A classic theme in analysis is the relation between the smoothness of a function and the speed by which it can be approximated by polynomials. For functions of operators, we can measure smoothness in the operator norm and the relation between smoothness and approximation is expressed by Peller's Bernstein inequality. There are other norms on operators, however, in particular the $p$-Schatten ideal $\mathcal{L}_p$ is the set of compact operators $T$ such that $\|T\|_p := \mathrm{Tr}(|T|^p)^{1/p} < \infty$ is a Banach space when $1\leq p<\infty,$ and a quasi-Banach space when $0<p<1.$ Particularly in the quasi-Banach range, most familiar tools are not useful and we have needed to employ techniques from wavelet analysis. I will describe some of my results in this area including Besov-type conditions for Lipschitz and Holder inequalities for functions of operators.
