If $T$ is a bounded self-adjoint operator on a Hilbert space, is the set of convergent power series in $T$ closed in the norm topology?
I ask because I'm reading some spectral theorems and I was wondering what the space $\overline{ \mathbb{C}[T] }$ is like. It would be nice if it were just the set of convergent power series in $T$.
Edit: Actually this seems sort of unlikely since then every complex valued continuous functions on the spectrum of $T$ would be given by a power series, but I still haven't seen enough examples of spectra to rule this out. Even a self adjoint $T$ whose spectrum has a limit point would suffice to rule it out.
Jonas Meyer's comment gives an example of a self-adjoint $T$ whose spectrum is $[-1,1]$. Then we may take the absolute value function, for instance, or even any continuous function whose zeros accumulate in $(-1,1)$ to see that that the space of continuous functions on the spectrum contains functions which are not power series. Since $C(\sigma (T))$ is isomorphic as a $C^\ast$-algebra to $\overline{\mathbb{C}[T]}$, we see that we have $$\mathbb{C}[T] \subset \{ \text{power series in T} \} \subsetneq \overline{\mathbb{C}[T]}$$ so that the set of power series in $T$ is not closed.