Let


* $U$, $H$, $\tilde H$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
* $Q$ be a self-adjoint and nonnegative nuclear linear operator on $U$
* $\Psi$ be a Hilbert-Schmidt operator from$^1$ $Q^{1/2}U$ to $H$
* $\tilde Q:=\left(\Psi Q^{1/2}\right)\left(\Psi Q^{1/2}\right)^\ast$
* $\Phi$ be a Hilbert-Schmidt operator from $\tilde Q^{1/2}H$ to $\tilde H$


Note that $\tilde Q$ is a self-adjoint and nonnegative nuclear linear operator on $H$ and $$\Psi Q^{1/2}U=\tilde Q^{1/2}H.\tag1$$



> 
> By $(1)$, the composition $\Phi\Psi$ is well-defined. How can we show that $$\left\|\Phi\Psi\right\|\_{\operatorname{HS}\left(Q^{1/2}U,\:\tilde H\right)}^2\le\left\|\Phi\right\|\_{\operatorname{HS}\left(\tilde Q^{1/2}H,\:\tilde H\right)}^2\operatorname{tr}\tilde Q\tag2,$$ where $\left\|\;\cdot\;\right\|\_{\operatorname{HS}}$ denotes the Hilbert-Schmidt norm and $\operatorname{tr}$ the trace functional?
> 
> 
> 


**EDIT**:
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Maybe it's useful to note that $$\operatorname{tr}\tilde Q=\left\|\Psi\right\|\_{\operatorname{HS}(Q^{1/2}U,\:H)}^2=\left\|\Psi Q^{1/2}\right\|\_{\operatorname{HS}(U,\:H)}^2\tag3.$$




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$^1$ As usual, $Q^{1/2}U$ is equipped with $$\langle u,v\rangle\_{Q^{1/2}U}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle\_U\;\;\;\text{for }u,v\in Q^{1/2}U,$$ where $Q^{-1/2}$ denotes the pseudo-inverse of $Q^{1/2}$.



Estimate for the composition of two Hilbert-Schmidt operators
We'll need the following fact: Let $U\_i,H$ be $\mathbb R$-Hilbert spaces and $A\_i\in\mathfrak L(U\_i,H)$ with $$A\_1A\_1^\ast=A\_2A\_2^\ast.\tag4$$ Then, $$\iota\_{12}:=A\_2^{-1}A\_1\operatorname P\_{(\ker A\_1)^\perp}$$ and $$\iota\_{21}:=A\_1^{-1}A\_2\operatorname P\_{(\ker A\_2)^\perp}$$ are well-defined partial isometries from $U\_1$ to $U\_2$ and $U\_2$ to $U\_1$, respectively, with $$


1. $A\_1=A\_2\iota\_{12}$
2. $A\_2=A\_1\iota\_{21}$
3. $A\_1^{-1}=\iota\_{21}A\_2^{-1}$
4. $A\_2^{-1}=\iota\_{12}A\_1^{-1}$
5. $\iota\_{12}^\ast=\iota\_{21}$
6. $\iota\_{21}^\ast=\iota\_{12}$
7. $\iota\_{21}\iota\_{21}^\ast$ is the orthogonal projection from $U\_1$ onto $(\ker A\_1)^\perp$
8. $\iota\_{12}\iota\_{12}^\ast$ is the orthogonal projection from $U\_2$ onto $(\ker A\_2)^\perp$


In particular, if $U$ is a $\mathbb R$-Hilbert space and $A\in\mathfrak L(U,H)$, then $$\iota:=(AA^\ast)^{-1/2}A\operatorname P\_{(\ker A)^\perp}$$ is a well-defined isometry from $U$ to $H$ with 


9. $A=(AA^\ast)^{1/2}\iota$
10. $(AA^\ast)^{1/2}=A\iota^\ast$
11. $\iota^\ast=A^{-1}(AA^\ast)^{1/2}\operatorname P\_{\overline{AU}}$




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In the situation of the question, we choose $A=\Psi Q^{1/2}$ and obtain $$\left\|\Phi\Psi Q^{1/2}\right\|\_{\operatorname{HS}(U,\:\tilde H)}\le\left\|\Phi\tilde Q^{1/2}\right\|\_{\operatorname{HS}(U,\:H)}\left\|\iota\right\|\_{\mathfrak L(U,\:H)}\tag5$$ and clearly $\left\|\iota\right\|\_{\mathfrak L(U,\:H)}\le1$.


