Real L2 multipliers and unitary transport

Null-equivalent finite representatives give null-equivalent products; this applies to changes of $m$ as well as $f$, so domain and value are well-defined. The domain is a vector subspace. With $E_N=\{\mathrm{\mid }m\mathrm{\mid }\le N\}$ and $f_N={\mathbf{1}}_{E_N}f$, one has $\mathrm{\parallel }m{f}_N{\mathrm{\parallel }}_2\le N\mathrm{\parallel }f{\mathrm{\parallel }}_2$, so $f_N\in D(M)$. Finiteness of $m$ gives $E_N\uparrow {\mathbb{R}}^n$, and [F2] applied to ${\mathbf{1}}_{E_N^c}\mathrm{\mid }f{\mathrm{\mid }}^2\le \mathrm{\mid }f{\mathrm{\mid }}^2$ gives $f_N\to f$ in norm. This proves density. If $h,h^{\mathrm{\prime }}$ both satisfy the adjoint identity for $g$, then $⟨f,h-h^{\mathrm{\prime }}⟩=0$ on this dense domain; continuity in [F1] extends it to every $f\in H$, including $h-h^{\mathrm{\prime }}$, forcing $h=h^{\mathrm{\prime }}$.

By [F5], $V_t{V}_s=V_{t+s}$, $V_0=I$, $V_{-t}=V_t^{-1}$ and $⟨V_{t}f,V_{t}g⟩=⟨f,g⟩$. For fixed $f$, $\mathrm{\mid }{e}^{itm}-1{\mathrm{\mid }}^2\mathrm{\mid }f{\mathrm{\mid }}^2\to 0$ pointwise as $t\to 0$, with majorant $4\mathrm{\mid }f{\mathrm{\mid }}^2$. [F2] gives strong continuity at zero; the isometry and group law give it at every $t$. Countable choice permits the sequential criterion for these real-parameter norm limits.

Real-valuedness of $m$ and [F1] give $⟨Mf,g⟩=⟨f,Mg⟩$ for $f,g\in D(M)$, with both integrals absolutely convergent. Thus $D(M)\subseteq D(M^{\ast })$ with the same value. Conversely let $M^{\ast }g=h$. On $E_N$, $w_N={\mathbf{1}}_{E_N}(mg-h)$ is in $H$, and $m{w}_N\in H$ because $\mathrm{\mid }m\mathrm{\mid }\le N$ there, so $w_N\in D(M)$. Inserting $f=w_N$ into the adjoint identity gives $0=⟨M{w}_N,g⟩-⟨w_N,h⟩={\int }_{E_N}\mathrm{\mid }mg-h{\mathrm{\mid }}^2$. By [F4], $mg=h$ a.e. on each $E_N$. Their countable union is the whole space, so $mg=h$ a.e. globally; in particular $mg\in H$. Thus $D(M^{\ast })=D(M)$ and the operators agree.

If $mf\in H$, [F6] gives pointwise $(e^{itm}-1)f\mathrm{/}t\to imf$ and the squared error is at most $4\mathrm{\mid }mf{\mathrm{\mid }}^2$. [F2] proves norm convergence to $iMf$. Conversely, if the norm derivative exists, the quotients at $t=1\mathrm{/}(N+1)$ have bounded norms for $N\in \mathbb{N}$. Their squared moduli tend pointwise to $\mathrm{\mid }mf{\mathrm{\mid }}^2$ by [F6]. [F3] gives $\int \mathrm{\mid }mf{\mathrm{\mid }}^2\le {\mathrm{lim\hspace{0.17em}inf}}_N\mathrm{\parallel }(V_{1\mathrm{/}(N+1)}f-f)\mathrm{/}(1\mathrm{/}(N+1)){\mathrm{\parallel }}_2^2<\mathrm{\infty }$. Thus $f\in D(M)$, and the forward part identifies the derivative. This proves both directions, including points where $m=0$.

Since $U$ and $U^{-1}$ preserve norms and pairings, $U^{-1}D(M)$ is dense. For $g,h\in H$, the assertion $⟨Pf,g⟩=⟨f,h⟩$ for every $f\in D(P)$ is equivalent, by writing $v=Uf$, to $⟨Mv,Ug⟩=⟨v,Uh⟩$ for every $v\in D(M)$. By step 2.1 this holds exactly when $Ug\in D(M)$ and $Uh=MUg$. Therefore $D(P^{\ast })=D(P)$ and $P^{\ast }=P$. Conjugating the group identities and norm limits of steps 1.2 and 2.2 by $U$ proves the asserted unitary group, continuity, and both directions of the transported derivative-domain criterion. No spectral theorem or choice of a basis is used.