]> 2.3 Semigroups and Riesz bases

### 2.3 Semigroups and Riesz bases

Deﬁnition 2.3.1. Let $\text{X}$ be a Banach space. A family ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ of operators in $L\left(\text{X}\right)$ is said to be a ${\text{C}}_{0}$semigroup on $\text{X}$ if the following conditions are satisﬁed:

1. $S\left(0\right)=I$, $S\left(t+s\right)=S\left(t\right)S\left(s\right)$ for all $t,s\ge 0$,
2. ${lim}_{t\to 0+}S\left(t\right)x=x$ for all $x\in \text{X}$.

1. For each ﬁxed $x\in \text{X}$, the map $t↦S\left(t\right)x$ is real analytic function on $\left(0,\infty \right)$,

then ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is an analytic semigroup on $\text{X}$. The linear operator

$\mathcal{𝒜}x={lim}_{t\to 0+}\frac{S\left(t\right)x-x}{t},\phantom{\rule{2em}{0ex}}D\left(\mathcal{𝒜}\right)=\left\{x\in \text{X}:\phantom{\rule{0ex}{0ex}}\exists {lim}_{t\to 0+}\frac{S\left(t\right)x-x}{t}\right\}$

is called the inﬁnitesimal generator of a linear ${\text{C}}_{0}$–semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ on $\text{X}$.

The following two theorems characterize the generators of analytic and ${\text{C}}_{0}$–semigroups.

Theorem 2.3.1 (Hille). , . A linear operator $\mathcal{𝒜}:\phantom{\rule{0ex}{0ex}}\left(D\left(\mathcal{𝒜}\right)\subset \text{X}\right)\to \text{X}$ acting on a Banach space $\text{X}$ is the inﬁnitesimal generator of a linear analytic semigroup on $\text{X}$ iff $\mathcal{𝒜}$ is closed and densely deﬁned and there exist $\omega \in \mathbb{ℝ}$, $\varphi \in \left(\frac{\pi }{2},\pi \right)$, and $M\ge 1$ such that the sector ${S}_{\omega ,\varphi }=\left\{\lambda \in \mathbb{ℂ}:\phantom{\rule{0ex}{0ex}}\varphi \ge \left|arg\left(\lambda -\omega \right)\right|,\phantom{\rule{0ex}{0ex}}\lambda \ne \omega \right\}$ is contained in the resolvent set $\rho \left(\mathcal{𝒜}\right)$ and the resolvent of $\mathcal{𝒜}$ satisﬁes the estimate

 ${∥{\left(\lambda I-\mathcal{𝒜}\right)}^{-1}∥}_{L\left(\text{X}\right)}\le \frac{M}{\left|\lambda -\omega \right|}\phantom{\rule{2em}{0ex}}\forall \lambda \in {S}_{\omega ,\varphi }$ (2.10)

Theorem 2.3.2 (Hille–Phillips–Yosida). A linear operator $\mathcal{𝒜}:\phantom{\rule{0ex}{0ex}}\left(D\left(\mathcal{𝒜}\right)\subset \text{X}\right)\to \text{X}$ acting on a Banach space $\text{X}$ is the inﬁnitesimal generator of a linear ${\text{C}}_{0}$–semigroup on $\text{X}$ iff $\mathcal{𝒜}$ is closed and densely deﬁned, and there exist $\omega \in \mathbb{ℝ}$ and $M\ge 1$ such that the halfplane ${\Pi }_{\omega }=\left\{\lambda \in \mathbb{ℂ}:\phantom{\rule{0ex}{0ex}}Re\lambda >\omega \right\}$ is contained in the resolvent set $\rho \left(\mathcal{𝒜}\right)$ and the resolvent of $\mathcal{𝒜}$ satisﬁes the estimate

 ${∥{\left(\lambda I-\mathcal{𝒜}\right)}^{-n}∥}_{L\left(\text{X}\right)}\le \frac{M}{{\left(Re\lambda -\omega \right)}^{n}}\phantom{\rule{2em}{0ex}}\forall \lambda \in {\Pi }_{\omega }\phantom{\rule{1em}{0ex}}\forall n\in \mathbb{ℕ}$ (2.11)

If the above conditions are satisﬁed then for any ${x}_{0}\in D\left(\mathcal{𝒜}\right)$, $T>0$ there exists a unique classical solution $x\in {\text{C}}^{1}\left(\left[0,T\right],\text{X}\right)\cap \text{C}\left(\left[0,T\right],{D}_{\mathcal{𝒜}}\right)$, $x\left(t\right)=S\left(t\right){x}_{0}$ of problem (2.3), where ${D}_{\mathcal{𝒜}}$ denotes the Banach space $D\left(\mathcal{𝒜}\right)$ equipped with the norm ${∥x∥}_{\mathcal{𝒜}}={∥x∥}_{\text{X}}+{∥\mathcal{𝒜}x∥}_{\text{X}}$. The function $x\left(t\right)=S\left(t\right){x}_{0}$, where ${x}_{0}\in \text{X}$, is called a weak solution of (2.3).

In what follows $\text{H}$ will denote a Hilbert space with a scalar product ${〈\cdot ,\cdot 〉}_{\text{H}}$.

Deﬁnition 2.3.2. The system ${\left\{{\phi }_{i}\right\}}_{i\in J}\subset \text{H}$ is called a Riesz basis of $\text{H}$ if there exist $T\in L\left(\text{H}\right)$ with ${T}^{-1}\in L\left(\text{H}\right)$ and an orthonormal basis ${\left\{{e}_{i}\right\}}_{i\in J}$ such that ${\phi }_{i}=T{e}_{i}$.

For other characterizations of Riesz bases, see , ,  and .

Deﬁnition 2.3.3. Assume that ${\text{H}}_{1}$ and ${\text{H}}_{2}$ are Hilbert spaces with scalar products ${〈\cdot ,\cdot 〉}_{{\text{H}}_{1}}$, ${〈\cdot ,\cdot 〉}_{{\text{H}}_{2}}$, respectively. Let $\mathcal{𝒯}:\left(D\left(\mathcal{𝒯}\right)\subset {\text{H}}_{1}\right)\to {\text{H}}_{2}$ be a closed, densely deﬁned linear operator.

$D\left({\mathcal{𝒯}}^{\ast }\right):=\left\{v\in {\text{H}}_{2}:\phantom{\rule{0ex}{0ex}}\exists \left(!\right)\phantom{\rule{0ex}{0ex}}{h}_{v}\in {\text{H}}_{1}:\phantom{\rule{0ex}{0ex}}{〈\mathcal{𝒯}x,v〉}_{{\text{H}}_{2}}={〈x,{h}_{v}〉}_{{\text{H}}_{1}}\phantom{\rule{1em}{0ex}}\forall x\in D\left(\mathcal{𝒯}\right)\right\}$

is the domain of the adjoint operator ${\mathcal{𝒯}}^{\ast }:\phantom{\rule{0ex}{0ex}}\left(D\left({\mathcal{𝒯}}^{\ast }\right)\subset {\text{H}}_{2}\right)\to {\text{H}}_{1}$ with respect to $\mathcal{𝒯}$, deﬁned as ${\mathcal{𝒯}}^{\ast }v:={h}_{v}$, $v\in D\left({\mathcal{𝒯}}^{\ast }\right)$.

Recall that the operator $\mathcal{𝒜}$ is called discrete if its resolvent ${\left(\lambda I-\mathcal{𝒜}\right)}^{-1}$ is a compact operator for some, or equivalently, for all $\lambda \in \rho \left(\mathcal{𝒜}\right)$ [69, p. 49]. The name discrete is motivated by the fact that operators with a compact resolvent have the spectrum consisting entirely of eigenvalues, however it is not ensured that the set of eigenvalues (point spectrum) is nonvoid. There exist also discrete operators with a ﬁnite number of eigenvalues. The special situation holds for discrete normal operators. The operator $\mathcal{𝒜}:\left(D\left(\mathcal{𝒜}\right)\subset \text{H}\right)↦\text{H}$ is called normal if $D\left(\mathcal{𝒜}\right)=D\left({\mathcal{𝒜}}^{\ast }\right)$ and ${∥\mathcal{𝒜}x∥}_{\text{H}}={∥{\mathcal{𝒜}}^{\ast }x∥}_{\text{H}}$ for all $x\in D\left(\mathcal{𝒜}\right)$.

Theorem 2.3.3 (Spectral theorem for discrete normal operators). An operator with a compact resolvent is normal iff it possesses a system of eigenvectors forming an orthonormal basis of $\text{H}$.

Proof. For the proof of necessity see [53, pp. 260 - 263 and pp. 276 - 277], [76, pp. 250 - 255] or, less explicitly [86, Theorem 7.2, p. 167]. Sufficiency can be deduced from [86, Theorem 7.2, p. 167]. □

Deﬁnition 2.3.4. $\mathcal{𝒜}:\left(D\left(\mathcal{𝒜}\right)\subset \text{H}\right)\to \text{H}$ is similar to a normal operator if there exists a similarity transformation $T\in L\left(\text{H}\right)$ with ${T}^{-1}\in L\left(\text{H}\right)$ such that $\mathcal{𝒩}={T}^{-1}\mathcal{𝒜}T$ is normal.

The following conclusion can be drawn from Theorem 2.3.3 and Deﬁnition 2.3.4.

Corollary 2.3.1. An operator $\mathcal{𝒜}:\phantom{\rule{0ex}{0ex}}\left(D\left(\mathcal{𝒜}\right)\subset \text{H}\right)\to \text{H}$ with a compact resolvent is similar to a normal operator iff $\mathcal{𝒜}$ possesses a system of eigenvectors forming a Riesz basis of $\text{H}$. To be more precise, ${\left\{{e}_{i}\right\}}_{i\in J}$ is an orthonormal basis formed by the system of eigenvectors of a discrete normal operator $\mathcal{𝒩}$, corresponding to its eigenvalues ${\left\{{\lambda }_{i}\right\}}_{i\in J}$ iff the system ${\left\{{\phi }_{i}\right\}}_{i\in J}$, ${\phi }_{i}=T{e}_{i}$, where $T$ stands for the similarity transformation, is the Riesz basis of eigenvectors of $\mathcal{𝒜}$, corresponding to its eigenvalues ${\left\{{\lambda }_{i}\right\}}_{i\in J}$, or equivalently, the system ${\left\{{\psi }_{i}\right\}}_{i\in J}$, ${\psi }_{i}={\left({T}^{\ast }\right)}^{-1}{e}_{i}$ is the Riesz basis of eigenvectors of ${\mathcal{𝒜}}^{\ast }$, corresponding to its eigenvalues ${\left\{\overline{{\lambda }_{i}}\right\}}_{i\in J}$.

Deﬁnition 2.3.5. The semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is asymptotically stable (AS) if

 ${lim}_{t\to \infty }S\left(t\right)x=0\phantom{\rule{2em}{0ex}}\forall x\in \text{H}$ (2.12)

and weakly asymptotically stable, if

${lim}_{t\to \infty }{〈S\left(t\right){x}_{1},{x}_{2}〉}_{\text{H}}=0\phantom{\rule{2em}{0ex}}\forall {x}_{1},{x}_{2}\in \text{H}.$

It is exponentially stable (EXS), if there exist $M\ge 1$, $\alpha >0$ such that

${∥S\left(t\right)∥}_{L\left(\text{H}\right)}\le M{e}^{-\alpha t}\phantom{\rule{2em}{0ex}}\forall t\ge 0.$

Clearly EXS implies AS, and for compact semigroups both concepts are equivalent. In turn, AS implies weak asymptotic stability, and for semigroups having discrete inﬁnitesimal generator both concepts are equivalent , [4, p. 25]. The semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$, generated by $\mathcal{𝒜}$, is EXS if the adjoint semigroup ${\left\{{S}^{\ast }\left(t\right)\right\}}_{t\ge 0}$, generated by ${\mathcal{𝒜}}^{\ast }$, is EXS. AS of ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ implies weak asymptotic stability of ${\left\{{S}^{\ast }\left(t\right)\right\}}_{t\ge 0}$.

Exercise 2.3.1. Consider the Hilbert space $\text{H}={\text{L}}^{2}\left(0,\infty \right)$ with the standard scalar product

${〈{x}_{1},{x}_{2}〉}_{\text{H}}={\int }_{0}^{\infty }{x}_{1}\left(\theta \right){x}_{2}\left(\theta \right)d\theta .$

A family of operators ${\left\{S\left(t\right)\right\}}_{t\ge 0}$,

 $\left(S\left(t\right)g\right)\left(\theta \right):=\left\{\begin{array}{cc}g\left(\theta -t\right),\hfill & \hfill \theta >t\\ 0,\hfill & \hfill t>\theta >0\end{array}\right\}$ (2.13)

deﬁnes a ${\text{C}}_{0}$–semigroup of right–shifts on $\text{H}$. The most difficult fact to be veriﬁed is the axiom (ii) of the semigroup. By the Parseval theorem

${∥S\left(h\right)g-g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|\left(\stackrel{̂}{S\left(h\right)g}\right)\left(j\omega \right)-ĝ\left(j\omega \right)\right|}^{2}d\omega .$

But

$\left(\stackrel{̂}{S\left(h\right)g}\right)\left(s\right)={\int }_{h}^{\infty }{e}^{-s\theta }g\left(\theta -h\right)d\theta ={\int }_{0}^{\infty }{e}^{-s\left(\xi +h\right)}g\left(\xi \right)d\xi ={e}^{-sh}ĝ\left(s\right)$

and hence, by the Lebesgue dominated convergence theorem, we have

${lim}_{h\to 0+}{∥S\left(h\right)g-g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}={lim}_{h\to 0+}\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|{e}^{-j\omega h}-1\right|}^{2}{\left|ĝ\left(j\omega \right)\right|}^{2}d\omega =0.$

The inﬁnitesimal generator of this semigroup is

$\left\{\begin{array}{ccc}\hfill \mathcal{𝒜}x& \hfill =\hfill & -{x}^{\prime }\hfill \\ \hfill D\left(\mathcal{𝒜}\right)& \hfill =\hfill & \left\{x\in {\text{L}}^{2}\left(0,\infty \right):\phantom{\rule{0ex}{0ex}}{x}^{\prime }\in {\text{L}}^{2}\left(0,\infty \right),\phantom{\rule{0ex}{0ex}}x\left(0\right)=0\right\}:={\text{W}}_{0}^{1,2}\left[0,\infty \right)\hfill \end{array}\right\}.$

Indeed, if $x\in D\left(\mathcal{𝒜}\right)$ then $x$ is continuous on $\left[0,\infty \right)$ and ${lim}_{\theta \to \infty }x\left(\theta \right)=0$.

$2{〈x,\mathcal{𝒜}x〉}_{{\text{L}}^{2}\left(0,\infty \right)}=-2{\int }_{0}^{\infty }x\left(\theta \right){x}^{\prime }\left(\theta \right)d\theta =-2{lim}_{t\to \infty }{\int }_{0}^{t}x\left(\theta \right){x}^{\prime }\left(\theta \right)d\theta =$

$=-{lim}_{t\to \infty }{x}^{2}\left(t\right)⇒\exists \phantom{\rule{0ex}{0ex}}{lim}_{t\to \infty }x\left(t\right).$

The limit cannot be nonzero as otherwise the integral ${\int }_{0}^{\infty }{x}^{2}\left(t\right)dt$ would be divergent. Now, applying the Parseval theorem and the Lebesgue dominated convergence theorem we obtain

${lim}_{h\to 0+}{∥\frac{1}{h}\left[S\left(h\right)g-g\right]-\mathcal{𝒜}g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}=$

$={lim}_{h\to 0+}\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|\frac{1}{h}\left[{e}^{-j\omega h}-1\right]+j\omega \right|}^{2}{\left|ĝ\left(j\omega \right)\right|}^{2}d\omega =0\phantom{\rule{2em}{0ex}}\forall g\in D\left(\mathcal{𝒜}\right).$

For comparison observe that $\mathcal{𝒜}$ is densely deﬁned and closed, and satisﬁes (2.11). Indeed, $x\left(\theta \right)={\int }_{0}^{\theta }{e}^{-\lambda \left(\theta -\tau \right)}g\left(\tau \right)d\tau$ solves the system

$\left\{\begin{array}{ccc}\hfill {x}^{\prime }\left(\theta \right)& \hfill =\hfill & -\lambda x\left(\theta \right)+g\left(\theta \right)\hfill \\ \hfill x\left(0\right)& \hfill =\hfill & 0\hfill \end{array}\right\}$

uniquely and thus we have

${∥{\left(\lambda I-\mathcal{𝒜}\right)}^{-1}g∥}_{{\text{L}}^{2}\left(0,\infty \right)}={∥{e}^{-\lambda \left(\cdot \right)}\star g∥}_{{\text{L}}^{2}\left(0,\infty \right)}\le$

$\le {∥{e}^{-\lambda \left(\cdot \right)}∥}_{{\text{L}}^{1}\left(0,\infty \right)}{∥g∥}_{{\text{L}}^{2}\left(0,\infty \right)}\le \frac{1}{Re\lambda }{∥g∥}_{{\text{L}}^{2}\left(0,\infty \right)}$

provided that $g\in \text{H}$, $Re\lambda >0$. By iterations one gets (2.11) with $\omega =0$. Notice that ${\left(sI-\mathcal{𝒜}\right)}^{-1}g$ is the Laplace transform of the function $t↦S\left(t\right)g$ as we have

$\left({\left(sI-\mathcal{𝒜}\right)}^{-1}g\right)\left(\theta \right)={\int }_{0}^{\theta }{e}^{-s\left(\theta -\xi \right)}g\left(\xi \right)d\xi ={\int }_{0}^{\theta }{e}^{-st}g\left(\theta -t\right)dt=$

$={\int }_{0}^{\infty }{e}^{-st}\left\{\begin{array}{cc}g\left(\theta -t\right),\hfill & \hfill 0\theta \end{array}\right\}dt={\int }_{0}^{\infty }{e}^{-st}\left(S\left(t\right)g\right)\left(\theta \right)dt.$

This conﬁrms (2.13).

By Deﬁnition 2.3.3 the adjoint operator with respect to $\mathcal{𝒜}$ is

$\left\{\begin{array}{ccc}\hfill {\mathcal{𝒜}}^{\ast }v& \hfill =\hfill & {v}^{\prime }\hfill \\ \hfill D\left({\mathcal{𝒜}}^{\ast }\right)& \hfill =\hfill & \left\{v\in {\text{L}}^{2}\left(0,\infty \right):\phantom{\rule{0ex}{0ex}}{v}^{\prime }\in {\text{L}}^{2}\left(0,\infty \right)\right\}:={\text{W}}^{1,2}\left[0,\infty \right)\hfill \end{array}\right\}.$

Since $D\left({\mathcal{𝒜}}^{\ast }\right)⊉D\left(\mathcal{𝒜}\right)$ the operator $\mathcal{𝒜}$ is not normal. The operator ${\mathcal{𝒜}}^{\ast }$ generates the adjoint semigroup ${\left\{{S}^{\ast }\left(t\right)\right\}}_{t\ge 0}$ with respect to ${\left\{S\left(t\right)\right\}}_{t\ge 0}$. By Deﬁnition 2.3.3,

 $\begin{array}{c}{〈S\left(t\right)x,v〉}_{{\text{L}}^{2}\left(0,\infty \right)}={\int }_{0}^{\infty }\left\{\begin{array}{cc}x\left(\theta -t\right),\hfill & \hfill \theta >t\\ 0,\hfill & \hfill t>\theta >0\end{array}\right\}v\left(\theta \right)d\theta =\hfill \\ ={\int }_{t}^{\infty }x\left(\theta -t\right)v\left(\theta \right)d\theta ={\int }_{0}^{\infty }x\left(\xi \right)v\left(\xi +t\right)d\xi ={〈x,{S}^{\ast }\left(t\right)v〉}_{{\text{L}}^{2}\left(0,\infty \right)}\hfill \end{array}$ (2.14)

for all $x,v\in \text{H}$. This means that the adjoint semigroup

 $\left({S}^{\ast }\left(t\right)v\right)\left(\theta \right)=v\left(t+\theta \right),\phantom{\rule{2em}{0ex}}\theta ,t\ge 0$ (2.15)

is the semigroup of left–shifts. Prove by direct solving of the equation $\left(\lambda I-{\mathcal{𝒜}}^{\ast }\right)v=g$

$\left({\left(\lambda I-{\mathcal{𝒜}}^{\ast }\right)}^{-1}g\right)\left(\theta \right)={\int }_{\theta }^{\infty }{e}^{\lambda \left(\theta -\xi \right)}g\left(\xi \right)d\xi$

and verify the result applying the Laplace transform to (2.15). Here $g\in \text{H}$ and $Re\lambda >0$.

The resolvent of $\mathcal{𝒜}$ is not compact. Indeed, if the resolvent of $\mathcal{𝒜}$ were compact then the number of eigenvectors of $\mathcal{𝒜}$ would be equal to the number of eigenvectors of ${\mathcal{𝒜}}^{\ast }$. However, this is not the case as $\mathcal{𝒜}$ has no eigenvectors while ${\mathcal{𝒜}}^{\ast }$ has them. By the same arguments $\mathcal{𝒜}$ is not similar to a normal operator.

The semigroup ${\left\{{S}^{\ast }\left(t\right)\right\}}_{t\ge 0}$ is AS as we have

${lim}_{t\to \infty }{∥{S}^{\ast }\left(t\right)g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}={lim}_{t\to \infty }{\int }_{0}^{\infty }{g}^{2}\left(t+\theta \right)d\theta ={lim}_{t\to \infty }{\int }_{t}^{\infty }{g}^{2}\left(\theta \right)d\theta =0\phantom{\rule{2em}{0ex}}\forall g\in \text{H}.$

This semigroup does not decay exponentially (consider $g\left(\theta \right)=\frac{1}{\theta +1}$ as a counterexample). The semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is not AS. Indeed, by (2.13) we have

${∥S\left(t\right)g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}={\int }_{t}^{\infty }{g}^{2}\left(\theta -t\right)d\theta ={\int }_{0}^{\infty }{g}^{2}\left(\xi \right)d\xi ={∥g∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2},$

so any nonzero trajectory does not tend to $0$ as $t\to \infty$. However, this semigroup tends weakly to $0$ which follows from (2.14) and AS of the semigroup ${\left\{{S}^{\ast }\left(t\right)\right\}}_{t\ge 0}$.

The next theorem is a particular case of known results [13, Theorem 2], [72, pp. 25 - 28], however, the proof given here is elementary.

Theorem 2.3.4. Let $\text{H}$ be a Hilbert space with an inner product ${〈\cdot ,\cdot 〉}_{\text{H}}$ and let $\mathcal{𝒜}:\phantom{\rule{0ex}{0ex}}\left(D\left(\mathcal{𝒜}\right)\subset \text{H}\right)\to \text{H}$ be a linear discrete operator which is similar to a normal one.

A necessary and sufficient conditon for $\mathcal{𝒜}$ to be the inﬁnitesimal generator of a linear analytic semigroup on $\text{H}$ is the existence of $\omega \in \mathbb{ℝ}$ and ${\varphi }_{0}\in \left(\frac{\pi }{2},\pi \right)$ such that $\rho \left(\mathcal{𝒜}\right)$ contains the sector ${S}_{\omega ,{\varphi }_{0}}=\left\{\lambda \in \mathbb{ℂ}:\phantom{\rule{0ex}{0ex}}{\varphi }_{0}\ge \left|arg\left(\lambda -\omega \right)\right|,\phantom{\rule{0ex}{0ex}}\lambda \ne \omega \right\}$.

A necessary and sufficient condition for $\mathcal{𝒜}$ to be the inﬁnitesimal generator of a linear ${\text{C}}_{0}$–semigroup on $\text{H}$ is the existence of $\omega \in \mathbb{ℝ}$ such that $\rho \left(\mathcal{𝒜}\right)$ contains the halfplane ${\Pi }_{\omega }=\left\{\lambda \in \mathbb{ℂ}:\phantom{\rule{0ex}{0ex}}Re\lambda >\omega \right\}$. Moreover, this semigroup is EXS iff

${sup}_{{\lambda }_{i}\in \sigma \left(\mathcal{𝒜}\right)}Re{\lambda }_{i}<0,$

where $\sigma \left(\mathcal{𝒜}\right)$ denotes the spectrum of $\mathcal{𝒜}$.

Proof. The necessary conditions easily follow from Theorems 2.3.1, 2.3.2, and Deﬁnition 2.3.5. We now prove the sufficient condition for the generation of analytic semigroup. $\mathcal{𝒜}$ is closed because ${\left(\lambda I-\mathcal{𝒜}\right)}^{-1}\in L\left(\text{H}\right)$ [86, p. 89]. Also, span ${\left\{{\phi }_{i}\right\}}_{i\in J}\subset D\left(\mathcal{𝒜}\right)$, and ${\left\{{\phi }_{i}\right\}}_{i\in J}$ is complete, which implies the density of $D\left(\mathcal{𝒜}\right)$ in $\text{H}$. Now consider the sector ${S}_{\omega ,\varphi }$, deﬁned as ${S}_{\omega ,{\varphi }_{0}}$ but with the angle of obtuseness $\varphi <{\varphi }_{0}$, with $\varphi \in \left(\frac{\pi }{2},\pi \right)$. Then ${S}_{\omega ,\varphi }{\subset }_{\omega ,{\varphi }_{0}}\subset \rho \left(\mathcal{𝒜}\right)$. So, by Hille’s theorem, it suffices to show the existence of $M\ge 1$ for which (2.10) holds. Let us inspect Figure 2.2. Figure 2.2: The auxiliary diagram for the proof of analytic–semigroup generation

Since $0\le {\varphi }_{0}-\frac{\pi }{2}\le \frac{\pi }{2}<\varphi$ we have ${S}_{\omega ,{\varphi }_{0}-\frac{\pi }{2}}\subset {S}_{\omega ,\varphi }$. For $\lambda$ belonging to the cone ${S}_{\omega ,{\varphi }_{0}-\frac{\pi }{2}}$, the inequalities

$\left|\lambda -\omega \right|\le \left|\lambda -{\lambda }_{i}\right|,\phantom{\rule{2em}{0ex}}i\in J$

hold. If $\lambda$ belongs to the remaining part of ${S}_{\omega ,\varphi }$, then

$\left|\lambda -\omega \right|sin\left({\varphi }_{0}-\varphi \right)\le \left|\lambda -{\lambda }_{i}\right|\phantom{\rule{2em}{0ex}}\forall i\in J.$

Theses estimates together yield

$\frac{1}{{\left|\lambda -{\lambda }_{i}\right|}^{2}}\le \frac{{csc}^{2}\left({\varphi }_{0}-\varphi \right)}{{\left|\lambda -\omega \right|}^{2}}\phantom{\rule{2em}{0ex}}\forall i\in J\phantom{\rule{1em}{0ex}}\forall \lambda \in {S}_{\omega ,\varphi }.$

By virtue of Corollary 2.3.1,

 $\begin{array}{c}{∥{\left(\lambda I-\mathcal{𝒜}\right)}^{-1}f∥}_{\text{H}}^{2}={∥T{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}{T}^{-1}f∥}_{\text{H}}^{2}\le \hfill \\ \le {∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}{∥{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}f∥}_{\text{H}}^{2}=\hfill \\ ={∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}\sum _{i\in J}{\left|{〈{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}f,{e}_{i}〉}_{\text{H}}\right|}^{2}=\hfill \\ ={∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}\sum _{i\in J}{\left|{〈f,{\left(\overline{\lambda }I-{\mathcal{𝒩}}^{\ast }\right)}^{-1}{e}_{i}〉}_{\text{H}}\right|}^{2}=\hfill \\ ={∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}\sum _{i\in J}\frac{1}{{\left|\lambda -{\lambda }_{i}\right|}^{2}}{\left|{〈f,{e}_{i}〉}_{\text{H}}\right|}^{2}\le \hfill \\ \le {∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}\frac{{csc}^{2}\left({\varphi }_{0}-\varphi \right)}{{\left|\lambda -\omega \right|}^{2}}\sum _{i\in J}{\left|{〈f,{e}_{i}〉}_{\text{H}}\right|}^{2}=\hfill \\ ={∥T∥}_{L\left(\text{H}\right)}^{2}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}^{2}\frac{{csc}^{2}\left({\varphi }_{0}-\varphi \right)}{{\left|\lambda -\omega \right|}^{2}}{∥f∥}_{\text{H}}^{2}\phantom{\rule{2em}{0ex}}\forall f\in \text{H}\phantom{\rule{1em}{0ex}}\forall \lambda \in {S}_{\omega ,\varphi }\hfill \end{array}$ (2.16)

Hence (2.10) holds with $M={∥T∥}_{L\left(\text{H}\right)}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}csc\left({\varphi }_{0}-\varphi \right)\ge 1$.

To prove the sufficient condition for generation of a semigroup we consider the sequence of identities

${\left(\lambda I-\mathcal{𝒜}\right)}^{-n}={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n}={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+1}{\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-1}=$

$={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+1}{\left(T\left(\lambda I-\mathcal{𝒩}\right){T}^{-1}\right)}^{-1}={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+1}T{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}{T}^{-1}=$

$={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+2}{\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-1}T{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}{T}^{-1}=$

$={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+2}T{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}{T}^{-1}T{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}{T}^{-1}=$

$={\left(\lambda I-T\mathcal{𝒩}{T}^{-1}\right)}^{-n+2}T{\left(\lambda I-\mathcal{𝒩}\right)}^{-2}{T}^{-1}=\dots =T{\left(\lambda I-\mathcal{𝒩}\right)}^{-n}{T}^{-1}.$

Hence

 $\begin{array}{cc}\hfill {∥{\left(\lambda I-\mathcal{𝒜}\right)}^{-n}∥}_{L\left(\text{H}\right)}=& {∥T{\left(\lambda I-\mathcal{𝒩}\right)}^{-n}{T}^{-1}∥}_{L\left(\text{H}\right)}\le \hfill \\ \hfill \le & {∥T∥}_{L\left(\text{H}\right)}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}{∥{\left(\lambda I-\mathcal{𝒩}\right)}^{-n}∥}_{L\left(\text{H}\right)}\hfill \end{array}$ (2.17)

Now,

$\begin{array}{ccc}\hfill {∥{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}x∥}_{\text{H}}^{2}& \hfill =\hfill & \sum _{i\in J}{\left|{〈{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}x,{e}_{i}〉}_{\text{H}}\right|}^{2}=\sum _{i\in J}{\left|{〈x,{\left(\overline{\lambda }I-{\mathcal{𝒩}}^{\ast }\right)}^{-1}{e}_{i}〉}_{\text{H}}\right|}^{2}=\hfill \\ \hfill & \hfill =\hfill & \sum _{i\in J}\frac{1}{{\left|\lambda -{\lambda }_{i}\right|}^{2}}{\left|{〈x,{e}_{i}〉}_{\text{H}}\right|}^{2}\phantom{\rule{2em}{0ex}}\forall x\in \text{H}.\hfill \end{array}$

Let us inspect Figure 2.3. Figure 2.3: The auxiliary diagram for the proof of semigroup generation

If $\lambda \in {\Pi }_{\omega }$ then

$\left|\lambda -{\lambda }_{i}\right|\ge Re\lambda -\omega >0,\phantom{\rule{2em}{0ex}}\omega :={sup}_{i\in J}Re{\lambda }_{i}$

and therefore

${∥{\left(\lambda I-\mathcal{𝒩}\right)}^{-1}x∥}_{\text{H}}\le \frac{1}{Re\lambda -\omega }{∥x∥}_{\text{H}}\phantom{\rule{2em}{0ex}}\forall x\in \text{H}.$

By iterations we get

 ${∥{\left(\lambda I-\mathcal{𝒩}\right)}^{-n}x∥}_{\text{H}}\le \frac{1}{{\left(Re\lambda -\omega \right)}^{n}}{∥x∥}_{\text{H}}\phantom{\rule{2em}{0ex}}\forall x\in \text{H}$ (2.18)

Finally, (2.11) with $M={∥T∥}_{L\left(\text{H}\right)}{∥{T}^{-1}∥}_{L\left(\text{H}\right)}\ge 1$ follows from (2.17) and (2.18). □ □