Help with the notation $A-i\eta$, where $A$ is an operator and $\eta\in\mathbb{R}$

Question

Consider an operator $A:X\rightarrow Y$ and let $\eta\in\mathbb{R}$ with $\eta\neq 0$. I'm interested in understanding the meaning of this notation. I mean, what is $A-i\eta$? It is an operator or something else?

Furthermore, let $A-i\eta$ (whatever it is) be invertible, and let $(A-i\eta)^{-1}$ be the inverse. What is $(A-i\eta)^{-1}$? It reminds me of the resolvent of the operator $A$, but I am confused because of the imaginary unit. Could anyone help me or give some references?

Thank you in advance!

Answer 1

It's an operator. They mean $$A-i\eta=A-i\eta I,$$ where $I$ is the identity operator.

And yes, $(A-i\eta)^{-1}$ is the resolvent. The resolvent set is a subset of $\mathbb{C},$ so the imaginary unit doesn't violate anything (provided, of course, $i\eta$ is in the resolvent set).