Example of semigroup of mappings

Question

Want some examples of semigroup of mappings.

Let $X$ be a Banach space and consider the collection of mappings $\tau =\{T_t:t\ge 0\}$. Define, $T_t:X \to X$ by $$T_t(x)=xe^t \text{ for all }x\in X.$$ Then for $s,t\ge 0$ we have $T_{s+t}(x)=xe^{s+t}=(xe^t)e^s=(T_tx)e^s=T_s(T_tx)$. Hence the collection $\tau$ forms a semigroup of mappings.

I want some examples of another new type of semigroup of mappings.

Can anyone help me please?

Answer 1

Here are a few simple examples for such one-parameter semigroups of operators on a Banach space $X$:

1. Operator exponential functions: Let $A: X \to X$ be a bounded linear operator and set $T_t := e^{tA} = \sum_{k=0}^\infty \frac{t^kA^k}{k!}$ for each $t \ge 0$ (or even each $t \in \mathbb{R}$). As a special case, one obtains matrix semigroups when $X$ is finite-dimensional.

2. Shift semigroup: Let $X = L^p(\mathbb{R})$ and $(T_tf)(\omega) = f(\omega-t)$ for $f \in X$ and $\omega \in \mathbb{R}$.

3. Nilpotent shift semigroup: Let $X = L^p([0,1])$, and set $$ (T_tf)(\omega) = \begin{cases} f(\omega-t) \quad & \text{if } \omega \ge t, \\ 0 \quad & \text{if } \omega < t. \end{cases} $$ for $f \in X$ and $\omega \in [0,1]$.

4. Periodic shift semigroup: Let $X = L^p(\mathbb{T})$, where $\mathbb{T}$ denotes the complex unit circle, and set $(T_tf)(\omega) = f(e^{it}\omega)$ for $f \in X$ and $\omega \in \mathbb{T}$.

5. Heat semigroup (also called Gausian semigroup): Let $X = L^p(\mathbb{R}^d)$ and set $T_tf = k_t \star f$ for $f \in X$, where $\star$ denotes convolution over $\mathbb{R}^d$ and $k_t$ denotes the heat kernel on $\mathbb{R}^d$. This semigroup describes the solutions of the heat equation on $L^p(\mathbb{R}^d)$.

6. Convolution with the probability mass function of the Poisson distribution: Let $X = \ell^1(\mathbb{N}_0)$ and set $T_t x = k_t \star x$ for each $x \in X$, where $\star$ now denotes the convolution over $\mathbb{N_0}$, and $k_t(k) := e^{-t} \frac{t^k}{k!}$ denotes the density of the Poisson distribution with expected value $t$.

There are many more examples, and things become interesting when the semigroups can't be written down explicitly anymore, but their so-called generators can. This is the main idea of $C_0$-semigroup theory, which is at the heart of infinite dimensional linear evolution equations. An excellent reference for this topic is the book by Engel and Nagel (available online for free on the hompage of Rainer Nagel: link).