Proposition 11

Let $\mathscr{T}$ be a locally convex topology on $\mathscr{L}(E)$ which is less fine than the Banach-space topology of $\mathscr{L}(E)$.

a) Let $f\in \mathscr{C}_0(\mathbf{R})$. For every sequence $(u_n)_{n\in\mathbf{N}}$ in $\mathscr{A}(E)$ which converges to $u$ for the $\mathscr{T}$ -resolvent topology, the sequence $(f(u_n))_{n\in\mathbf{N}}$ converges to $f(u)$ in the space $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}$;

b) Suppose that every $u\in \mathscr{A}(E)$ admits a countable fundamental system of neighbourhoods for the $\mathscr{T}$ -resolvent topology. The mapping of $\mathscr{C}_0(\mathbf{R})\times \mathscr{A}(E)$ into the space $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}$ defined by $(f, u)\mapsto f(u)$ is continuous.

Let us prove a). The topology $\mathscr{T}$ on $\mathscr{L}(E)$ is the supremum of the topologies defined by the seminorms continuous for the topology $\mathscr{T}$ (EVT, II, p. 26, Cor. and following remark). It is therefore enough to prove that for every seminorm $p$ on $\mathscr{L}(E)$ which is continuous for the topology $\mathscr{T}$, the sequence $p(f(u_n)-f(u))$ converges to 0 (TG, I, p. 51, Prop. 10).

Let $p$ be such a seminorm. Let $\mathscr{L}(E)_p$ denote the separated completed Banach space of the space $\mathscr{L}(E)$ endowed with the seminorm $p$, and let $\varpi$ denote the canonical mapping of $\mathscr{L}(E)$ into $\mathscr{L}(E)_p$; it is continuous, and one has $p(u) =\|\varpi (u)\|_p$ for every $u\in \mathscr{L}$ (E), where the norm is that of the space $\mathscr{L}(E)_p$.

Since $\mathscr{T}$ is less fine than the Banach-space topology of $\mathscr{L}$ (E), there exists $c\geqslant 0$ such that $p(u)\leqslant c\|u\|$ for every $u\in \mathscr{L}(E)$.

Suppose first that $f\in \mathscr{D}(\mathbf{R})$. Let $\widetilde{f}$ be an almost analytic extension of $f$. Let us denote by $\mu$ the Lebesgue measure on $\mathbf{C}$. By the Helffer–Sjöstrand formula (Theorem 2 of IV, p. 284), we have

for every $n\in \mathbf{N}$.

Let $\lambda \in \mathbf{C}-\mathbf{R}$. By assumption, the sequence of resolvents $R(u_n, \lambda )$ converges to $R(u, \lambda )$ in $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}$, hence the sequence $(\varpi (R(u_n, \lambda )))_n$ converges to $\varpi (R(u, \lambda ))$ in the Banach space $\mathscr{L}(E)_p$.

For every $n\in \mathbf{N}$, we have

(Proposition 17 of IV, p. 248), hence

Estimate (13) of IV, p. 281 shows that the right-hand side of this inequality is a bounded function for $\lambda \in \mathbf{C}-\mathbf{R}$; it is integrable on $\mathbf{C}$ since $\widetilde{f}$ has compact support. It follows from Lebesgue's theorem (INT, IV, p. 137, § 3, n$^o7$, Theorem 6) and from the Helffer–Sjöstrand formula applied to $f$ that $\varpi (f(u_n))$ converges to $\varpi (f(u))$, hence $p(f(u_n)-f(u))$ tends to 0.

Suppose now that $f\in \mathscr{C}_0(\mathbf{R})$. Let $\varepsilon >0$. There exists a function $f_{\varepsilon}$ in $\mathscr{D}(\mathbf{R})$ such that $\|f_{\varepsilon}-f\|_{\infty}\leqslant \varepsilon ($cf. Proposition 4, a) of IV, p. 202 and INT, III, p. 45, § 1, n$^o2$, Proposition 3). By Proposition 5, c) of IV, p. 275, we then have $\|f(u)-f_{\varepsilon}(u)\|\leqslant \varepsilon$ and $\|f(u_n)-f_{\varepsilon}(u_n)\|\leqslant \varepsilon$ for every $n\in \mathbf{N}$. Consequently, we obtain

for every $n\in \mathbf{N}$. Since $f_{\varepsilon}\in \mathscr{D}(\mathbf{R})$, the sequence $(p(f_{\varepsilon}(u_n)-f_{\varepsilon}(u)))_{n\in\mathbf{N}}$ converges to 0 by the preceding case, and therefore $p(f(u_n)-f(u))$ converges to 0. This completes the proof of a).

Let us prove assertion b). Under the assumption concerning $\mathscr{T}$, it follows from TG, IX, p. 17, Proposition 10 and the following remark, and from assertion a), that the mapping $u\mapsto f(u)$ from $\mathscr{A}(E)$ into $\mathscr{L}(E)$ is continuous when $f\in \mathscr{C}_0(\mathbf{R})$. Since the mappings $f\mapsto f(u)$ from $\mathscr{C}_0(\mathbf{R})$ into $\mathscr{L}(E)$ are continuous of norm $\leqslant 1$ for every $u\in \mathscr{A}(E)$ (Proposition 5, c) of IV, p. 275), assertion b) follows from TG, X, p. 13, Corollary 3.