  Given a symmetric Dirichlet form $(\mathcal{E},\mathcal{F})$ on a
(non-trivial) $\sigma$-finite measure space $(E,\mathcal{B},m)$ with associated
Markovian semigroup $\{T_{t}\}_{t\in(0,\infty)}$, we prove that
$(\mathcal{E},\mathcal{F})$ is both irreducible and recurrent if and only if
there is no non-constant $\mathcal{B}$-measurable function $u:E\to[0,\infty]$
that is \emph{$\mathcal{E}$-excessive}, i.e., such that $T_{t}u\leq u$
$m$-a.e.\ for any $t\in(0,\infty)$. We also prove that these conditions are
equivalent to the equality $\{u\in\mathcal{F}_{e}\mid
\mathcal{E}(u,u)=0\}=\mathbb{R}\mathbf{1}$, where $\mathcal{F}_{e}$ denotes the
extended Dirichlet space associated with $(\mathcal{E},\mathcal{F})$. The proof
is based on simple analytic arguments and requires no additional assumption on
the state space or on the form. In the course of the proof we also present a
characterization of the $\mathcal{E}$-excessiveness in terms of
$\mathcal{F}_{e}$ and $\mathcal{E}$, which is valid for any symmetric
positivity preserving form.
