  Consider --- for the generator \({-}A\) of a symmetric contraction semigroup
over some measure space $\mathrm{X}$, $1\le p < \infty$, $q$ the dual exponent
and given measurable functions $F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ ---
the statement: $$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A
F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ {\em for all
$\mathbb{C}^d$-valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such
that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in
\mathrm{L}^q(\mathrm{X})$ for all $j$.}
  It is shown that this statement is valid in general if it is valid for
$\mathrm{X}$ being a two-point Bernoulli $(\frac{1}{2}, \frac{1}{2})$-space and
$A$ being of a special form. As a consequence we obtain a new proof for the
optimal angle of $\mathrm{L}^{p}$-analyticity for such semigroups, which is
essentially the same as in the well-known sub-Markovian case.
  The proof of the main theorem is a combination of well-known reduction
techniques and some representation results about operators on
$\mathrm{C}(K)$-spaces. One focus of the paper lies on presenting these
auxiliary techniques and results in great detail.

Functional Analysis;Probability;Spectral Theory