Generators, resolvents and the Hille-Yosida framework

Prove that the assertion of first-edition Problem 363 ceases to be true in the case $\dim L=\infty$.

Show that $P_{t}$ and $R_{\lambda }$ commute and that

Show that for any $a<b$ we have

Hint: Approximate $R_{\lambda } f$ by a Riemann sum.

Show that if $P_{t}$ is a contraction semigroup and $R_{\lambda }$ is the resolvent, then

Show that if $A$ is a bounded operator and $T_{t}=e^{t A}$, then $T_{t}$ is a strongly continuous semigroup of operators with infinitesimal generator $A$. (We cannot assert that the $T_{t}$ are contractions.)

Prove that if $\mathcal{L}$ is dissipative, the domain of $\mathcal{L}$ is dense in $\mathcal{B}$, and the range of $\lambda -\mathcal{L}$ is dense in $\mathcal{B}$, then $\overline{\mathcal{L}}$ defined in (37.12) is a closed extension of $\mathcal{L}$ that is dissipative and the range of $\lambda -\overline{\mathcal{L}}$ is equal to $\mathcal{B}$. Show there is only one such closed extension of $\mathcal{L}$.

If the range of $\lambda -\mathcal{L}$ equals $\mathcal{B}$ for a single value of $\lambda$, then the range of $\lambda -\mathcal{L}$ equals $\mathcal{B}$ for every value of $\lambda$.

Hint: Define $R_{\lambda }$ as the inverse of $\lambda -\mathcal{L}$, then use (37.3) to define $R_{a}$ for other values of $a$.

Let $P_{t}$ be a strongly continuous semigroup of contraction operators and let $\mathcal{L}$ be the infinitesimal generator. Show that $\mathcal{D}\left(\mathcal{L}^{n}\right)$ is dense in $\mathcal{B}$ for every positive integer $n$.

Suppose that $P_{t}$ is a strongly continuous semigroup, but not necessarily of contractions. Thus $P_{t+s}=P_{t} P_{s}$ and $P_{t} f \rightarrow f$ in norm if $f \in \mathcal{B}$, but we do not assume $\left\| P_{t}\right\| \leq 1$. Prove that there exist constants $K, b>0$ such that $\left\| P_{t}\right\| \leq K e^{b t}$ for all $t \geq 0$.

Hint: Use the uniform boundedness principle from functional analysis to prove there exists $c, t_{0}$ such that $\left\| P_{t}\right\| \leq c$ if $t \leq t_{0}$. Then use the semigroup property.

Show that if the resolvent operator $R_{\lambda }$ is a symmetric operator, then the transition operators $P_{t}$ are also symmetric: if $f, g \in \mathcal{B}$, then $\left\langle P_{t} f, g \right\rangle = \left\langle f, P_{t} g \right\rangle$.