CC spectral calculus on an arbitrary complex Hilbert space by compatible vector-local Cayley subspaces

Vector-local Cayley subspaces: the adjoint inverse equations give $Q_{-i}^{\ast }=Q_i$; inserting $A+i=(A-i)+2i$ into the inverse equations gives $Q_i-Q_{-i}=2i{Q}_i{Q}_{-i}$ and commutation of the resolvents. Consequently $(I+2i{Q}_i)(I-2i{Q}_{-i})=I$, with the reverse product also $I$, so the $U$ in [F2] is unitary. For a finite tuple $\mathbf{x}=(x_1,\dots ,x_m)$ put $K_{\mathbf{x}}=\stackrel{}{\mathrm{span}}\{U^n{x}_j:n\in \mathbb{Z},1\le j\le m\}$, permitting the zero space. The rational-complex finite linear combinations of this explicitly enumerated countable orbit give a specified dense sequence in $K_{\mathbf{x}}$. This says nothing about separability of the whole $H$. The orbit span and its closure are invariant under both $U$ and $U^{\ast }$, hence reduce $U$; [F3] gives a unique projection $P_{\mathbf{x}}$ commuting with $U$. Thus it commutes with $Q_{-i}=(I-U)\mathrm{/}(2i)$ and, by taking adjoints, with $Q_i$. For $x=Q_{-i}y\in D(A)$, $P_{\mathbf{x}}x=Q_{-i}{P}_{\mathbf{x}}y\in D(A)$ and $A{P}_{\mathbf{x}}x=P_{\mathbf{x}}Ax$ by $(A+i)Q_{-i}=I$. Hence every $K_{\mathbf{x}}$ reduces the unbounded operator with its actual domain. No decomposition indexed by an arbitrary family has been selected.

Actual self-adjoint restrictions: for any $K=K_{\mathbf{x}}$ define $A_K$ on $D(A)\cap K$ by restriction. This domain is dense in $K$: projecting a norm-approximating sequence from $D(A)$ onto $K$ preserves the domain by step 1.1 and converges to any chosen vector of $K$. The restricted operator is symmetric, and $Q_{\pm i}{\mathrm{\mid }}_K$ are two-sided inverses from $K$ onto $D(A)\cap K$ of $A_K\mp i$, because both the resolvents and $A$ preserve $K$. If $y\in D(A_K^{\ast })$, solve $(A_K-i)x=(A_K^{\ast }-i)y$ with $x\in D(A_K)$ by that surjectivity. Then $y-x\in \ker (A_K^{\ast }-i)=\mathrm{ran}(A_K+i{)}^{\perp }=\{0\}$, so $y=x\in D(A_K)$ and $A_K^{\ast }y=A_{K}y$. This proves $A_K$ self-adjoint including domains. We may therefore apply [F1] using the specified orbit-dense sequence from step 1.1. Write its uniquely determined PVM as $E_K$ and its calculus as $b(A_K)$.

Compatibility inside a common finite-tuple space: suppose $M=K_{\mathbf{y}}\subseteq K_{\mathbf{x}}=K$; the restriction to $K$ of the projection onto $M$ commutes with $A_K$ including its domain, by step 1.1. The reflection $J=2P_M-I_K$ is unitary and $J{A}_{K}J=A_K$ on exactly $D(A_K)$. The PVM $J{E}_K(B)J$ is consequently another regular representing PVM for $A_K$, so uniqueness in [F1] gives $J{E}_K(B)J=E_K(B)$ for each Borel $B$. Thus $E_K(B)$ preserves $M$ and $M^{\perp }\cap K$. Its restriction to $M$ is a regular PVM representing $A_M$, including the exact domain: both the norm integrals and the operator action are the restrictions of those on $K$. Uniqueness in [F1] therefore gives $E_K(B){\mathrm{\mid }}_M=E_M(B)$. Simple-function approximations imply the same restriction identity for bounded Borel calculus, and its norm-convergent cutoffs give it for unbounded Borel functions on their precise domains. In particular, for any $x,y$ and scalars $c,d$, all of $K_x$, $K_y$ and $K_{cx+dy}$ lie in $K_{(x,y)}$ and use the same calculus there. The local objects are unique; their assembly never chooses among possible PVMs or representations.

Global projections and bounded calculus by compatible vectors: define $E_A(B)x=E_{K_x}(B)x$, with zero at $x=0$. Within $K_{(x,y)}$, step 3.1 identifies these values with the single linear projection $E_{K_{(x,y)}}(B)$; hence $E_A(B)$ is linear on all of $H$ and has norm at most $1$. The same common space gives $⟨E_A(B)x,y⟩=⟨x,E_A(B)y⟩$. Moreover $E_A(B)x\in K_x$, so $K_{E_A(B)x}\subseteq K_x$ and compatibility gives $E_A(C)E_A(B)x=E_{K_x}(C)E_{K_x}(B)x=E_A(C\cap B)x$. Thus the global operators are orthogonal projections. They have $E_A(\mathbb{R})=I$; for disjoint $B_n$, strong countable additivity on each fixed vector is exactly the local additivity inside $K_x$. Its finite scalar measure ${\mu }_x$ equals the local scalar measure and is regular. Likewise define $b(A)x=b(A_{K_x})x$ for bounded Borel $b$. The common spaces give linearity, norm bound, product and adjoint laws. Every value is uniquely determined by its property, so replacement assembles a single operator-valued function on the set of vectors; arbitrary-index choice and a global Hilbert basis are absent.

Exact global unbounded domains and closedness: for finite-valued Borel $b$ put $D_b=\{x:\int \mathrm{\mid }b{\mathrm{\mid }}^{2}d{\mu }_x<\mathrm{\infty }\}$ and define $b(A)x=b(A_{K_x})x$ there. The local unbounded restriction identity in step 3.1 shows that this domain is linear and that the operation is linear: for $x,y\in D_b$ use the common space $K_{(x,y)}$, where the norm estimate for $b(x+y)$ follows from the local linear operator and its domain. Its norm is exactly the stated integral. Let $P_n=E_A(\{\mathrm{\mid }b\mathrm{\mid }\le n\})$ and $b_n=b{1}_{\mathrm{\mid }b\mathrm{\mid }\le n}$. For every $x$, $P_{n}x\to x$ and $P_{n}x\in D_b$ by its local measure and boundedness of $b$ on that set, so $D_b$ is dense. For $x\in D_b$, $b_n(A)x\to b(A)x$ in norm, again by the local square-integrable tail. If $x_j\to x$ and $b(A)x_j\to y$ in $H$, bounded calculus gives $P_{n}y=b_n(A)x$ for each fixed $n$. Hence ${\int }_{\mathrm{\mid }b\mathrm{\mid }\le n}\mathrm{\mid }b{\mathrm{\mid }}^{2}d{\mu }_x=\mathrm{\parallel }{P}_{n}y{\mathrm{\parallel }}^2\le \mathrm{\parallel }y{\mathrm{\parallel }}^2$; monotone convergence gives $x\in D_b$, and increasing $n$ gives $b(A)x=y$. Thus $b(A)$ is closed without putting the whole $H$ into one multiplication representation. For $b(\lambda )=\lambda$, $D_b=D(A)$: by step 2.1 the local domain is exactly $D(A)\cap K_x$, and $x\in K_x$ always, so finite local second moment is equivalent to $x\in D(A)$. Its action is the original $Ax$.

Spectrum support and the exact distance bound: if $E_A((\lambda -\epsilon ,\lambda +\epsilon ))=0$, the bounded multiplier $b(t)={\mathbf{1}}_{\{\mathrm{\mid }t-\lambda \mathrm{\mid }\ge \epsilon \}}\mathrm{/}(t-\lambda )$ satisfies $tb(t)$ bounded. Hence $b(A)H\subset D(A)$ and $(A-\lambda )b(A)=I$, $b(A)(A-\lambda )=I$ on $D(A)$, by the exact domains and product identities in step 5.1; thus $\lambda \mathrm{\notin }\sigma (A)$. Conversely, if $(A-\lambda {)}^{-1}$ has norm $M$, every $u$ in the projection of a bounded interval $\mathrm{\mid }t-\lambda \mathrm{\mid }<\epsilon$ belongs to $D(A)$ and satisfies $\mathrm{\parallel }(A-\lambda )u\mathrm{\parallel }\le \epsilon \mathrm{\parallel }u\mathrm{\parallel }$. When $M\epsilon <1$ this forces $u=0$, so that projection is zero. These two implications prove the stated neighborhood characterization. Each point of the real resolvent set therefore has a zero-projection neighborhood; the countable rational intervals whose closures lie in such neighborhoods cover that open set. Their projections are zero, and countable PVM additivity shows $E_A(\mathbb{R}\setminus \sigma (A))=0$. Nonreal points are in the resolvent by [F2], so $\sigma (A)$ is real. For any $z\mathrm{\notin }\sigma (A)$ let $d=\mathrm{dist}(z,\sigma (A))>0$, and define $b(t)=(t-z{)}^{-1}$ on the spectrum and zero off it. Both $b$ and $tb$ are bounded, with $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}\le d^{-1}$. The same exact-domain identities give the two-sided inverse $(A-z{)}^{-1}=b(A)$ and its distance bound. Its range is $D(A)$ by the inverse identities. The zero Hilbert space case has zero PVM/operator and empty spectrum, consistent with the stated convention. No arbitrary global spectral representation or DC inverse theorem has been used.

Global uniqueness by scalar cyclic moments: let $\tilde{E}$ be any other regular PVM representing $A$ with its exact domain. Its simple-function calculus and truncations make $(A-i)(A+i{)}^{-1}$ the bounded multiplier $(\lambda -i)\mathrm{/}(\lambda +i)$, so pushing its scalar measure at $x$ to the circle gives moments $⟨U^{n}x,x⟩$ for every integer $n$ and no mass at $1$. The same is true for $E_A$, because on $K_x$ its Cayley unitary is $U{\mathrm{\mid }}_{K_x}$. These circle scalar measures are regular: compact inner approximation on $\mathbb{R}$ followed by continuous Cayley images gives compact inner approximation on the circle, and complements give outer approximation. The scalar moment uniqueness proved in [F1] therefore forces equality for every $x$. Inverse Cayley transport gives $⟨\tilde{E}(B)x,x⟩=⟨E_A(B)x,x⟩$ for every real Borel $B$. Applying complex polarization to the common quadratic forms gives equality of all pairings and hence of the bounded projection operators. Thus the global PVM, and the bounded/unbounded calculus built from its truncations, are unique. This uniqueness uses only the scalar moment argument, never an arbitrary orthogonal cyclic decomposition of $H$.

Strong groups, precise derivative domain and group uniqueness: define $U_A(t)=b_t(A)$ with $b_t(\lambda )=e^{it\lambda }$. Step 4.1 gives unitary group laws on $H$. For every fixed $x$, this is the local strongly continuous group on $K_x$; hence it is strongly continuous on all of $H$. If $x\in D(A)$, the exact local derivative in [F1] gives $(U_A(t)x-x)\mathrm{/}t\to iAx$. Conversely, if those quotients converge in $H$, all belong to the closed space $K_x$, so their limit belongs to $K_x$ and the local derivative criterion gives $x\in D(A_{K_x})=D(A)\cap K_x$. Domain invariance, commutation with $A$ and preservation of both $\mathrm{\parallel }x\mathrm{\parallel }$ and $\mathrm{\parallel }Ax\mathrm{\parallel }$ also follow in $K_x$, proving graph-norm invariance. If another strongly continuous unitary group $V(t)$ has this exact generator $iA$, its group law gives $V(t)D(A)=D(A)$ and $AV(t)x=V(t)Ax$ by taking the derivative at zero in $V(s)V(t)x=V(t)V(s)x$. For $x\in D(A)$, differentiate $U_A(-t)V(t)x$: the two derivatives cancel, so every scalar pairing of this vector-valued function has zero derivative and is constant by the scalar fundamental theorem. Therefore $V(t)x=U_A(t)x$ on the dense domain; boundedness extends equality to all $H$. This supplies the arbitrary-H autonomous group and exact generator without a global separability assumption or an AC Stone theorem.

Transport under intertwiners: for a unitary $W$ carrying $A$ with its full domain to another self-adjoint $B$, the PVM $W{E}_A(C)W^{\ast }$ is a regular representing PVM for $B$ with the correct domain, so step 6.2 gives $E_B(C)W=W{E}_A(C)$ and all bounded and unbounded calculus transports. If instead an isometry $W$ intertwines their groups for every real $t$, its closed range $M$ reduces the target group because it is invariant for both time signs. The unique orthogonal projection onto $M$ commutes with the target group; the derivative criterion in step 6.3 shows that it preserves $D(B)$ and commutes with $B$ there. The restrictions of the nonreal resolvents to $M$ are two-sided inverses on the dense restricted domain, so the elementary adjoint argument of step 2.1 makes $B{\mathrm{\mid }}_M$ self-adjoint. Both directions of the derivative criterion give $D(B{\mathrm{\mid }}_M)=WD(A)$ and $B{\mathrm{\mid }}_M=WA{W}^{\ast }$. Uniqueness on this arbitrary closed subspace, already proved in step 6.2, gives the PVM intertwining and its bounded/unbounded versions. The Statement is proved for arbitrary $H$ under CC.