Image of a bounded linear operator under an odd continuous function.

Question

I have the following exercise to solve:

Let $H$ be a complex Hilbert-space and $\mathbf{B}(H)$ denote the set of bounded linear operators $H \to H$. Let $a, b, x \in \mathbf{B}(H)$ with $x^* = x$ and $ax = xb$ and $xa = bx$.

Let $f : \sigma (x) \to \mathbf{C}$ be an odd continuous function. Then $a f(x) = f(x) b$.

I have no idea whatsoever how to start this exercise. Can anyone give me a hint how to tackle it?

Thanks!

Answer 1

Hint: Show that this holds when $f$ is a polynomial whose terms are all odd. Next, show that any continuous odd $f$ can be approximated uniformly by such polynomials. Then use the definition of the functional calculus to conclude.