# Is the following operator a positive operator?

Let $$H_{1}$$ and $$H_{2}$$ be two Hilbert spaces. Now let $$A \in B(H_{1})$$ and $$B_{ij} \in B(H_{2})$$ for all i and j be positive operators. i.e. $$\forall \phi \in H_{1}$$ and $$\forall \psi \in H_{2}$$

i.e. $$\langle \phi ,A \phi\rangle \geq 0$$ and $$\langle \psi, B_{ij}\psi\rangle\geq 0$$ for all i and j. I am wondering if the following operator is also a positive operator. Let us write $$A$$ using some arbitrary basis that spans $$H_{1} \{\nu_{n}\}_{n}$$. $$A \rightarrow \sum_{nm}a_{nm}\nu_{n}\otimes \nu_{m}.$$

I am wondering if the following operator acting over $$H_{1}\otimes H_{2}$$ is also a positive operator.

$$OP:=\sum_{nm}a_{nm}\big(\nu_{n}\otimes \nu_{m}\big)\otimes B_{nm}.$$

I have tried starting with an arbitrary element of $$H_{1}\otimes H_{2}$$ call it $$\omega$$ and using the fact that it may be represented as a schmidt decomposition. Let us assume infinite dimensional Hilbert spaces.

$$\omega = \sum_{i} \beta_{i}f_{i}\otimes g_{i}$$ where $$\{f_{i}\}_{i}$$ and $$\{g_{i}\}_{i}$$ are respectively basis whose span is respectively dense in $$H_{1}$$ and $$H_{2}$$.

Computing $$\langle \omega , OP \omega\rangle$$ I am not able to attain anythig useful. I get $$\langle \omega , OP \omega\rangle = \sum_{i,n,m} a_{nm}|\beta_{i}|^{2}\langle f_{i},\nu_{n}\rangle \langle \nu_{m},f_{i}\rangle \langle g_{i}, B_{nm}, g_{i}\rangle$$

Thank you very much in advance for any help.

In case $$H_1=\mathbb C^n$$, $$A$$ is the matrix with $$a_{i,j}=1$$, for all $$i$$ and $$j$$, and $$H_2=\mathbb C$$, your question is asking whether a matrix with positive entries is a positive operator. The answer to this is of course no!
• A is a positive operator. Also, all of the operators $B_{ij}$ are positive. I never said anything about their entries. The A you defined is not a positive operator while the one I have defined is so your example is urelated. Thank you for your attention nevertheless. Jun 19, 2022 at 0:35