Why and how can any admissible unbounded self-adjoint operator be converted to a bounded one by taking its arctangent?

Question

My quantum information theory professor mentioned as a passing remark that:

You can convert any admissible unbounded, self-adjoint operator to a bounded one by taking its arctangent.

Could someone explain this to me? And how to prove this? Or could someone provide me a reference where I can look this up? I'm looking for something at the beginners level, because I do not have formal training in functional analysis.

Answer 1

Borel functional calculus for (potentially unbounded) self-adjoint operators ensures that for every such operator $D$ there is a self-adjoint operator $\arctan D$. Now, one of the properties of functional calculus is that $\lVert f(D)\rVert = \mathop{\mathrm{ess\,sup}}_{\lambda\in\sigma(D)}|f(\lambda)|$, so $\lVert \arctan D\rVert\leqslant \pi/2$.