Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let V \mathtt {V} be a standard subspace in the complex Hilbert space alttext="script H"> class="MJX-tex-caligraphic" mathvariant="script">H encoding="application/x-tex">\mathcal {H} and alttext="upper G"> G encoding="application/x-tex">G finite dimensional Lie group of unitary antiunitary operators on containing modular alttext="left-parenthesis normal Delta Subscript monospace V Superscript i t Baseline right-parenthesis element-of double-struck R"> ( mathvariant="normal">Δ i t stretchy="false">) ∈ mathvariant="double-struck">R encoding="application/x-tex">(\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} corresponding conjugation J J encoding="application/x-tex">J_{\mathtt {V}} . We study semigroup S = fence="false" stretchy="false">{ g ∩<!-- ∩ mathvariant="normal">U ⁡<!-- ⁡ <mml:mo>:<!-- : <mml:mo>⊆<!-- ⊆ stretchy="false">} encoding="application/x-tex">S_{\mathtt {V}} = \{ g\in \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] determine its wedge alttext="bold L x German exp R plus class="MJX-TeXAtom-OP MJX-fixedlimits"> L x mathvariant="fraktur">g exp + encoding="application/x-tex">\operatorname {\textbf {L}}(S_{\mathtt {V}}) \mathfrak {g} \colon \exp (\mathbb {R}_+ x) S_{\mathtt {V}}\} , i.e., generators one-parameter subsemigroups algebra alttext="German g"> encoding="application/x-tex">\mathfrak {g} The is analyzed terms representations their analytic extension to semigroups form C right-parenthesis"> C encoding="application/x-tex">G (iC) where encoding="application/x-tex">C an A d Ad {Ad}(G) -invariant closed convex cone. Our main results assert that {V}}) spans alttext="3"> 3 encoding="application/x-tex">3 -graded subalgebra which it can described explicitly involution alttext="tau"> τ<!-- τ encoding="application/x-tex">\tau induced by generator alttext="h tau"> h encoding="application/x-tex">h {g}^\tau group, positive cone representation. also derive some global information itself.