This problem is from "A Course on Functional Analysis II" by Zhang Gongqing and Guo Maozheng, which is the textbook of a functional analysis course I'm taking.
Precisely speaking, let T(t) be a strong continuous operator semigroup on Hilbert space H, with T(t) normal for any t. Let A be the generator of the semigroup and prove that A is also normal.
The hint on the book suggests us to use Gelfand representation, but I have no idea about how to use it. Other ways seem also difficult to me. Does anyone have some idea?