CC spectral calculus on a Hilbert space with a fixed countable dense sequence

Finite positive moment approximants: first let $U$ be any unitary on $H$ and fix $x\in H$. For $N\ge 0$ define $q_N(\theta )=\mathrm{\parallel }{\sum }_{j=0}^N{e}^{-ij\theta }{U}^{j}x{\mathrm{\parallel }}^2\mathrm{/}(2\pi (N+1))$ for $0\le \theta \le 2\pi$. These nonnegative continuous densities define finite measures ${\nu }_N$ on $\mathbb{R}$ supported in $[0,2\pi ]$. Expansion of the squared norm and $⟨U^{j}x,U^{l}x⟩=⟨U^{j-l}x,x⟩$ give total mass $M=\mathrm{\parallel }x{\mathrm{\parallel }}^2$ and $\int e^{in\theta }d{\nu }_N=(1-\mathrm{\mid }n\mathrm{\mid }\mathrm{/}(N+1))⟨U^{n}x,x⟩$ when $\mathrm{\mid }n\mathrm{\mid }\le N$, and zero for the other integer Fourier coefficients. This is a finite sum identity and uses no spectral representation.

Actual scalar measure existence: enumerate $\mathbb{Q}=(r_j)$. Write $F_N(t)={\nu }_N((-\mathrm{\infty },t])$. For each bounded real sequence use the following canonical subsequence operation: start in $[0,M]$ (if $M=0$ all measures are zero); repeatedly bisect the current closed interval and retain its first half containing infinitely many remaining indices, then choose the least eligible increasing index at each stage. The nested intervals have lengths tending to zero, so the selected values converge. Apply this uniquely specified operation to $F_N(r_1)$, then to $F_N(r_2)$ within the previous subsequence, and so forth, and take increasing diagonal indices. The resulting sequence $N_j$ tends to infinity and $F_{N_j}(q)$ converges to $G(q)$ for every rational $q$. Put $F(t)={\inf }_{q\in \mathbb{Q},q>t}G(q)$. Monotonicity of $G$ gives monotonicity of $F$; if $t_n\downarrow t$, any rational $q>t$ eventually satisfies $q>t_n$, so $F(t_n)\le G(q)$ and hence $F(t_n)\to F(t)$. Also $F(t)=0$ for $t<0$ and $F(t)=M$ for $t\ge 2\pi$. Thus [F1] applies to this bounded finite-valued function and supplies a measure $\nu$; continuity from below applied to expanding intervals shows $\nu (\mathbb{R})=M$ and support in $[0,2\pi ]$. For a continuity point $t$ of $F$, rational bounds $q_{-}<t<q_{+}$ show $F_{N_j}(q_{-})\le F_{N_j}(t)\le F_{N_j}(q_{+})$; taking limits and then taking rational bounds toward $t$ gives $F_{N_j}(t)\to F(t)$. Here ${\sup }_{q<t}G(q)=F(t-)$ and ${\inf }_{q>t}G(q)=F(t)$ follow directly from the definition and rational interpolation. A monotone bounded function has at most countably many jumps: the points with jump at least $1\mathrm{/}n$ are finite, because the sum of any finite set of jumps is at most $M$, and their increasing order provides a canonical enumeration. Consequently finite partitions of an interval containing $[0,2\pi ]$ can have arbitrarily small mesh and only continuity-point interior endpoints; finite choices suffice for each partition, and CC suffices for a sequence of partitions. Upper/lower step bounds and uniform continuity of a continuous test function show $\int gd{\nu }_{N_j}\to \int gd\nu$. In particular the pushforward ${\mu }_x$ under $\theta \mapsto e^{i\theta }$ satisfies $\int z^{n}d{\mu }_x=⟨U^{n}x,x⟩$ for every integer $n$. This is a construction from [F1], not an invocation of a measure representation theorem.

Regularity and its precise choice cost: every finite Borel measure on a compact metric space $X$ is regular by the following direct argument. For closed $C$, the open sets $\{\mathrm{dist}(x,C)<1\mathrm{/}n\}$ decrease to $C$, so their measures decrease to $\mu (C)$; for open $O\mathrm{\ne }X$, the compact sets $\{\mathrm{dist}(x,X\setminus O)\ge 1\mathrm{/}n\}$ increase to $O$, so their measures increase to $\mu (O)$; $O=X$ uses $X$ itself. Let $\mathcal{R}$ be the Borel sets $B$ such that for every $\epsilon >0$ there are compact $C\subseteq B\subseteq O$ open with $\mu (O\setminus C)<\epsilon$. Closed and open sets are in $\mathcal{R}$, complements exchange the pair $(C,O)$ with $(X\setminus O,X\setminus C)$, and countable unions remain in $\mathcal{R}$: choose pairs for each $B_j$ with errors less than $\epsilon 2^{-j-2}$ by CC, use $O=\bigcup O_j$, and choose a finite initial union of the $C_j$ whose measure is within $\epsilon \mathrm{/}2$ of the full union, by continuity from below. Thus $\mathcal{R}$ contains the Borel sigma-algebra. Apply this first to the restriction of $\nu$ to $[0,2\pi ]$ and then to its compact continuous pushforward; pushforward regularity follows by compact images for inner approximation, and complements for outer approximation. The measure ${\mu }_x$ is therefore regular. The same argument on $\mathbb{R}$ uses compact truncation $[-n,n]$ to make the omitted mass small before applying the compact-metric construction. It also applies to every finite measure used subsequently. No unspecified regularity property of [F1] is assumed.

Fejer approximation and uniqueness: for continuous $g$ on $S^1$ let ${\mathrm{\Phi }}_N(t)=\mathrm{\mid }{\sum }_{j=0}^N{e}^{ijt}{\mathrm{\mid }}^2\mathrm{/}(N+1)$ and put $g_N(e^{i\theta })=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }{\mathrm{\Phi }}_N(t)g(e^{i(\theta -t)})dt$. Finite expansion makes $g_N$ a Laurent polynomial; ${\mathrm{\Phi }}_N\ge 0$ and its normalized integral is $1$. For $\delta \le \mathrm{\mid }t\mathrm{\mid }\le \pi$, the geometric-series formula gives ${\mathrm{\Phi }}_N(t)\le [(N+1){\sin }^2(\delta \mathrm{/}2){]}^{-1}$. Split the convolution at $\mathrm{\mid }t\mathrm{\mid }=\delta$: uniform continuity makes the first contribution to $\mathrm{\mid }{g}_N-g\mathrm{\mid }$ arbitrarily small, and the displayed bound makes the second uniformly tend to zero. Thus Laurent polynomials are uniformly dense in $C(S^1)$. Any two finite regular positive measures with the moments of step 2.1 agree on continuous integrals. They agree on closed sets by dominated convergence applied to $g_n(z)=\max (0,1-n\mathrm{dist}(z,C))\downarrow 1_C$. Their compact inner approximations from step 3.1 then give equality on every Borel set. Thus ${\mu }_x$ is unique; the construction does not select among multiple representing measures. In particular the assignment $x\mapsto {\mu }_x$ is defined by a unique property, and its use for a countable family requires no additional selection of measures.

Cyclic isometry is onto: for a Laurent polynomial $p$, the moment identities give $\mathrm{\parallel }p(U)x{\mathrm{\parallel }}^2=\int \mathrm{\mid }p(z){\mathrm{\mid }}^{2}d{\mu }_x(z)$; polarization gives the corresponding inner products. Laurent polynomials are dense in $L^2({\mu }_x)$, with the density proved as follows. For a Borel $B$, regularity gives compact $C\subseteq B\subseteq O$ with small ${\mu }_x(O\setminus C)$; when $C$ is nonempty and $O\mathrm{\ne }{S}^1$, the continuous function $\mathrm{dist}(z,S^1\setminus O)\mathrm{/}(\mathrm{dist}(z,C)+\mathrm{dist}(z,S^1\setminus O))$ is $1$ on $C$, $0$ outside $O$, and between $0$ and $1$, so its squared $L^2$ distance from $1_B$ is at most ${\mu }_x(O\setminus C)$. The cases $C=\mathrm{\varnothing }$ or $O=S^1$ use the constants $0$ or $1$, respectively, with the same error bound. Truncation and quantization of any measurable square-integrable function approximate it by bounded finite simple functions in $L^2$, so continuous functions are dense; step 4.1 then supplies Laurent-polynomial density. For completeness of this $L^2$ space under CC, take a Cauchy sequence and choose the least successive subsequence indices giving norm differences below $2^{-n}$, then choose measurable representatives of its countably many classes using CC. Finite triangle inequalities and Fatou [F4] bound the $L^2$ norm of the sum of their absolute differences by ${\sum }_n{2}^{-n}$; that sum is finite almost everywhere. The representative series therefore converges almost everywhere, and Fatou on each tail gives $L^2$ convergence with tail norm at most ${\sum }_{j\ge n}{2}^{-j}$. Cauchyness gives convergence of the original sequence. Thus the isometry $p\mapsto p(U)x$ extends uniquely to a unitary from $L^2({\mu }_x)$ onto $K_x=\stackrel{}{\mathrm{span}}\{U^{n}x:n\in \mathbb{Z}\}$: choose approximating sequences using CC if necessary, and equality of their distances proves the extension independent of that choice; the range is closed by completeness and dense by definition. It carries multiplication by $z$ to $U{\mathrm{\mid }}_{K_x}$. If $x=0$, both spaces are zero.

Countable cyclic decomposition: set $S_0=\{0\}$ and recursively define $x_j=(I-P_{S_{j-1}})d_j$, $K_j=K_{x_j}$ and $S_j=S_{j-1}\oplus K_j$. Each $S_{j-1}$ reduces $U$: this holds initially, and a cyclic space spanned by all integer powers reduces $U$ and $U^{\ast }$; its orthogonal complement reduces both as well. Thus $K_j\subseteq S_{j-1}^{\perp }$, the sum is orthogonal and closed, and [F3] supplies its uniquely determined projection. Moreover $d_j\in S_j$. The closure of the countable orthogonal sum is therefore all $H$, since it contains the specified dense sequence. These are unique recursions from fixed data, not relation-following dependent choices. Step 5.1 identifies $H$ with ${⨁}_{j\ge 1}{L}^2({\mu }_{x_j})$ and $U$ with multiplication by $z$ in each block. To verify onto explicitly, decompose $h$ by the block projections: the orthogonal norm sums are bounded by $\mathrm{\parallel }h{\mathrm{\parallel }}^2$, their partial sums converge, and the remaining vector is orthogonal to each $K_j$, hence zero by density. The inverse representation is the convergent orthogonal sum of the unique block inverses.

Circle PVM and bounded calculus: in the representation of step 6.1 define $F(B)$ blockwise as multiplication by $1_B$. Then $F(S^1)=I$, $F(B{)}^{\ast }=F(B)$ and $F(B)F(C)=F(B\cap C)$. For disjoint $B_n$, the norm square of the omitted tail on $h=(h_j)$ is the corresponding tail of ${\sum }_j\int \mathrm{\mid }{h}_j{\mathrm{\mid }}^{2}d{\mu }_{x_j}$, which tends to zero by countable additivity; hence $F$ is strongly countably additive. Its scalar measures are ${\mu }_h(B)={\sum }_j{\int }_B\mathrm{\mid }{h}_j{\mathrm{\mid }}^{2}d{\mu }_{x_j}$, finite positive Borel measures, regular by step 3.1. Define $b(U)$ by block multiplication for bounded Borel $b$; the norm bound, adjoint and product laws follow by integrating pointwise identities. Also $U=\int zdF$. Any other regular circle PVM representing $U$ gives scalar moments $⟨U^{n}h,h⟩$ for every $n\in \mathbb{Z}$; step 4.1 forces equality of all scalar measures, and complex polarization forces equality of the projection operators on every Borel set. Thus $F$ and the bounded calculus are unique, without an AC spectral supplier.

Cayley transform of the actual self-adjoint operator: take $U=(A-i)(A+i{)}^{-1}=I-2i{Q}_{-i}$ using [F2]. The adjoint identity $Q_{-i}^{\ast }=Q_i$ follows from testing $⟨(A+i)x,y⟩=⟨x,(A-i)y⟩$ on the respective exact domains and the bijections in [F2]. The resolvent identity $Q_i-Q_{-i}=2i{Q}_i{Q}_{-i}$ follows by inserting $A+i=(A-i)+2i$ into the inverse equations, and the same computation shows the resolvents commute. Therefore $(I+2i{Q}_i)(I-2i{Q}_{-i})=I$ and the reverse product is $I$, so $U$ is unitary. Also $I-U=2i{Q}_{-i}$ is injective, and its range is exactly $D(A)$. In the circle representation of step 7.1, $\ker (I-U)$ consists precisely of the functions supported on $\{1\}$; hence $F(\{1\})=0$. For $z\mathrm{\ne }1$ put $\lambda (z)=i(1+z)\mathrm{/}(1-z)\in \mathbb{R}$. Multiplication by $1-z$ has range exactly the $h$ for which $h\mathrm{/}(1-z)$ has square-summable block $L^2$ norm. Since $1\mathrm{/}(1-z)=(\lambda (z)+i)\mathrm{/}(2i)$, this range is exactly $\{h:{\sum }_j\int {\lambda }^2\mathrm{\mid }{h}_j{\mathrm{\mid }}^{2}d{\mu }_{x_j}<\mathrm{\infty }\}$. If $h=(I-U)g=2i{Q}_{-i}g$, then $Ah=2i(I-i{Q}_{-i})g=i(I+U)g$; in the multiplication representation this says $Ah=\lambda h$. Thus the inverse Cayley identification proves the exact unbounded domain, rather than merely identifying an extension of $A$.

Real PVM and unbounded calculus: transport $F$ under the Borel bijection $z\mapsto \lambda (z)$ from $S^1\setminus \{1\}$ to $\mathbb{R}$, defining $E(B)=F({\lambda }^{-1}(B))$. Step 8.1 proves the representation and exact domain of $A$, its norm formula and $E(\mathbb{R})=I$. Strong additivity and the bounded calculus transfer from step 7.1; finite scalar measures are regular on $\mathbb{R}$ by the truncation argument in step 3.1. For finite-valued Borel $b$, define multiplication on the exact square-integrability domain; truncations $1_{\mathrm{\mid }b\mathrm{\mid }\le n}h$ prove density of the domain. To prove closedness, if $h_n\to h$ and $b{h}_n\to g$ in the representation, restrict to $\{\mathrm{\mid }b\mathrm{\mid }\le N\}$: bounded multiplication gives $1_{\mathrm{\mid }b\mathrm{\mid }\le N}g=b{1}_{\mathrm{\mid }b\mathrm{\mid }\le N}h$. Increasing $N$ shows $g=bh$ a.e., and $g\in L^2$ gives $h$ in the domain. Dominated convergence proves the stated truncation convergence. Uniqueness of a regular PVM representing $A$ follows by inverse Cayley transport: its multiplication realization of $A$ with the prescribed domain makes $(A-i)(A+i{)}^{-1}$ multiplication by $(\lambda -i)\mathrm{/}(\lambda +i)$, so it gives a regular circle PVM representing the same $U$ (regularity also follows directly for its finite scalar measures). Step 7.1 gives equality there, and inverse transport gives equality on $\mathbb{R}$.

Strong groups and exact generator: define $e^{itA}$ by multiplication by $e^{it\lambda }$. The group law and unitarity are pointwise; dominated convergence with bound $4\mathrm{\mid }{h}_j{\mathrm{\mid }}^2$, summed over the blocks, proves strong continuity. If $h\in D(A)$, then $\mathrm{\mid }(e^{it\lambda }-1)\mathrm{/}t\mathrm{\mid }\le \mathrm{\mid }\lambda \mathrm{\mid }$ and pointwise convergence to $i\lambda$ give norm convergence of the difference quotient by dominated convergence against the finite weighted norm measure. Conversely, existence of the strong derivative makes the difference quotients at $t=1\mathrm{/}n$ norm bounded. Fatou applied to their squared pointwise limits gives ${\sum }_j\int {\lambda }^2\mathrm{\mid }{h}_j{\mathrm{\mid }}^{2}d{\mu }_{x_j}<\mathrm{\infty }$, exactly $h\in D(A)$ by step 8.1; its derivative is then $iAh$. This proves the group and its precise generator without a Stone existence theorem.

Transport and isometric group intertwiners: if a unitary $W$ intertwines the self-adjoint operators including their domains, it intertwines both nonreal inverses, hence their Cayley unitaries; the transported circle PVM is a regular representing PVM, so uniqueness in step 7.1, followed by step 9.1, gives $E_B(C)W=W{E}_A(C)$ and transport of every multiplier with its exact domain. More generally, let an isometry $W$ intertwine the two groups for every real $t$. Its closed range $M$ is invariant under the target group and its inverse, so $M$ and $M^{\perp }$ reduce the group. The projection $P_M$ commutes with the group; taking difference quotients and using step 10.1 shows $P_{M}D(B)\subseteq D(B)$ and $B{P}_M=P_{M}B$ there, so it commutes with the resolvents by their inverse equations. Thus $M$ reduces $B$ and its restriction is self-adjoint: its nonreal inverses are the restricted full inverses, or equivalently it is the block restriction of step 9.1. Difference quotients also identify that restriction with $WA{W}^{\ast }$ on exactly $WD(A)$, using both directions of step 10.1. The range has dense sequence $(W{d}_j)$; apply uniqueness to this restriction to obtain $E_B(C)W=W{E}_A(C)$ and the asserted calculus transport. All conclusions of the Statement follow.