# Banach holomorphy, generators, and resolvent limits ## OA-MOD-SG-04 — Stone's theorem with the derivative domain **Theorem.** Let \(U:\mathbb R\to B(H)\) be a strongly continuous unitary group. There is a unique self-adjoint operator \(A\) such that \[ U_t=e^{itA}\qquad(t\in\mathbb R). \tag{SG.8} \] Its domain and action are exactly \[ \begin{aligned} D(A)&=\left\{x:\lim_{h\to0}\frac{U_hx-x}{h} \text{ exists in }H\right\},\\ iAx&=\lim_{h\to0}\frac{U_hx-x}{h}. \end{aligned} \tag{SG.9} \] Conversely, the spectral group of every self-adjoint \(A\) is strongly continuous and has (SG.9) as its derivative domain. **Proof.** Define \(A\) by (SG.9). For \(\varepsilon>0\), set \[ T_\varepsilon x=\frac1\varepsilon\int_0^\varepsilon U_t x\,dt. \tag{SG.10} \] A difference quotient and translation of the integral give \[ T_\varepsilon H\subseteq D(A),\qquad iAT_\varepsilon=\frac{U_\varepsilon-I}{\varepsilon}. \tag{SG.11} \] Strong continuity gives \(T_\varepsilon x\to x\), so \(D(A)\) is dense. For \(x\in D(A)\), the group law gives \[ \frac d{dt}U_tx=iU_tAx=iAU_tx. \tag{SG.12} \] If \(x_j\to x\) and \(Ax_j\to y\), integrate (SG.12) and pass to the limit: \[ U_tx-x=i\int_0^tU_sy\,ds. \] The derivative at zero puts \(x\in D(A)\) and gives \(Ax=y\). Thus \(A\) is closed. Differentiating \(\langle U_tx,U_ty\rangle\) at zero shows \[ \langle Ax,y\rangle=\langle x,Ay\rangle \quad(x,y\in D(A)), \] so \(A\) is symmetric. The two norm integrals \[ B_+x=i\int_0^\infty e^{-t}U_{-t}x\,dt, \qquad B_-x=-i\int_0^\infty e^{-t}U_tx\,dt \tag{SG.13} \] exist and have norm at most one. Changing variables in the difference quotients gives \[ \frac{U_hB_+x-B_+x}{h}\longrightarrow-B_+x+ix, \qquad \frac{U_hB_-x-B_-x}{h}\longrightarrow B_-x+ix. \] Therefore \[ (A-i)B_+=I,\qquad (A+i)B_-=I. \tag{SG.14} \] Both \(A-i\) and \(A+i\) are onto. If \(y\in D(A^*)\), choose \(x\in D(A)\) with \((A-i)x=(A^*-i)y\). Then \(y-x\in\ker(A^*-i)=\operatorname{ran}(A+i)^\perp=0\). Hence \(D(A^*)\subseteq D(A)\), and symmetry gives \(A=A^*\). Let \(V_t=e^{itA}\), using the spectral calculus. Both \(U_t\) and \(V_t\) preserve \(D(A)\), commute there with \(A\), and have derivative \(iA\). For \(x\in D(A)\), differentiation of \(U_{t-s}V_sx\) with respect to \(s\) gives zero. Thus \(U_tx=V_tx\); density proves (SG.8) on all of \(H\). Conversely, scalar dominated convergence proves strong continuity of \(e^{itA}\). It also gives the derivative \(iAx\) for \(x\in D(A)\). If the difference quotient converges for an arbitrary \(x\), its norms are bounded along a sequence \(h\to0\). Fatou's lemma applied to \[ \left|\frac{e^{ih\lambda}-1}{h}\right|^2 \longrightarrow\lambda^2 \] puts \(x\) in the spectral domain of \(A\). This proves the converse domain in (SG.9). Uniqueness follows from that formula. \(\square\)