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.
