New product formulæ and quantum Zeno dynamics with generalized observables

We demonstrate a pair of new product formulæ which combine a projection with a resolvent of a positive operator, or with an exponential function and spectral projections, respectively. The convergence is strong or even operator-norm under more restrictive assumptions. The second mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observable in the sense of Davies.