Let $T$ be a bounded operator on a Hilbert space. Let $L$ be another bounded linear operator satisfying $\|L\|<\varepsilon$. What can be said about the relationship between the spectra of $f(T),f(T+L)$ for $f\in \mathbb C[x]$ or between their operator norms?
I am motivated by the fact that for continuous $f$ and small $h$ we have $f(t+h)\approx f(t)$.
Since $L$ has a small spectral radius it seems reasonable to expect that $f(T+L)$ will have a similar spectrum to $f(T)$. I don't know how to tackle this though. Maybe some clever applications of the spectral mapping theorem and the functional calculus?