I'm doing work with repelling (that is, non-contracting) linear operators on hilbert spaces, and I wondered if it might be worth my while to study my operators on a projectivization of my hilbert spaces, with the hopes that the repelling qualities of my operators will become contractions about the point at infinity. However, everything I've found on the subject is incomprehensible to me, being deeply enmeshed in either quantum mechanics or algebraic jibber-jabber. All I want is to know if the things I would like to do can indeed be done, and—if they can be done—the formulae/written expressions for how to do so.
Let $G$ be a compact abelian group, and let $\hat{G}$ be its pontryagin dual (a discrete group). I'm working with the hilbert spaces $L^{2}\left(G\right)$ and $L^{2}\left(\hat{G}\right)$ of square-integrable complex-valued functions on $G$ and $\hat{G}$, respectively (note: both spaces are of countably infinite dimension) with a bounded linear operator $A:L^{2}\left(G\right)\rightarrow L^{2}\left(G\right)$ and its “fourier conjugate” $\hat{A}=\mathscr{F}\circ A\circ\mathscr{F}^{-1}$, where $\mathscr{F}:L^{2}\left(G\right)\rightarrow L^{2}\left(\hat{G}\right)$ is the fourier transform. I write $\left\langle f\mid g\right\rangle _{G}$ and $\left\langle \hat{f}\mid\hat{g}\right\rangle _{\hat{G}}$ to denote the inner products on $L^{2}\left(G\right)$ and $L^{2}\left(\hat{G}\right)$, and write $\left\Vert f\right\Vert _{G}$ and $\left\Vert \hat{f}\right\Vert _{\hat{G}}$ to denote the norms. Finally, I write $\left\langle \cdot,\cdot\right\rangle :G\times\hat{G}\rightarrow\mathbb{R}/\mathbb{Z}$ to denote the pontryagin duality bracket between $G$ and $\hat{G}$, so that every continuous character on $G$ is of the form:$$\mathfrak{z}\in G\mapsto e^{2\pi i\left\langle x,\mathfrak{z}\right\rangle }\in\partial\mathbb{D}$$ for some $x\in\hat{G}$, and vice-versa for characters on $\hat{G}$.
Let me be clear: in all of the following questions, I'm looking for the formulae for how to do concrete, non-abstract computations (such as computing inner products, norms, fourier transforms, etc.), ideally in terms of how to do them over $L^{2}\left(G\right)$ and $L^{2}\left(\hat{G}\right)$, respectively.
Following the expected definition, define $P$ as the set of equivalence classes of $L^{2}\left(G\right)$ under the relation $f\sim g\Leftrightarrow\exists\lambda\in\mathbb{C}\backslash\left\{ 0\right\} :\textrm{ }f=\lambda g$. Likewise, let $\hat{P}$ denote the set of equivalence classes of $L^{2}\left(\hat{G}\right)$ under the relation $\hat{f}\sim\hat{g}\Leftrightarrow\exists\lambda\in\mathbb{C}\backslash\left\{ 0\right\} :\textrm{ }\hat{f}=\lambda\hat{g}$.
(1) Let $\mathcal{B}=\left\{ \beta_{n}\left(\mathfrak{z}\right)\right\} _{n\geq1}\subseteq L^{2}\left(G\right)$ be an orthogonal (but not necessarily orthonormal) basis for $L^{2}\left(G\right)$. How do I physically write an element of $P$ in terms of $\mathcal{B}$? What kinds of algebraic operations can I do with points in $P$ (addition, scalar multiplication, inner products, norms, etc.), and, given the aforementioned concrete expression for points in $P $ with respect to $\mathcal{B}$, how do I write out the results of these operations (that is, what are the formulae for the results in terms of the inputs)?
(2) I have a formula that gives $A\left\{ \beta_{n}\right\} \left(\mathfrak{z}\right)$ as a linear combination of the elements of $\mathcal{B}$. How would I go about physically writing the analogue of this formula in $P$?
(3) Wikipedia says that there is a “natural” way to define a metric on $P$ and $\hat{P}$, respectively. What are the formulae for these metrics (preferably, in terms of the inner products/norms on $L^{2}\left(G\right)$ and $L^{2}\left(\hat{G}\right)$, respectively?)
(4) Does the pontryagin duality formulation of characters and the fourier transform carry over to $P$ and $\hat{P}$ in any way? If so, what are the formulae for the analogues of these things in the projectivization? Do things like plancherel's theorem and parseval's theorem have analogues, as well?
For reference, I write the fourier and inverse fourier transforms like so:$$\mathscr{F}\left\{ f\right\} \left(x\right)=\int_{G}f\left(\mathfrak{z}\right)e^{-2\pi i\left\langle x,\mathfrak{z}\right\rangle }d\mathfrak{z} $$
$$\mathscr{F}^{-1}\left\{ \hat{f}\right\} \left(\mathfrak{z}\right)=\sum_{x\in\hat{G}}\hat{f}\left(x\right)e^{2\pi i\left\langle x,\mathfrak{z}\right\rangle }$$
Finally, let me be clear, I am not asking to be given an exercise in projective geometry or a condescending lecture about how I need to "toughen up". I am simply looking for an explanation (or reference) for how to do something that I do not know how to do, so that I move on with my investigations. As an example of what I mean, if I were asking "how do I take the fourier transform of a complex-valued functions on the $p$-adic integers", an ideal answer would be something along the lines of these wonderful notes by Jordan Bell.
Finally, for reference, I'm doing analytic number theory. As such, in your answers, you can assume that I know absolutely nothing about differential geometry, and even less about abstract algebra.
Anyhow, thanks in advance for any assistance with answering some or all of these, admittedly, rather pedantic, questions of mine.
