Solution Operator for Non-Autonomous Perturbation of Gibbs Semigroup

To this end we consider on separable Hilbert space ${\mathfrak{H}}$ a linear non-autonomous dynamics given by evolution equation of the type:

where ${\mathbb{R}}_{0}^{+}=\{0\}\cup{\mathbb{R}}^{+}$ and linear operator $A$ is generator of a Gibbs semigroup. Note that for the autonomous Cauchy problem (ACP) (1.1), when $B(t)=B$, the outlined programme corresponds to the Trotter product formula approximation of the Gibbs semigroup generated by a closure of operator $A+B$, Zag2019 Ch.5.

The main result of the present paper concerns the non-autonomous Cauchy problem (nACP) (1.1) under the following

Assumptions:

(A1) The operator $A\geq\mathds{1}$ in a separable Hilbert space ${\mathfrak{H}}$ is self-adjoint. The family $\{B(t)\}_{t\in{\mathcal{I}}}$ of non-negative self-adjoint operators in ${\mathfrak{H}}$ is such that the bounded operator-valued function $(\mathds{1}+B(\cdot))^{-1}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is strongly measurable.

(A2) There exists ${\alpha}\in[0,1)$ such that inclusion: $\mathrm{dom}(A^{\alpha})\subseteq\mathrm{dom}(B(t))$, holds for a.e. $t\in{\mathcal{I}}$. Moreover, the function $B(\cdot)A^{-{\alpha}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is strongly measurable and essentially bounded in the operator norm:

(A3) The map $A^{-{\alpha}}B(\cdot)A^{-{\alpha}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is Hölder continuous in the operator norm: for some ${\beta}\in(0,1]$ there is a constant $L_{{\alpha},{\beta}}>0$ such that one has estimate

(A4) The operator $A$ is generator of the Gibbs semigroup $\{G(t)=e^{-tA}\}_{t\geq 0}$, that is, a strongly continuous semigroup such that $G(t)|_{t>0}\in\mathcal{C}_{1}(\mathfrak{H})$. Here $\mathcal{C}_{1}(\mathfrak{H})$ denotes the $\ast$-ideal of trace-class operators in $C^{*}$-algebra $\mathcal{L}(\mathfrak{H})$ of bounded operators on $\mathfrak{H}$.

In the present paper we essentially focus on convergence of the product approximants $\{U_{n}(t,s)\}_{(t,s)\in{\Delta},n\geq 1}$ to solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$. Let

for $k\in\{1,2,\ldots,n\}$, $n\in{\mathbb{N}}$, be partition of the interval $[s,t]$. Then corresponding approximants may be defined as follows:

It turns out that if the assumptions (A1)-(A3), adapted to a Banach space $\mathfrak{X}$, are satisfied for $\alpha\in(0,1)$, $\beta\in(0,1)$ and in addition the condition $\alpha<\beta$ holds, then solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$ admits the operator-norm approximation

for some constant $R_{\beta,\alpha}>0$. This result shows that convergence of the approximants $\{U_{n}(t,s)\}_{(t,s)\in{\Delta},n\geq 1}$ is determined by the smoothness of the perturbation $B(\cdot)$ in (A3) and by the parameter of inclusion in (A2), see NSZ2020 .

The Lipschitz case $\beta=1$ was considered for $\mathfrak{X}$ in NSZ2017 . There it was shown that if $\alpha\in(1/2,1)$, then one gets estimate

For the Lipschitz case in a Hilbert space $\mathfrak{H}$ the assumptions (A1)-(A3) yield a stronger result IT1998 :

Note that actually it is the best of known estimates for operator-norm rates of convergence under conditions (A1)-(A3).

The estimate (1.7) was improved in NSZ2019 for $\alpha\in(1/2,1)$ in a Hilbert space using the evolution semigroup approach Ev1976 ; How1974 ; Nei1981 . This approach is quite different from technique used for (1.9) in IT1998 , but it is the same as that employed in NSZ2017 .

Note that the condition ${\beta}>2{\alpha}-1$ is weaker than ${\beta}>{\alpha}$ (1.7), but it does not cover the Lipschitz case (1.8) because of condition $\beta<1$.

The main result of the present paper is the lifting of any known operator-norm bounds (1.7)-(1.10) (we denote them by $R_{\alpha,\beta}\varepsilon_{\alpha,\beta}(n)$) to estimate in the trace-norm topology $\|\cdot\|_{1}$. This is a subtle matter even for ACP, see Zag2019 Ch.5.4.
- The first step is the construction for nACP (1.1) a trace-norm continuous solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$, see Theorem 2.2 and Corollary 2.3.
- Then in Section 3 for assumptions (A1)-(A4) we prove (Theorem 1.4) the corresponding trace-norm estimate $R_{\alpha,\beta}(t,s)\varepsilon_{\alpha,\beta}(n)$ for difference $\|U_{n}(t,s)-U(t,s)\|_{1}$.

Note that solution of (1.1) is called classical if $u(t)\in C([0,T],\mathfrak{H})\cap C^{1}([0,T],\mathfrak{H})$, $u(t)\in\mathrm{dom}(C(t))$, $u(s)=u_{s}$, and $C(t)u(t)\in C([0,T],\mathfrak{H})$ for all $t\geq s$, with convention that $(\partial_{t}u)(s)$ is the right-derivative, see, e.g., Sob1961 , Theorem 1, or EngNag2000 , Ch.VI.9.

Since the involved into (A1), (A2) operators are non-negative and self-adjoint, equation (2.1) implies that the solution operator consists of contractions:

By (A1) $G(t)=e^{-tA}:\mathfrak{H}\rightarrow\mathrm{dom}(A)$. Applying to (2.1) the variation of constants argument we obtain for $U(t,s)$ the integral equation :

Hence evolution family $\{U(t,s)\}_{(t,s)\in{\Delta}}$, which is defined by equation (2.3), can be considered as a mild solution of nACP (2.1) for $0\leq s\leq t\leq T$ in the Banach space $\mathcal{L}(\mathfrak{H})$ of bounded operators, cf. EngNag2000 , Ch.VI.7.

Note that assumptions (A1)-(A4) yield for $0\leq s<t\leq T$, $\tau\in(s,t)$ and for the closure $\overline{A^{-\alpha}B(\tau)}$:

Then (2.2), (2.4) give the trace-norm estimate

and by (2.3) we ascertain that $\{U(t,s)\}_{(t,s)\in{\Delta}}\in\mathcal{C}_{1}(\mathfrak{H})$ for $t>s$.

Therefore, we can construct solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$ as a trace-norm convergent Dyson-Phillips series $\sum_{n=0}^{\infty}S_{n}(t,s)$ by iteration of the integral formula (2.3) for $t>s$. To this aim we define the recurrence relation

Since in (2.6) the operators $S_{n\geq 1}(t,s)$ are the $n$-fold trace-norm convergent Bochner integrals

by contraction property (2.2) and by estimate (2.5) there exit $0\leq s\leq t$ such that ${{M_{\alpha}}C_{\alpha}}{(t-s)^{1-\alpha}}/{(1-\alpha)}=:\xi<1$ and

Consequently $\sum_{n=0}^{\infty}S_{n}(t,s)$ converges for $t>s$ in the trace-norm and satisfies the integral equation (2.3). Thus we get for solution operator of nACP the representation

This result can be extended to any $0\leq s<t\leq T$ using (1.4).

We note that for $s\leq t$ the above arguments yield the proof of assertions in the next Theorem 2.2 and Corollary 2.3, but only in the strong (Yagi1989 Proposition 3.1, main Theorem in VWZ2009 ) and in the operator-norm topology, IT1998 Lemma 2.1. While for $t>s$ these arguments prove a generalisation of Theorem 2.2 and Corollary 2.3 to the trace-norm topology in Banach space $\mathcal{C}_{1}(\mathfrak{H})$:

We note that these results for ACP in Banach space $\mathcal{C}_{1}(\mathfrak{H})$ are well-known for Gibbs semigroups, see Zag2019 , Chapter 4.

Now, to proceed with the proof of Theorem 1.4 about trace-norm convergence of the solution operator approximants (1.6) we need the following preparatory lemma.

1. By virtue of (1.4) and (1.6) we obtain for difference in (1.11) formula:

Let integer $k_{n}\in(1,n)$. Then (3.1) yields the representation:

which implies the trace-norm estimate

2. Now we assume that $\lim_{n\rightarrow\infty}k_{n}/n=1/2$. Then (1.5) yields $\lim_{n\rightarrow\infty}t_{k_{n}}=(t+s)/2$, $\lim_{n\rightarrow\infty}t_{{n}}=t$ and uniform estimates (1.7)-(1.10) with the bound $R_{\alpha,\beta}\varepsilon_{\alpha,\beta}(n)$ imply

for $n\in{\mathbb{N}}$ and for some constants $R^{(1,2)}_{\alpha,\beta}>0$.

3. Since $\lim_{n\rightarrow\infty}k_{n}/n=1/2$ and $t>s$, by definition (1.6) and by Lemma 2.4 for contractions $\{V_{k}=e^{-\tfrac{t-s}{n}B(t_{k})}\}_{k=1}^{n}$ there exists $a_{1}>0$ such that

Similarly there is $a_{2}>0$ such that

4. Since for $t>s$ the trace-norm $c(t-s):=\|e^{-\tfrac{t-s}{2}A}\|_{1}<\infty$, by (3.2)-(3.6) we obtain the proof of the estimate (1.11) for

and $0\leq s<t\leq T$. $\Box$