The frequency theorem for continuous one-parameter semigroups

The following is proved under certain, not very restrictive, assumptions. For the existence of a bounded linear operator $H=H^*$ such that the quadratic form $\operatorname{Re}(Ax+bu,Hx)+F(x,u)$ is positive definite on $X\times U$, it is necessary and sufficient that the form $F[(i\omega I-A)^{-1}bu,u]$ $\forall\omega\in R^1$ be positive definite, where $A$ is the infinitesimal generating operator of a strongly continuous semigroup in a Hilbert space $X$, $b$ is a bounded linear operator acting from a Hilbert space $U$ into $X$, and $F(x,u)$ is a quadratic form on $X$. Moreover, there exist bounded linear operators $H_0,h$, and $\varkappa$ such that the representation $\operatorname{Re}(Ax+bu,Hx)+F(x,u)=[\varkappa u-hx]^2$ holds. A similar assertion is proved in the “degenerate” case.
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