lu=\lambda mu\) in a space of Hölder continuous T-periodic functions on \({\bar \Omega}\times {\mathbb{R}}\), where \(m\neq 0\) is a (not necessarily positive) weight function.
Main result: (*) has a positive eigenvalue \(\lambda_ 1(m)\) having a positive eigenfunction u if and only if \(\int^{T}_{0}(\max_{x}m(x,t))dt\) is positive. The result generalizes previous results for (a) \(m=1\), (b) elliptic eigenvalue problems with indefinite weight functions.