$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey paper Well-posedness and Asymptotic Behaviour of Non-autonomous Linear Evolution Equations.

In this section we review existence results for the non-autonomous Cauchy problem $$ \mathrm{(C P)} \begin{cases} \frac{\diff}{\diff t} u(t) & =A(t) u(t)+f(t) \qtextq{for} t \in J , \\ u(s) & = x, \end{cases} $$ on a Banach space $X$, where $A(t)$ are linear operators on $X$; $x \in X, f \in L_{\mathrm{loc}}^1(J, X)$; and $J \subseteq \mathbb{R}$ is a closed interval. The homogeneous problem with $f=0$ is denoted by $\mathrm{(C P)_0}$.

Definition 2.1. The homogeneous problem $\mathrm{(C P)_0}$ is called well-posed (on spaces $Y_s$ ) if there are dense subspaces $Y_s, s \in J$, of $X$ with $Y_s \subseteq D(A(s))$ such that for each $x \in Y_s$ there is a unique solution $u=u(\cdot ; s, x) \in C^1\left(J_s, X\right)$ of $\mathrm{(C P)_0}$ with $u(t) \in Y_t$ for $t \in J_s$ and if $s_n \rightarrow s$ and $x_n \rightarrow x$ in $X$ for $s_n \in J, x_n \in Y_{s_n}, x \in Y_s$, then $\hat{u}\left(t ; s_n, x_n\right) \rightarrow \hat{u}(t ; s, x)$ in $X$ uniformly for $t$ in compact subsets of $J$.

Above, we set $\hat{u}(t ; r, y):=u(t ; r, y)$ for $t \geq r$ and $\hat{u}(t ; r, y):=y$ for $t \leq r$ and $y \in Y_r$. Starting with a well-posed Cauchy problem, we define $U(t, s) x:=u(t ; s, x)$ for $t, s \in J ; t \geq s$, and $x \in Y_s$. It is not difficult to show that $U(t, s)$ can be extended to a unique bounded linear operator on $X$ (denoted by the same symbol) such that

• (E1) $U(t, s)=U(t, r) U(r, s), U(s, s)=I$, and
• (E2) $(t, s) \mapsto U(t, s)$ is strongly continuous for $t \geq r \geq s$ and $t, s, r \in J$.

Definition 2.2. A collection $U(\cdot, \cdot)=(U(t, s))_{t \geq s ; t, s \in J} \subseteq \mathcal{L}(X)$ satisfying (E1) and (E2) is called an evolution family. If $\mathrm{(C P)_0}$ is well-posed (on $Y_t$ ) with solutions $u=U(\cdot, s) x$, we say that $U(\cdot, \cdot)$ solves $\mathrm{(C P)_0}$ on ($Y_t$) or that $A(\cdot)$ generates $U(\cdot, \cdot)$.

The parabolic case: Here one assumes that the operators $A(t)$ generate analytic $C_0$-semigroups of the same type and that $t \mapsto A(t)$ is regular in a sense specified below. Then there exists an evolution family $U(\cdot, \cdot)$ on $X$ solving $\mathrm{(C P)_0}$ on $D(A(t))$ such that $U(t, s) X \subseteq D(A(t)), \frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ in $\mathcal{L}(X)$, and $\|A(t) U(t, s)\| \leq \frac{C}{t-s}$ for $0 \leq s<$ $t \leq T$. The operators $U(t, s)$ can be constructed as solutions to certain integral equations like $$ U(t, s)=e^{(t-s) A(s)}+\int_s^t U(t, \tau)(A(\tau)-A(s)) e^{(\tau-s) A(s)} \diff \tau $$ (in the case $D(A(t)) \equiv D(A(0))$).

I am interested in $\mathrm{(C P)_0}$ where $f=0$. Let $\frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ hold (e.g. backward Kolmogorov equation in this question).

Could you provide a proof or a reference for the representation $U(t, s)=e^{(t-s) A(s)}$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey paper Well-posedness and Asymptotic Behaviour of Non-autonomous Linear Evolution Equations.

In this section we review existence results for the non-autonomous Cauchy problem $$ \mathrm{(C P)} \begin{cases} \frac{\diff}{\diff t} u(t) & =A(t) u(t)+f(t) \qtextq{for} t \in J , \\ u(s) & = x, \end{cases} $$ on a Banach space $X$, where $A(t)$ are linear operators on $X$; $x \in X, f \in L_{\mathrm{loc}}^1(J, X)$; and $J \subseteq \mathbb{R}$ is a closed interval. The homogeneous problem with $f=0$ is denoted by $\mathrm{(C P)_0}$.

Definition 2.1. The homogeneous problem $\mathrm{(C P)_0}$ is called well-posed (on spaces $Y_s$ ) if there are dense subspaces $Y_s, s \in J$, of $X$ with $Y_s \subseteq D(A(s))$ such that for each $x \in Y_s$ there is a unique solution $u=u(\cdot ; s, x) \in C^1\left(J_s, X\right)$ of $\mathrm{(C P)_0}$ with $u(t) \in Y_t$ for $t \in J_s$ and if $s_n \rightarrow s$ and $x_n \rightarrow x$ in $X$ for $s_n \in J, x_n \in Y_{s_n}, x \in Y_s$, then $\hat{u}\left(t ; s_n, x_n\right) \rightarrow \hat{u}(t ; s, x)$ in $X$ uniformly for $t$ in compact subsets of $J$.

Above, we set $\hat{u}(t ; r, y):=u(t ; r, y)$ for $t \geq r$ and $\hat{u}(t ; r, y):=y$ for $t \leq r$ and $y \in Y_r$. Starting with a well-posed Cauchy problem, we define $U(t, s) x:=u(t ; s, x)$ for $t, s \in J ; t \geq s$, and $x \in Y_s$. It is not difficult to show that $U(t, s)$ can be extended to a unique bounded linear operator on $X$ (denoted by the same symbol) such that

• (E1) $U(t, s)=U(t, r) U(r, s), U(s, s)=I$, and
• (E2) $(t, s) \mapsto U(t, s)$ is strongly continuous for $t \geq r \geq s$ and $t, s, r \in J$.

Definition 2.2. A collection $U(\cdot, \cdot)=(U(t, s))_{t \geq s ; t, s \in J} \subseteq \mathcal{L}(X)$ satisfying (E1) and (E2) is called an evolution family. If $\mathrm{(C P)_0}$ is well-posed (on $Y_t$ ) with solutions $u=U(\cdot, s) x$, we say that $U(\cdot, \cdot)$ solves $\mathrm{(C P)_0}$ on ($Y_t$) or that $A(\cdot)$ generates $U(\cdot, \cdot)$.

The parabolic case: Here one assumes that the operators $A(t)$ generate analytic $C_0$-semigroups of the same type and that $t \mapsto A(t)$ is regular in a sense specified below. Then there exists an evolution family $U(\cdot, \cdot)$ on $X$ solving $\mathrm{(C P)_0}$ on $D(A(t))$ such that $U(t, s) X \subseteq D(A(t)), \frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ in $\mathcal{L}(X)$, and $\|A(t) U(t, s)\| \leq \frac{C}{t-s}$ for $0 \leq s<$ $t \leq T$. The operators $U(t, s)$ can be constructed as solutions to certain integral equations like $$ U(t, s)=e^{(t-s) A(s)}+\int_s^t U(t, \tau)(A(\tau)-A(s)) e^{(\tau-s) A(s)} \diff \tau $$ (in the case $D(A(t)) \equiv D(A(0))$).

I am interested in $\mathrm{(C P)_0}$ where $f=0$. Let $\frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ hold (e.g. backward Kolmogorov equation in this question). Assume that $X$ is a Hilbert space and that $A(t)$ is self-adjoint for $t \in J$.

Is it true that $U(t, s)$ is self-adjoint? Any reference is appreciated.

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading the survey paper Well-posedness and Asymptotic Behaviour of Non-autonomous Linear Evolution Equations.

In this section we review existence results for the non-autonomous Cauchy problem $$ \mathrm{(C P)} \begin{cases} \frac{\diff}{\diff t} u(t) & =A(t) u(t)+f(t) \qtextq{for} t \in J , \\ u(s) & = x, \end{cases} $$ on a Banach space $X$, where $A(t)$ are linear operators on $X$; $x \in X, f \in L_{\mathrm{loc}}^1(J, X)$; and $J \subseteq \mathbb{R}$ is a closed interval. The homogeneous problem with $f=0$ is denoted by $\mathrm{(C P)_0}$.

Definition 2.1. The homogeneous problem $\mathrm{(C P)_0}$ is called well-posed (on spaces $Y_s$ ) if there are dense subspaces $Y_s, s \in J$, of $X$ with $Y_s \subseteq D(A(s))$ such that for each $x \in Y_s$ there is a unique solution $u=u(\cdot ; s, x) \in C^1\left(J_s, X\right)$ of $\mathrm{(C P)_0}$ with $u(t) \in Y_t$ for $t \in J_s$ and if $s_n \rightarrow s$ and $x_n \rightarrow x$ in $X$ for $s_n \in J, x_n \in Y_{s_n}, x \in Y_s$, then $\hat{u}\left(t ; s_n, x_n\right) \rightarrow \hat{u}(t ; s, x)$ in $X$ uniformly for $t$ in compact subsets of $J$.

Above, we set $\hat{u}(t ; r, y):=u(t ; r, y)$ for $t \geq r$ and $\hat{u}(t ; r, y):=y$ for $t \leq r$ and $y \in Y_r$. Starting with a well-posed Cauchy problem, we define $U(t, s) x:=u(t ; s, x)$ for $t, s \in J ; t \geq s$, and $x \in Y_s$. It is not difficult to show that $U(t, s)$ can be extended to a unique bounded linear operator on $X$ (denoted by the same symbol) such that

• (E1) $U(t, s)=U(t, r) U(r, s), U(s, s)=I$, and
• (E2) $(t, s) \mapsto U(t, s)$ is strongly continuous for $t \geq r \geq s$ and $t, s, r \in J$.

Definition 2.2. A collection $U(\cdot, \cdot)=(U(t, s))_{t \geq s ; t, s \in J} \subseteq \mathcal{L}(X)$ satisfying (E1) and (E2) is called an evolution family. If $\mathrm{(C P)_0}$ is well-posed (on $Y_t$ ) with solutions $u=U(\cdot, s) x$, we say that $U(\cdot, \cdot)$ solves $\mathrm{(C P)_0}$ on ($Y_t$) or that $A(\cdot)$ generates $U(\cdot, \cdot)$.

The parabolic case: Here one assumes that the operators $A(t)$ generate analytic $C_0$-semigroups of the same type and that $t \mapsto A(t)$ is regular in a sense specified below. Then there exists an evolution family $U(\cdot, \cdot)$ on $X$ solving $\mathrm{(C P)_0}$ on $D(A(t))$ such that $U(t, s) X \subseteq D(A(t)), \frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ in $\mathcal{L}(X)$, and $\|A(t) U(t, s)\| \leq \frac{C}{t-s}$ for $0 \leq s<$ $t \leq T$. The operators $U(t, s)$ can be constructed as solutions to certain integral equations like $$ U(t, s)=e^{(t-s) A(s)}+\int_s^t U(t, \tau)(A(\tau)-A(s)) e^{(\tau-s) A(s)} \diff \tau $$ (in the case $D(A(t)) \equiv D(A(0))$).

I am interested in $\mathrm{(C P)_0}$ where $f=0$. Let $\frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ hold (e.g. backward Kolmogorov equation in this question).

Could you provide a proof or a reference for the representation $U(t, s)=e^{(t-s) A(s)}$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey paper Well-posedness and Asymptotic Behaviour of Non-autonomous Linear Evolution Equations.

In this section we review existence results for the non-autonomous Cauchy problem $$ \mathrm{(C P)} \begin{cases} \frac{\diff}{\diff t} u(t) & =A(t) u(t)+f(t) \qtextq{for} t \in J , \\ u(s) & = x, \end{cases} $$ on a Banach space $X$, where $A(t)$ are linear operators on $X$; $x \in X, f \in L_{\mathrm{loc}}^1(J, X)$; and $J \subseteq \mathbb{R}$ is a closed interval. The homogeneous problem with $f=0$ is denoted by $\mathrm{(C P)_0}$.

Definition 2.1. The homogeneous problem $\mathrm{(C P)_0}$ is called well-posed (on spaces $Y_s$ ) if there are dense subspaces $Y_s, s \in J$, of $X$ with $Y_s \subseteq D(A(s))$ such that for each $x \in Y_s$ there is a unique solution $u=u(\cdot ; s, x) \in C^1\left(J_s, X\right)$ of $\mathrm{(C P)_0}$ with $u(t) \in Y_t$ for $t \in J_s$ and if $s_n \rightarrow s$ and $x_n \rightarrow x$ in $X$ for $s_n \in J, x_n \in Y_{s_n}, x \in Y_s$, then $\hat{u}\left(t ; s_n, x_n\right) \rightarrow \hat{u}(t ; s, x)$ in $X$ uniformly for $t$ in compact subsets of $J$.

Above, we set $\hat{u}(t ; r, y):=u(t ; r, y)$ for $t \geq r$ and $\hat{u}(t ; r, y):=y$ for $t \leq r$ and $y \in Y_r$. Starting with a well-posed Cauchy problem, we define $U(t, s) x:=u(t ; s, x)$ for $t, s \in J ; t \geq s$, and $x \in Y_s$. It is not difficult to show that $U(t, s)$ can be extended to a unique bounded linear operator on $X$ (denoted by the same symbol) such that

• (E1) $U(t, s)=U(t, r) U(r, s), U(s, s)=I$, and
• (E2) $(t, s) \mapsto U(t, s)$ is strongly continuous for $t \geq r \geq s$ and $t, s, r \in J$.

Definition 2.2. A collection $U(\cdot, \cdot)=(U(t, s))_{t \geq s ; t, s \in J} \subseteq \mathcal{L}(X)$ satisfying (E1) and (E2) is called an evolution family. If $\mathrm{(C P)_0}$ is well-posed (on $Y_t$ ) with solutions $u=U(\cdot, s) x$, we say that $U(\cdot, \cdot)$ solves $\mathrm{(C P)_0}$ on ($Y_t$) or that $A(\cdot)$ generates $U(\cdot, \cdot)$.

The parabolic case: Here one assumes that the operators $A(t)$ generate analytic $C_0$-semigroups of the same type and that $t \mapsto A(t)$ is regular in a sense specified below. Then there exists an evolution family $U(\cdot, \cdot)$ on $X$ solving $\mathrm{(C P)_0}$ on $D(A(t))$ such that $U(t, s) X \subseteq D(A(t)), \frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ in $\mathcal{L}(X)$, and $\|A(t) U(t, s)\| \leq \frac{C}{t-s}$ for $0 \leq s<$ $t \leq T$. The operators $U(t, s)$ can be constructed as solutions to certain integral equations like $$ U(t, s)=e^{(t-s) A(s)}+\int_s^t U(t, \tau)(A(\tau)-A(s)) e^{(\tau-s) A(s)} \diff \tau $$ (in the case $D(A(t)) \equiv D(A(0))$).

I am interested in $\mathrm{(C P)_0}$ where $f=0$. Let $\frac{\diff}{\diff t} U(t, s)=A(t) U(t, s)$ hold (e.g. backward Kolmogorov equation in this question).

Could you provide a proof or a reference for the representation $U(t, s)=e^{(t-s) A(s)}$?