Exercise 29, § 5

Let E be a complex Hilbert space. We denote by $\mathscr{T}_b$ (resp. $\mathscr{T}_s,\mathscr{T}_f$) the Banach-space topology of $\mathscr{L}(E)$ (resp. the topology of simple convergence, the locally convex topology defined by the seminorms $u\mapsto$ $\langle x|u(y)\rangle$ for $(x, y)\in E\times E$).

a) Let $\lambda_0\in \mathbf{C}-\mathbf{R}$. The $\mathscr{T}_b$-resolvent topology (resp. the $\mathscr{T}_s$-resolvent topology) is the coarsest topology on $\mathscr{A}(E)$ for which the mapping $u\mapsto R(u, \lambda_0)$ of $\mathscr{A}(E)$ into $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}_b$ (resp. $\mathscr{T}_s$) is continuous.

b) The $\mathscr{T}_f$-resolvent topology coincides with the $\mathscr{T}_s$-resolvent topology.

c) Let $X\subset \mathscr{L}(E)$ be a bounded set of Hermitian endomorphisms of E. The topology induced on X by the $\mathscr{T}_b$-resolvent topology (resp. the $\mathscr{T}_s$-resolvent topology) coincides with the topology induced by the topology $\mathscr{T}_b$ ((esp. the topology $\mathscr{T}_s$).

d) The property of question a) does not hold for the $\mathscr{T}_f$-resolvent topology. e) Give an example of a sequence $(u_n)$ of partial self-adjoint operators which converges for the $\mathscr{T}_b$-resolvent topology and of a function $f\in \mathscr{C}_b(\mathbf{R})$ such that $f(u_n)$ does not converge to $f(u)$ in $\mathscr{L}(E)$.

f) Let $(u_n)$ be a sequence of partial self-adjoint operators on E and let $u$ be a partial self-adjoint operator on E. If there exists a dense subspace F of $E_u$, contained in dom($u_n$) for every $n$, such that $u_n(x)$ converges to $u(x)$ for every $x\in F$, then $(u_n)$ converges to $u$ for the $\mathscr{T}_s$-resolvent topology.