and ${\mathfrak{m}_\varphi}= {\mathfrak{n}_\varphi}^\ast {\mathfrak{n}_\varphi}$.

Integral operators
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Let $(X,d x)$ be a measure space. A (bounded) integral operator $A$ on $L^2(X,d x)$ is a bounded operator for which there is a measurable function $(s,t) \mapsto A_{s,t}$ (called the kernel of $A$) on $X \times X$ such that, for all $\xi \in L^2(X,d x)$,

-   $t \mapsto A_{s,t} \xi(t) \in L^1(X,d x)$ for almost every $s \in X$.

-   $A \xi (s) = \int_X A_{s,t} \xi(t)$ for almost every $s \in X$.

By Fubini the kernel is only defined up to an almost everywhere zero function. The set of integral operators is a self-adjoint subspace of $B(L^2(X))$, and the kernel of the adjoint is given by $(A^*)_{s,t} = \overline{A_{t,s}}$. The Schatten ${\mathcal{S}}_2$-class consists of all integral operators with kernel belonging to $L^2(X \times X)$, and for such operator, $$\|A\|_{{\mathcal{S}}_2(L^2X)} = \left(\iint |A_{s,t}|^2 ds dt\right)^{1/2}.$$ More generally if $A,B \in {\mathcal{S}}^2(L^2X)$, $$\label{eq=formula_for_trace} {\mathrm{Tr}}(AB) =\iint A_{s,t} B_{t,s} dt ds.$$

Schur multipliers {#subsection=Schur}
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Let $(X, \mu)$ be a measure space and let $\psi: X \times X \rightarrow \mathbb{C}$ be an essentially bounded measurable function. We identify ${\mathcal{S}}_2(L^2 X )$ linearly with $L^2(X \times X)$ by the previous paragraph. Then, we obtain a bounded 