*(English)*Zbl 0756.46038

Let $E$ be a complex Banach space and ${E}_{0}$ a closed subspace with involution. Let $(x,a,y)\mapsto \left\{x{a}^{*}y\right\}$ be a continuous real trilinear map $E\times {E}_{0}\times E\to E$, which is symmetric complex bilinear in $x$, $y$ and conjugate linear in $a$. Certain algebraic postulates for $\left\{x{a}^{*}y\right\}$ are assumed, including $a\square {a}^{*}\in \text{Her}\left(E\right)$ ($\forall a\in {E}_{0}$), where $a\square {a}^{*}$ is the operator $x\mapsto \left\{a{a}^{*}x\right\}$ and $\text{Her}\left(E\right)$ is the set of all operators on $E$ which are Hermitian in the sense of Vidav. Such systems are called here Partial ${J}^{*}$-triples.

The main result is that when the system is geometric (all vector fields $a-\left\{x{a}^{*}x\right\}\partial /\partial x$ ($a\in {E}_{0}$) are complete in some bounded balanced domain in $E$), then every Hermitian operator $a\square {a}^{*}$ ($a\in {E}_{0}$) has a non-negative spectrum.