remains open. We demonstrate, however, the strong convergence in ${{\mathcal H}}$ for the particular case when the projections involved are finite-dimensional – cf. Theorem \[fin\]. On the other hand, from the physical point of view the result given in Theorem \[main\] is quite plausible taking into account that any real measurement is burdened with errors – see Remark \[phys-conv\] below.

We will formulate the theorems together with their corollaries in the next section. Theorem \[main\] will be then proven in Sections \[s-proof\] and \[s-proof-of-L1\], Theorem \[fin\] in Section \[finite\]. As an example we discuss in the concluding section reduction of a free dynamics to a domain in ${\mathbb{R}}^d$ by permanent observation. We will establish that the Zeno generator mentioned above is in this case the Dirichlet Laplacian, obtaining thus in a different way the result of the paper [@FP].

The main result {#s-main}
===============

Throughout the paper $H$ will be a nonnegative self-adjoint operator in a [*separable*]{} Hilbert space ${{\mathcal H}}$, and $P$ will be an orthogonal projection. The nonnegativity assumption is made for convenience; our main result extends easily to any self-adjoint operator $H$ bounded from below as well as one bounded from above, i.e. to each semi-bounded self-adjoint 