Complete and incomplete systems of exponentials in spaces with a power weight on a half-line

We essentially widen the class of sequences $\lambda_n$ for which the completeness (non-completeness) of system of exponentials $e^{-\lambda_nt},~{\rm Re}\lambda_n>0$ is proved in the spaces $L^p(\mathbb{R}_+,t^\alpha dt),~\alpha>-1$. The proof uses the invariance of completeness relative to the change of the weight $t^\alpha$ by the weight $(1+t)^\alpha$; this fact is also proved here.