  The interaction of a discrete state coupled to a continuum is a longstanding
problem of major interest in different areas of quantum and classical physics.
In Hermitian models, several dynamical decoupling schemes have been suggested,
in which the discrete-continuum interaction can be substantially reduced and
even suppressed. In this work we consider a discrete state interacting with a
continuum via a time-dependent {\em non-Hermitian} coupling with finite (albeit
arbitrarily long) duration, and show rather generally that for a wide class of
coupling temporal shapes, in which the real and imaginary parts of the coupling
are related each other by a Hilbert transform, the discrete state returns to
its initial condition after the interaction with the continuum, while the
continuum keeps trace of the interaction. Such a behavior, which does not have
any counterpart in Hermitian dynamics, can be referred to as non-Hermitian {\em
pseudo} decoupling. Non-Hermitian pseudo decoupling is illustrated by
considering a non-Hermitian extension of the Fano-Anderson model in a
one-dimensional tight-binding lattice. Scuh a non-Hermitian model can describe,
for example, photonic hopping dynamics in a tight-binding chain of optical
microrings or resonators, in which non-Hermitian coupling can be realized by
fast modulation of the real and imaginary (gain/loss) parts of the refractive
index of the edge microring.
