pseudo-inverses in convenient ways, since such questions are linked to efficient finite rank approximations.

One way to measure the decay of singular values is to consider Schatten-von Neumann classes. More precisely, let $T$ be a linear operator. Then $T$ belongs to ${\mathscr I}_p$, the set of Schatten-von Neumann operators of order $p\in (0,\infty ]$, if and only if its singular values $\sigma _1(T),\sigma _2(T),\dots$ belong to $\ell ^p$. Since the singular values are non-negative and non-increasing, it follows that $$\label{SiVaSchattProp}
\begin{alignedat}{3}
\sigma _j(T) &= o (j^{-1/p}),&
\quad &\text{when} &\quad
T&\in {\mathscr I}_p,\ p<\infty ,
\\[1ex]
\sigma _j(T) &\neq o (j^{-1/p}),&
\quad &\text{when} &\quad
T&\notin {\mathscr I}_{p+{\varepsilon}},\ p<\infty ,\ {\varepsilon}>0,
\end{alignedat}$$ which indicates the link between Schatten-von Neumann classes and the decays of singular values.

It is in general a difficult task to find exact and convenient characterizations of Schatten-von Neumann classes. One is therefore left to find suitable necessary or sufficient conditions when characterizing such classes. For example a Toeplitz operator, acting on $L^2$ belongs to ${\mathscr I}_p$, $p\in [1,\infty]$, when its symbol belongs to $L^p$ (cf. [@HeWo; @BaCoIs; @BaIs]). For pseudo-differential operators ${\operatorname{Op}}(a)$, the situation is slightly different since ${\operatorname{Op}}(a)$ might not be in ${\mathscr I}_p$, $p\neq 2$, when 