Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform?

Question

Pretty much the title says it all. For $f$, $g$ in $S(\mathbb{R})$ one has

$$ \langle f, g \rangle = \int_{-\infty}^{\infty} f(x)g(x) dx = \int_{-\infty}^\infty \hat{f}(k)\hat{g}(k) dk = \langle \hat{f},\hat{g} \rangle \quad \quad (1) $$

where $\hat{f}$ is the Fourier transform of $f$.

Are there other function transforms with the property (1) above? If there are, is there some kind of generalized Fourier transform that encompasses them all? If there aren't, could you provide a link to the proof?

What is a good reference to read about this subject?

Edit 1: As Yiorgos showed below, it is rather trivial to find a host of $T$s such that

$$\langle f,g\rangle = \langle Tf, Tg \rangle \,. $$

However, all these $T$s satisfy that there exists an $n$ such that

$$T^n = 1 \,. \quad\quad (2)$$

This is also true for the Fourier transform. A $T$ that does not satisfy (2) and still preserves the scalar product is simply

$$Tf = f(x+a) \,.$$

Are there any other examples of $T$ that do not satisfy $T^n = 1$? I would be specifically interested in $T$s that, in analogy with the Fourier transform, do NOT preserve the "shape" of $f$.

Edit 2: Maybe I should create another question given that the original one has been answered.

Anyhow, as Yiorgos points out, one can find unitary transformations $T(\alpha)$ in general parametrized by an infinite number of phases which I collectively represent here as $\alpha$. These $T$ do not satisfy $T^n = 1$.

On the other hand, any two such $T$ commute:

$$T(\alpha_1) \cdot T(\alpha_2) = T(\alpha_2) \cdot T(\alpha_1) $$

The new question is then, does there exist a family of unitary transformations on $S(\mathbb{R})$ as above - say parametrized by some parameters collectively denoted as $\beta$ - such that $T(\beta_1)$ and $T(\beta_2)$ do not commute for some values of $\beta_1$ and $\beta_2$?

Answer 1

A lot of them, i.e., $$ (Tf)(x)=f(-x), $$ $$ (Tf)=-f, $$ Take $g\in \mathscr{S}(\mathbb R)$, with $\langle g,g\rangle=1$, and define $$ Tf=f-2\langle f,g\rangle g. $$

Also, take $g_1,g_2\in\mathscr S(\mathbb R)$, with $$ \langle g_1,g_1\rangle=\langle g_2,g_2\rangle=1\quad\text{and}\quad\langle g_1,g_2\rangle=0. $$ Define $$ T_af=f-\langle f,g_1\rangle g_1-\langle f,g_2\rangle g_2 \\+(\cos a\langle f,g_1\rangle+\sin a\langle f,g_2\rangle)g_1+(-\sin a\langle f,g_1\rangle+\cos a\langle f,g_2\rangle)g_2. $$ Clearly $T^n_a=T_{na}$, and if $2\pi/a$ irrational, then $T^n\ne I$, for anny $n$.

Now, more generally, if $\{e_n(x)\}_{n\in\mathbb N}\subset\mathscr S(\mathbb R)$, is an orthnormal basis of $L^2(\mathbb R)$, then $$ u=\sum_{n\in\mathbb N}\langle u,e_n\rangle e_n. $$ Let $a_n\in\{-1,1\}$ (in general $a_n\in\mathbb C$ and $|a_n|=1$) and define $$ Tu=\sum_{n\in\mathbb N}a_n\langle u,e_n\rangle e_n. $$ Clearly $T$ is also unitary and $$ (Tu)(x)=\sum_{n\in\mathbb N}a_n\left(\int_{-\infty}^\infty u(y)\,\bar e_n(y)\,dy\right)e_n(x) =\int_{-\infty}^\infty K(x,y)\,u(y)\,dy, $$ where $$ K(x,y)=\sum_{n\in\mathbb N}a_n\, e_n(x)\,\bar e_n(y). $$

Answer 2

Upon reflection, I suspect that the question may not quite be the one intended.

That is, in my interpretation, the question is about maps $T$ of the Schwartz space to itself which preserve the $L^2$ norm. Such $T$ extends by continuity to a map $L^2\to L^2$ which preserves $L^2$-norm and stabilizes the Schwartz space.

Taking that as the genuine question, recall that (for example) the so-called "quantum harmonic oscillator" (operator) $H=-\Delta+x^2$ (with suitable discussion of self-adjoint extension...) has an orthonormal basis of eigenfunctions in $L^2$, of the form $u_n=$ hermite-polynomial times suitable Gaussian. These eigenfunctions are also in the Schwartz space.

Maps such as "interchange $u_{n_1}$ and $u_{n_2}$" are "unitary" in the sense of the question, and are continuous in the (projective limit) topology on Schwartz functions. Certainly not all such (finite) permutations of the $u_n$'s commute...

EDIT: Yes, these finite permutations are of finite order. Such permutations are obviously also continuous on the Schwartz space.

In fact, a one-sided shift $T(\sum_n c_n\cdot u_n)=\sum_n c_n\cdot u_{n+1}$ is unitary on $L^2$, stabilizes the Schwartz space, and is continuous on the Schwartz space (though not stabilizing individual Sobolev-like spaces). These operators are not of finite order, and do not commute with the finite permutations.