Form Inequalities for Symmetric Contraction Semigroups

Abstract

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.