I am trying to understand a statement that is in some notes that I am reading right now. It is the following.
"Let $T$ be a bounded, self-adjoint operator, $\eta\in\mathbb{R}, \eta\neq 0$ and let $H$ be an Hilbert space. It can easily proved that $(T-i\eta)^{-1}$ is bounded from $H$ to itself. Hence it is the resolvent of $T$."
I don't understand why the boundeness of $(T-i\eta)^{-1}$ implies that it is the resolvent of $T$. It is a sort of characterization?
I searched something on the web, but I didn't find anything. Could anyone please help? Also some references will be well accepted.
Thank you in advance!
If $T$ is bounded and self-adjoint, then $\sigma(T)\subseteq\mathbb{R}$ (see Spectrum of self-adjoint operator on Hilbert space real). However, the spectrum is defined as a subset of $\mathbb{C}.$ Taking complements, you get that $\mathbb{C}\setminus\mathbb{R}\subseteq \rho(T).$ That is, if $\eta\in\mathbb{R},$ then $i\eta\in\rho(T).$ Thus, the resolvent $R_\eta:=(T-i\eta)^{-1}$ is defined, and it is bounded by e.g. the bounded inverse theorem.
If $(T-i\eta)^{-1}$ exists, then it means that $i\eta$ is in the resolvent set of $T$, and $(T-i\eta)^{-1}$ is called the resolvent of $T$. It is the definition of the resolvent.