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I would like to turn an equation (function) like 4x

^{2}+2 into an operator, L, acting on x. Lx.

I want to get into operator algebra so that I can manipulate equations like √(4x

^{2}+2) using operators (e.g. the last equation could be RLx).

I'm finding difficulties:-

1. Many common functions/equations (like the above) add an integer at the end. I don't really know what I'm talking about but aren't these some kind of affine operation, making them nonlinear. And how would you go about turning functions into operators anyway?

2. Everytime I try to look up operator algebra on the web I am met with very advanced double dutch on banach spaces, operator spectra and such like.

3. No-one seems to be talking about manipulating collections of operators using commutativeness or other techniques and how to set those operators up. I think you can understand what I'm trying to get at. If I did succeed in turning function f and function g into two operators F and G. I could look at FG and, say, find that they commuted to make GF-x, then use that identity to simplify function compositions. This I what I've heard about operators; you can manipulate them without even worrying about the x or whatever they are operating upon.

I really hope someone out there can shed some light on this or point me in the right direction or even just make some suggestions for further researches.

Thanks for your time reading this. Cheers.