Isomorphic Hilbert Spaces and the Fourier Transform for the Circle

Verify the statements in Example 5.3: define $S : l^2 \to l^2$ by $S(\alpha_1, \alpha_2, \ldots ) = (0, \alpha_1, \alpha_2, \ldots )$. Then $S$ is an isometry that is not surjective.

Define $V : L^2(0, \infty ) \to L^2(0, \infty )$ by $(Vf)(t) = f(t-1)$ if $t > 1$ and $(Vf)(t) = 0$ if $t \leq 1$. Show that $V$ is an isometry that is not surjective.

Define $V : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ by $(Vf)(t) = f(t-1)$ and show that $V$ is an isomorphism (a unitary operator).

Let $\mathcal{H}$ be the Hilbert space of Example 1.8 (the collection of all absolutely continuous functions $f : [0,1] \to \mathbb {F}$ with $f(0) = 0$ and $f' \in L^2(0,1)$, with $\left\langle f, g \right\rangle = \int_0^1 f'(t)\overline{g'(t)} \, dt$) and define $U : \mathcal{H} \to L^2(0,1)$ by $Uf = f'$. Show that $U$ is an isomorphism and find a formula for $U^{-1}$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $u : X \to \mathbb {F}$ be an $\Omega$-measurable function such that $\sup \left\{ \left|u(x)\right| : x \in X\right\} < \infty$. Show that $U : L^2(X, \Omega , \mu ) \to L^2(X, \Omega , \mu )$ defined by $Uf = uf$ is an isometry if and only if $\left|u(x)\right| = 1$ a.e. $[\mu ]$, in which case $U$ is surjective.

Let $\mathscr {C} = \left\{ f \in C[0, 2\pi ] : f(0) = f(2\pi )\right\}$ and show that $\mathscr {C}$ is dense in $L^2[0, 2\pi ]$.

Show that $\left\{ (1/\sqrt{2\pi }), \, (1/\sqrt{\pi }) \cos nt, \, (1/\sqrt{\pi }) \sin nt : 1 \leq n < \infty \right\}$ is a basis for $L^2[-\pi , \pi ]$.

Let $(X, \Omega )$ be a measurable space and let $\mu , \nu$ be two $\sigma$-finite measures defined on $(X, \Omega )$. Suppose $\nu \ll \mu$ and $\phi$ is the Radon--Nikodym derivative of $\nu$ with respect to $\mu$ ($\phi = d\nu /d\mu$). Define $V : L^2(\nu ) \to L^2(\mu )$ by $Vf = \sqrt{\phi } f$. Show that $V$ is a well-defined linear isometry and $V$ is an isomorphism if and only if $\mu \ll \nu$ (that is, $\mu$ and $\nu$ are mutually absolutely continuous).

If $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces and $U : \mathcal{H} \to \mathcal{K}$ is a surjective function such that $\left\langle Uf, Ug \right\rangle = \left\langle f, g \right\rangle$ for all vectors $f$ and $g$ in $\mathcal{H}$, then $U$ is linear.