Exercise 31, § 5

Let E be a complex Hilbert space. Let $\mathscr{T}_s$ denote the topology of simple convergence on $\mathscr{L}(E)$. Let $\mathbf{H}$ denote the set of complex numbers of strictly positive imaginary part.

Let $(u_n)$ be a sequence in $\mathscr{A}(E)$. Suppose that

(i) There exists $\lambda_+\in \mathbf{H}$ such that $R(u_n, \lambda_+)$ converges in $\mathscr{L}(E)$ endowed with the

topology(ii) There exists$\mathscr{T}_s$; let$\lambda_-\in v\overline{\mathbf{H}}^+$ be its limit ;such that $R(u_n, \lambda_-)$ converges in $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}_s$;

(iii) The image F of the endomorphism $v_+$ is dense in E.

Let D be the open disk in $\mathbf{C}$ formed by the $\lambda \in \mathbf{H}$ such that

a) For every $\lambda \in D$, the series

converges in $\mathscr{L}(E)$. Moreover, for every $\lambda \in D$, the sequence $(R(u_n, \lambda ))_n$ converges to $r_{\lambda}$ in $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}_s$.

b) There exists a unique holomorphic mapping from $\mathbf{H}$ into $\mathscr{L}(E)$ which coincides on D with the mapping $\lambda \mapsto r_{\lambda}$. It is again denoted by $\lambda \mapsto r_{\lambda}$.

c) There exists a unique holomorphic mapping from $\mathbf{C}-\mathbf{R}$ into $\mathscr{L}(E)$ which coincides on $\mathbf{H}$ with the mapping $\lambda \mapsto r_{\lambda}$, and which satisfies $r^*_{\lambda}=r_{\overline{\lambda}}$ for every $\lambda \in \mathbf{C}-\mathbf{R}$. It satisfies $r_{\lambda_-}=v_-$, and for every $\lambda \in \mathbf{C}-\mathbf{R}$, the sequence $(R(u_n, \lambda ))$ converges to $r_{\lambda}$ in $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}_s$.

d) For all $\lambda$ and $\mu$ in C-R, one has $r_{\lambda}-r_\mu= (\mu-\lambda ) =r_\mu r_{\lambda}$ and $r_{\lambda}r_\mu=r_\mu r_{\lambda}$. e) The image of $r_{\lambda}$ is equal to F for every $\lambda \in \mathbf{C}-\mathbf{R}$.

f ) For every $\lambda \in \mathbf{C}-\mathbf{R}$, the endomorphism $r_{\lambda}$ is injective ; the partial operator $\lambda 1_E-r_{\lambda}^{-1}$ of domain F does not depend on the choice of $\lambda \in \mathbf{C}-\mathbf{R}$; it is denoted by $u$. g) The partial operator $u$ is self-adjoint ; for every $\lambda \in$ C-R, one has $R(u, \lambda ) =$ $r_{\lambda}$.

h) The sequence $(u_n)$ converges to $u$ for the $\mathscr{T}_s$-resolvent topology. (“Kato-Trotter theorem”)