the non-weighted case, essentially state that any matrix $A\in \mathbb U^{p_0}$ can be factorized as $$\label{MatrixFaktor0}
A=A_1\cdot A_2,
\quad \text{when}\quad
A_j\in \mathbb U^{p_j},\quad
\frac 1{p_1}+\frac 1{p_2}= \frac 1{p_0}.$$ From these relations we obtain $$\label{embSimpleCase}
\mathbb U^{p}\subseteq {\mathscr I}_{p},\quad
\text{when}\quad p\in (0,2].$$

In fact, the set of Hilbert-Schmidt operators on $\ell ^2$ agrees with $\mathbb U^2$, and with ${\mathscr I}_2$ (also in norms). Consequently, $\mathbb U^{2}
={\mathscr I}_2$, and H[ö]{}lder’s inequality for Schatten-von Neumann classes together with give that for every $A\in \mathbb U^{2/N}$, with integer $N\ge 1$, there are matrices $A_1,\dots ,A_N\in \mathbb U^{2}$ such that $$A = A_1\cdots A_N \in \mathbb U^{2}\circ \cdots \circ \mathbb U^{2}
= {\mathscr I}_2\circ \cdots \circ {\mathscr I}_2 = {\mathscr I}_{2/N}.$$ Hence $\mathbb U^{2/N}\subseteq {\mathscr I}_{2/N}$ for every integer $N\ge 1$. A (real) interpolation argument between the cases $$\mathbb U^{2/N}\subseteq {\mathscr I}_{2/N}\quad
\text{and}\quad
\mathbb U^{2} =  {\mathscr I}_{2}$$ now shows that that $\mathbb U^{p}\subseteq {\mathscr I}_{p}$ when $p\in [2/N,2]$. Since $2/N$ can be chosen arbitrarily close to $0$, follows.

In Section \[sec2\], the previous arguments are used to deduce more general versions of involving weighted spaces. (See Theorem \[MatrixSchatten\].)

In Section \[sec5\] we show some applications and other results for Schatten-von Neumann symbols. Here 