We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a \(\mathcal {C}\)-selfadjoint operator generates a contraction \(C_0\)-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of complex symmetric, unbounded semigroups and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all \(\mathcal {C}\)-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a \(\mathcal {C}\)-selfadjoint, unbounded semigroup is not necessarily \(\mathcal {C}\)-selfadjoint.

中文翻译：

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复杂的对称演化方程

我们研究了某些动力学系统，这些系统通过复杂对称Hilbert空间算子的半群或演化族而使不定式成为二次形式。在有界算子的设置中，我们证明\（\ mathcal {C} \）- selfadjoint算子在且仅当有耗散时才产生收缩\（C_0 \）- semigroup。此外，我们研究了具有复杂对称性的非自治线性微分方程的抽象柯西问题。在无界算子框架中，我们隔离了对称对称，无界半群的类，并研究了适用于它们的Stone型定理。在Fock空间实现上，我们表征所有\（\ mathcal {C} \）-selfadjoint，无界加权合成半群。作为副产品，我们证明\（\ mathcal {C} \）- selfadjoint无界半群的生成者不一定是\（\ mathcal {C} \）- selfadjoint。