(06/05/2021, 12:12 AM)MphLee Wrote: I'm a bit lost tbh. What H* can look like in your opinion? Whatever it is, it is applied to exponentiation so it should be a good news.
Also operators are linear, usually, so Idk how to manage non-linearity inside an ambient, that of vector spaces/Hilbert spaces, where things should be linear.
But I'm curious of what you can carve out of this. Even if you don't get result I believe that what we are doing here is usefull, brushing the dust from the hidden spots, making clear what the mechanisms are.
I went full speed on categories and non-commutative groups just for that reason. Because I wanted a landscape where I couldn't care less about linearity because we don't want to start with a god-given abelian ground (i.e. linearity) by default.
Hf's adjoint H* depends on f; that's really about the most of it.
So we'd be talking about \( H^*_f x^{-z} \).
We're really just looking for \( g_s^z \) such that,
\(
(f,g_s^z) = \uparrow^s f\\
\)
I've screwed up some of my capitals a bit; but think we're trying to solve the equation with an inner product....
Give me a bit Mphlee; I can explain what I mean by adjoint in full detail. This is pretty straightforward, I think this is really important.