Exercise 30, § 5

Let E be a complex Hilbert space. Denote by $\mathscr{T}_b$ the topology of bounded convergence on $\mathscr{L}(E)$.

a) Let $u$ be a partially defined self-adjoint operator on E. For every $\lambda \in \mathbf{C}-\mathbf{R}$ such that the imaginary part of $\lambda$ is negative, one has

b) The $\mathscr{T}_b$-resolvent topology is the coarsest topology on $\mathscr{A}(E)$ such that the mappings $u\mapsto e^{itu}$ of $\mathscr{A}(E)$ into $\mathscr{L}(E)$ are continuous for every $t\in \mathbf{R}$.