An integrability condition on the resolvent for hyperbolicity of the semigroup (Q1336332)

A strongly continuous semigroup \((T(t))_{t\geq 0}\) on the Banach space \(X\) is called hyperbolic if there exists a decomposition \(X= X_-\oplus X_+\) with projection \(P: X\to X_-\) along \(X_+\) such that (i) both spaces are invariant under the semigroup, (ii) \(T(t)|_{X_-}\) extends to a strongly continuous group, and (iii) there are positive constants \(K\), \(\alpha\), \(\beta\) with (a) \(|T(t)x|\leq K\exp(\beta t)|P(x)|\) for \(t\leq 0\) and (b) \(|T(t)(I- P)x|\leq K\exp(- \alpha t)\) \(|(I- P) x|\) for \(t\geq 0\). The main aim of the paper is to establish criteria for the generator \(A\) of the semigroup which imply hyperbolicity. In order to obtain the main results, the authors prove a generalization of Gearharts and Herbsts result on the connection between the boundedness of the resolvent \(\lambda\to R(\lambda)= (\lambda- A)^{- 1}\) on paralleles to the imaginary axis and the resolvent of \((T(t))\). From this generalization they get a new proof of this result. Here is a selection of the most important results (excluding the very interesting applications to concrete examples): (1) The semigroup is hyperbolic iff the following conditions are satisfied: (i) There exists an open strip containing the imaginary axis in the resolvent set of \(A\) on which \(R(\lambda)\) is bounded (ii) There exists \(\omega> 0\) such that \[ \lim_{N\to \infty} \int^N_0 {d\ell\over N} \int^\ell_{-\ell} R(h+ iv)xidv\tag{\(\alpha\)} \] exists for each \(x\) and \(0< |h|< \omega(\beta)\). For each \(x\in X\), \(x'\in X'\) and \(0< |h|< \omega\) the function \(r(v, h, x, x'):= \langle x', R(h+ iv)x\rangle\) satisfies \(|\langle r(.,h, x, x'),\Phi\rangle|\leq K|x|\cdot |x'||\widehat\Phi|_1\) for all \(\Phi\in {\mathcal S}(R)\), the Schwartz space on \(R\). Here \(\widehat \Phi\) denotes the Fourier transform of \(\Phi\) and \(r(., h, x, x')\) is viewed as a tempered distribution on \(\mathcal S\). (2) Let \(A\), \((T(t))_{t\geq 0}\) be as before. Suppose that there exists a \(\gamma> \omega(A)\), such that \[ \int^{\gamma+ i\infty}_{\gamma- i\infty} |R(z) x|^2 dz \] exists for each \(x\in X\), and \[ \int^{\gamma+i\infty}_{\gamma-i\infty} |R(z)' x'|^2 dz \] exists for each \(x'\in X'\). Then the semigroup is hyperbolic iff (i) There is \(\omega> 0\) such that \(\{z: |\text{Re}(z)|< \omega\}\subset \rho(A)\) (the resolvent set of \(A\)), (ii) \(\sup\{|R(z)|: |\text{Re}(z)|< \omega\}\leq M< \infty\).