Have there been any reasearch for these operators. Let's say we have an operator $\langle|\rangle$ (No it is not the bra-ket notation from quantum mechanics) such that-
$\textstyle\displaystyle{\begin{align}\sum_{i\in A}a_ib_i=&(\sum_{i\in A}a_i)\langle|\rangle(\sum_{i\in A}b_i)\end{align}}$
Where, $|A|\leq\aleph_0$
Let's call this operator,uhhhh..., Euler's Dream. I mean that's a cool name.
And, similarly let's say we have another operation $|||$ such that-
$\textstyle\displaystyle{(\sum_{i\in A}a_i)(\sum_{i\in A}b_i)=\sum_{i\in A}(a_i|||b_i)}$
Where, $|A|\leq\aleph_0$
Let's call this one Newton's Dream.
If we have the definition of these operators, and also the inverse of them. Then it would have applications almost everywhere. Summations are needed everywhere and having an operation which can decompose or combine them would be game changing.
I don't have enough expertise to figure out any property of $\langle|\rangle$. But it seems weird that no mathematician in the last few centuries has imagined such dream-like operations and has tried to find definitions.
So I am asking has there been any research on figuring out some properties or possible definitions for type of operators? And if there has been research then I would request the links for them.
And obviously, if there are some trivial properties that we can work out, then please put them in the answer.
 A: Those "operators" are not in fact well-defined - even if we restrict attention to two-element sums (i.e. $A=\{1,2\}$).
I'll use the notation "$\star_E$" and "$\star_N$" for your "$\langle\vert\rangle$" and "$\vert\vert\vert$" respectively.

Take $\star_E$ first. On the one hand, setting $$a_1=a_2=0, b_1=1,b_2=2$$ we get $a_1b_1+a_2b_2=0$, $a_1+a_2=0$, and $b_1+b_2=3$. This means we would need to have $$0\star_E3=0.$$ On the other hand, setting $$a_1'=1,a_2'=-1$$ (and keeping $b_1=1,b_2=2$) we get $a_1'b_1+a_2'b_2=-1$, $a_1'+a_2'=0$, and $b_1+b_2=3$, which means we would need to have $$0\star_E3=-1.$$

It's a good exercise to similarly show that there is no operation $\star_N$ satisfying $$(x+y)(z+w)=(x\star_N z)+(y\star_N w)$$ for all $w,x,y,z\in\mathbb{R}$. (HINT: first think about what happens if we set $x=y=0$ and allow $z,w$ to be arbitrary ...)

 Suppose $\star_N$ satisfied the above property. If $x=y=0$ then $(x+y)(z+w)=0$ and so $$(0\star_Nz)+(0\star_Nw)=0$$ regardless of what $z,w$ are. Consequently we get $$0\star_Na=0$$ for every $a$. Similarly we can show that $a\star_N0=0$ for every $a$. But now everything breaks down: for instance, we get $$1=(0+1)(1+0)=(0\star_N1)+(1\star_N0)=0+0=0.$$

