  We give necessary and sufficient conditions for a regular semi-Dirichlet form
to enjoy a new Feller type property, which we call \emph{weak Feller property}.
Our characterization involves potential theoretic as well as probabilistic
aspects and seems to be new even in the symmetric case. As a consequence, in
the symmetric case, we obtain a new variant of a decomposition principle of the
essential spectrum for (the self-adjoint operators induced by) regular
symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to
Cheeger forms on $\mathsf{RCD^*}$ spaces.
