  Assuming that neutrinos are spacelike (tachyonic) fermions, we calculate
width for the kinematically allowed, lepton number conserving, three-body decay
$\nu_{\alpha}\rightarrow \nu_{\alpha} \; \nu_{\beta} \bar{\nu}_{\beta}$ in the
Standard Model. Decays of tachyonic neutrinos over cosmological distances can
lead to a reduction of the neutrino flux in the high-energy end of the
spectrum. We estimate upper limits on the spacelike neutrino mass based on the
PeV-energy cosmological neutrino events observed in the IceCube experiment.
These limits are close to those deduced from the measurements of $m_{\nu}^2$ in
the tritium-decay experiment KATRIN.
