Suppose we have a normal matrix $A$ and a general matrix $B$. For a continuous function $f$ on the disk we can find upper-bounds on $\Vert[f(A),B]\Vert$ in terms of $\Vert[A,B]\Vert$. The more we know about $B$ and $f$ the more hope we have of a tight estimate. I work here with the operator norm, and add the restrictions $\left\Vert A\right\Vert \leq1$ and $\left\Vert B\right\Vert \leq1$ for simplicity.

If the spectrum of $A$ is in $[-1,1]$ then we are assuming $A$ is a Hermitian contraction. One method of attack uses the Fourier transform, where for nice $f$ we write $$ f^{\prime}(x)=\int_{-\infty}^{\infty}g(t)e^{itx}\, dx $$ and $$ \left[f(A),B\right]=\int_{-\infty}^{\infty}\frac{g(t)}{it}\left[e^{itA},B\right]\, dx. $$ From here we get quickly the estimate $$ \left\Vert \left[f(A),B\right]\right\Vert \leq\left\Vert \widehat{f^{\prime}}\right\Vert _{1}\left\Vert \left[A,B\right]\right\Vert . $$ Matt Hastings familiarized me with this trick. I suppose this is folklore. Uffe Haggerup thinks such a result and generalizations are in some old papers on derivations.

My current interest is in another special case, assuming $A$ has spectrum in the unit circle. Let's replace $A$ by $U$, now a unitary matrix. This is easier, as we can watch what happens for a trig-polynomial. Assume $f$ on the circle is given by $$ f(z)=\sum_{-N}^{N}a_{n}z^{n} $$ and use the estimate $$ \left\Vert \left[U^{n},B\right]\right\Vert \leq|n|\left\Vert \left[U,B\right]\right\Vert $$ to find $$ \left\Vert \left[f(U),B\right]\right\Vert \leq\sum_{-N}^{N}|n||a_{n}|\left\Vert \left[U,B\right]\right\Vert . $$ If we let $q(x)=f(e^{ix})$ then we can use classical Fourier series notation and can use the replacement $$ \left\Vert \widehat{q^{\prime}}\right\Vert _{1}=\sum_{-N}^{N}|n||a_{n}|. $$

This leads to a useful result.

For nice periodic functions $q$ we define $f$ so that $q(x)=f(e^{ix})$. The formula $$ \left\Vert \left[f(U),B\right]\right\Vert \leq\left\Vert \widehat{q^{\prime}}\right\Vert_{1}\left\Vert \left[U,B\right]\right\Vert $$ should hold true for at least all unitary matrices and all matrices $B$ of norm at most one. Where is this written down? I am flexible as to the meaning of nice here. Thrice differentiable handles most cases I find interesting.