I've been doing a lot of research in areas around linear operators and I've found the following theorem. If it's useful I'll prove how I got it. If not I won't.

We can express the following:

And so therefore we can write:

If you can't notice is a linear operator; so:

This result is quite elaborate to prove and requires knowledge of Hilbert spaces. I just found this expression recently of a more general result that I am more interested in. We must remember this is right hand composition.

Nonetheless; is it easier to iterate a linear operator than how we usually do it? I can apply these methods for pentation and every hyper operator; formally; without considering convergence of the integrals. I'm still in the baby steps.

Questions, comments?

I can do everything I just did for iteration of any base as well. Not sure about convergence though. We can actually turn every super-function into iteration of a linear transformation. However; again; formally; not sure about convergence. I'm working on a paper that proves all of this but feedback helps; maybe someone's seen this.

Edit: What's cool about this is we do not require a fixpoint!

We can express the following:

And so therefore we can write:

If you can't notice is a linear operator; so:

This result is quite elaborate to prove and requires knowledge of Hilbert spaces. I just found this expression recently of a more general result that I am more interested in. We must remember this is right hand composition.

Nonetheless; is it easier to iterate a linear operator than how we usually do it? I can apply these methods for pentation and every hyper operator; formally; without considering convergence of the integrals. I'm still in the baby steps.

Questions, comments?

I can do everything I just did for iteration of any base as well. Not sure about convergence though. We can actually turn every super-function into iteration of a linear transformation. However; again; formally; not sure about convergence. I'm working on a paper that proves all of this but feedback helps; maybe someone's seen this.

Edit: What's cool about this is we do not require a fixpoint!