]> 2.6 Semigroup generation

### 2.6 Semigroup generation

Observe, using Deﬁnition 2.3.3 that
${\mathcal{𝒜}}^{\ast }v={v}^{\prime \prime },\phantom{\rule{2em}{0ex}}D\left({\mathcal{𝒜}}^{\ast }\right)=\left\{v\in {\text{H}}^{2}\left(0,1\right):\phantom{\rule{0ex}{0ex}}v\left(0\right)=0,\phantom{\rule{0ex}{0ex}}{v}^{\prime }\left(1\right)=K{v}^{\prime }\left(0\right)\right\}.$

Solving the eigenproblem for ${\mathcal{𝒜}}^{\ast }$ we determine

 ${v}_{n}\left(\theta \right)=\frac{2}{{\overline{s}}^{2}-1}sinh\overline{{\mu }_{n}}\theta ,\phantom{\rule{2em}{0ex}}0\le \theta \le 1,\phantom{\rule{1em}{0ex}}n\in \mathbb{ℤ}$ (2.41)

Lemma 2.6.1. If $\left|K\right|\ne 1$, then the operator $\mathcal{𝒜}$ deﬁned by (2.4) is the inﬁnitesimal generator of a linear analytic semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ on $\text{H}$. It has the representation

$S\left(t\right)x=\sum _{n=-\infty }^{\infty }{e}^{{\lambda }_{n}t}{〈x,{v}_{n}〉}_{\text{H}}{x}_{n}=\sum _{n=-\infty }^{\infty }{e}^{{\lambda }_{n}t}{Res}_{\lambda ={\lambda }_{n}}{\left(\lambda I-\mathcal{𝒜}\right)}^{-1}x\phantom{\rule{2em}{0ex}}\forall t\ge 0,\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\forall x\in \text{H},$

where ${\left\{{x}_{n}\right\}}_{n\in \mathbb{ℤ}}$ is the system of eigenfunctions given by (2.34) and ${\left\{{v}_{n}\right\}}_{n\in \mathbb{ℤ}}$ is its biorthogonal system (2.41). Moreover, the condition

 $-cosh\pi (2.42)

is necessary and sufficient for EXS of the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$.

Proof. From Proposition 2.5.1 and Corollary 2.3.1 we know that the operator $\mathcal{𝒜}$ is similar to a normal discrete operator $\mathcal{𝒩}$ with a spectrum located on the negative real semiaxis or on the parabola having the branches directed to the left. Thus by Theorem 2.3.4 the operator $\mathcal{𝒜}$ generates an analytic semigroup. Using the boundedness of a semigroup and the basis property we get

$\begin{array}{cc}\hfill S\left(t\right)x=& T{e}^{t\mathcal{𝒩}}{T}^{-1}x=T{lim}_{l\to \infty }\sum _{n=-l}^{l}{e}^{t{\lambda }_{n}}{〈{T}^{-1}x,{e}_{n}〉}_{\text{H}}{e}_{n}=\sum _{n=-\infty }^{\infty }{e}^{t{\lambda }_{n}}{〈{T}^{-1}x,{e}_{n}〉}_{\text{H}}T{e}_{n}=\hfill \\ \hfill =& \sum _{n=-\infty }^{\infty }{e}^{t{\lambda }_{n}}{〈x,{\left({T}^{\ast }\right)}^{-1}{e}_{n}〉}_{\text{H}}T{e}_{n}=\sum _{n=-\infty }^{\infty }{e}^{t{\lambda }_{n}}{〈x,{v}_{n}〉}_{\text{H}}{x}_{n},\hfill \end{array}$

where ${\left\{{e}^{t\mathcal{𝒩}}\right\}}_{t\ge 0}$ denotes an analytic semigroup generated by $\mathcal{𝒩}$, ${\left\{{e}_{n}\right\}}_{n\in \mathbb{ℤ}}$ is a system of its eigenvectors being an orthonormal basis of $\text{H}$, $T$ stands for the similarity transformation, ${\left\{{v}_{n}\right\}}_{n\in \mathbb{ℤ}}$ is the (unique) systems biorthogonal with respect to ${\left\{{x}_{n}\right\}}_{n\in \mathbb{ℤ}}$, being the system of eigenvectors of ${\mathcal{𝒜}}^{\ast }$. The condition (2.42) follows easily from (2.33). □

Lemma 2.6.1 permits us to strengthen the results concerning the regularity of the functional $\mathcal{𝒞}$ deﬁned in (2.7).

Lemma 2.6.2. For $\left|K\right|\ne 1$ and $\alpha >\frac{1}{4}$, the functional $\mathcal{𝒞}$ deﬁned by (2.7) is linear and bounded on the Banach space ${\text{H}}^{\alpha }$, $0\le \alpha \le 1$, where ${\text{H}}^{\alpha }=D\left({\mathcal{𝒜}}^{\alpha }\right)$, is the domain of a fractional power of $\mathcal{𝒜}$, equipped with a topology induced by the norm ${∥x∥}_{{\text{H}}^{\alpha }}={∥{\mathcal{𝒜}}^{\alpha }x∥}_{\text{H}}$.

Proof. It follows from Lemma 2.6.1 that, for $\left|K\right|\ne 1$, the fractional powers ${\mathcal{𝒜}}^{\alpha }$ ($0\le \alpha \le 1$) of $\mathcal{𝒜}$ and the corresponding spaces ${\text{H}}^{\alpha }$ are well deﬁned [48, pp. 24 - 30]. Since for $\alpha >\frac{1}{4}$ we have ${\text{H}}^{\alpha }\subset \text{C}\left(0,1\right)$ [48, Theorem 1.6.1, p. 39], then there exists ${M}_{\alpha }>0$ such that $\left|\mathcal{𝒞}x\right|=\left|x\left(1\right)\right|\le {∥x∥}_{\text{C}\left(0,1\right)}\le {M}_{\alpha }{∥x∥}_{{\text{H}}^{\alpha }}$ for all $x\in {\text{H}}^{\alpha }$. □

Remark 2.6.1. There is another way to obtain the above result. By (2.8), (2.9), we have

$\mathcal{𝒞}x={〈{\left({\mathcal{𝒜}}^{\ast }\right)}^{1-\alpha }d,{\mathcal{𝒜}}^{\alpha }x〉}_{\text{H}}\phantom{\rule{2em}{0ex}}\forall x\in {\text{H}}^{\alpha },$

where $\alpha \in \left(0,1\right)$ is such that $d\in D\left[{\left({\mathcal{𝒜}}^{\ast }\right)}^{1-\alpha }\right]$. Now

${\mathcal{𝒜}}^{\ast }v=\sum _{n=-\infty }^{\infty }{\overline{\lambda }}_{n}{〈v,{x}_{n}〉}_{\text{H}}{v}_{n}\phantom{\rule{2em}{0ex}}\forall v\in D\left({\mathcal{𝒜}}^{\ast }\right),$

and $d\in D\left[{\left({\mathcal{𝒜}}^{\ast }\right)}^{\beta }\right]$ iff ${\sum }_{n=-\infty }^{\infty }{\left|{〈d,{x}_{n}〉}_{\text{H}}\right|}^{2}{\left|{\lambda }_{n}\right|}^{2\beta }<\infty$. Hence, taking (2.8), (2.37), and (2.41) into account, we get $\beta <\frac{3}{4}$.