Lifted spectral measure and isometric Bochner scale interpolation

Bounded linear maps preserve strong measurability by application to simple approximants. The usual integral pairing $⟨u,v⟩=\int ⟨u(t),v(t){⟩}_{H}dt$ is defined by the inner-product and scalar integral Cauchy–Schwarz inequalities in [F8], is positive definite on the almost-everywhere quotient, and has the displayed norm. Completeness follows by applying the real Bochner completeness theorem to the underlying real Banach space of $H$; the norm, measurable simple functions and Cauchy sequences are unchanged by this realification, as specified in [F8]. Hence $\mathcal{H}$ is a complex Hilbert space, without any separability assumption on $H$. For each Borel $C$, $\mathcal{E}(C)$ is a bounded orthogonal projection, as is checked pointwise. The identities $\mathcal{E}(\mathrm{\varnothing })=0$, $\mathcal{E}(\mathbb{R})=I$ and $\mathcal{E}(C\cap D)=\mathcal{E}(C)\mathcal{E}(D)$ also hold pointwise, by the corresponding identities for $E_A$. For disjoint $C_n$, the finite partial sums converge pointwise strongly to $E_A(\bigcup C_n)u(t)$; their squared differences are dominated by $\mathrm{\parallel }u(t){\mathrm{\parallel }}^2$. Scalar dominated convergence in [F9] proves strong countable additivity on $\mathcal{H}$.

The scalar spectral measure is ${\nu }_u(C)=\mathrm{\parallel }\mathcal{E}(C)u{\mathrm{\parallel }}^2={\int }_{I_{\tau }}\mathrm{\parallel }{E}_A(C)u(t){\mathrm{\parallel }}^{2}dt$. It is regular: approximate $u$ in $L^2$ by finite-valued measurable simple functions $v$. Such approximation follows by truncating the norm of $u$, taking its defining simple approximants, clipping them to a fixed ball, and using scalar dominated convergence from [F9], then letting that ball increase. For simple $v$ on disjoint time sets, ${\nu }_v$ is a finite positive linear combination of the regular scalar measures of $E_A$, so regular. For a finite disjoint Borel partition $(C_k)$, Cauchy–Schwarz from [F8] for the orthogonal projections gives ${\sum }_k\mathrm{\mid }{\nu }_u(C_k)-{\nu }_v(C_k)\mathrm{\mid }\le (\mathrm{\parallel }u\mathrm{\parallel }+\mathrm{\parallel }v\mathrm{\parallel })\mathrm{\parallel }u-v\mathrm{\parallel }$. Thus ${\nu }_v\to {\nu }_u$ in total variation. Given a Borel set and an error tolerance, first take such a $v$ with small variation error and then compact inner and open outer approximants for ${\nu }_v$; their errors for ${\nu }_u$ are also small. This proves regularity of each ${\nu }_u$ and hence of the PVM (polarization also gives regular cross measures). The converse spectral theorem now gives $\mathcal{A}$ as asserted.

For every nonnegative Borel $w$, the identity $\int wd{\nu }_u={\int }_{I_{\tau }}(\int wd(E_A{)}_{u(t)})dt$ follows first for indicators from step 2.1, then for nonnegative simple functions by finite addition, then for arbitrary $w$ by the increasing simple approximations and scalar monotone convergence in [F9] on both sides. In particular for $s\ge 0$ the exact norm formula with $w(\lambda )={\lambda }^s$ shows that $D({\mathcal{A}}^{s\mathrm{/}2})$ consists precisely of $u(t)\in D(A^{s\mathrm{/}2})$ almost everywhere with $\int \mathrm{\parallel }{A}^{s\mathrm{/}2}u(t){\mathrm{\parallel }}^{2}dt<\mathrm{\infty }$, and gives equality of the norms. The pointwise power is strongly measurable because the bounded spectral truncations are strongly measurable and converge in $H$ pointwise on this domain; conversely the inverse $A^{-s\mathrm{/}2}:H\to H_A^s$ is bounded, so the domain functions are strongly measurable also as $H_A^s$-valued functions. Truncated multipliers prove that ${\mathcal{A}}^{s\mathrm{/}2}u(t)=A^{s\mathrm{/}2}u(t)$ on this domain.

For any real $s$, the core isometry $u\mapsto {\mathcal{A}}^{s\mathrm{/}2}u$ extends onto $\mathcal{H}$ by the cutoff argument. The pointwise map $u(t)\mapsto J_{s}u(t)$ is also an onto isometry $L^2(I_{\tau };H_A^s)\to \mathcal{H}$, since $J_s$ and its inverse are bounded maps between their indicated Hilbert spaces and preserve strong measurability. On the ranges of $\mathcal{E}([1,N])$, these two maps agree by the bounded multiplier identities and step 3.1; these cutoff ranges lie in $D({\mathcal{A}}^{\mathrm{\infty }})$ and are dense at every completed level. In the Bochner space they are dense because $E_A([1,N])$ converges strongly and contractively at level $s$, followed by scalar dominated convergence from [F9] in time with majorant $\mathrm{\parallel }u(t){\mathrm{\parallel }}_s^2$. This proves the canonical isometric completion identity for all real $s$, consistently in common smaller-index spaces. Applying exact spectral interpolation to $\mathcal{A}$ proves the displayed identity.

Consistency makes $L$ well defined: two decompositions of $x$ differ by an element of $X_0\cap X_1$, on which the endpoint maps agree. For any decomposition $x=x_0+x_1$, the target sum norm satisfies $\mathrm{\parallel }Lx{\mathrm{\parallel }}_{Y_0+Y_1}\le M_0\mathrm{\parallel }{x}_0{\mathrm{\parallel }}_{X_0}+M_1\mathrm{\parallel }{x}_1{\mathrm{\parallel }}_{X_1}$. Infimizing gives $\mathrm{\parallel }Lx{\mathrm{\parallel }}_{Y_0+Y_1}\le \max (M_0,M_1)\mathrm{\parallel }x{\mathrm{\parallel }}_{X_0+X_1}$. Hence, for an admissible source function $F$, composition $LF$ is bounded and continuous into the target sum. Its interior difference quotients converge to $L{F}^{\mathrm{\prime }}(z)$ by this boundedness, so it is holomorphic. On boundary $j$, the restriction is $L_{j}F(j+it)$; boundedness of $L_j$ preserves endpoint continuity and decay, with supremum norm at most $M_j\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}$. This argument uses only the arbitrary compatible source couple, without requiring it to be a spectral couple.

Fix $x\in [X_0,X_1{]}_{\theta }$ and an admissible $F$ with $F(\theta )=x$. For $\delta >0$ put $m_j=M_j+\delta >0$ and $G(z)=(m_0\mathrm{/}{m}_1{)}^{z-\theta }LF(z)$, using the real logarithm of the positive ratio. This scalar factor is holomorphic and bounded on the closed strip, with boundary modulus $(m_0\mathrm{/}{m}_1{)}^{j-\theta }$, so $G$ is admissible in the target and $G(\theta )=Lx$. Its boundary supremum at $j=0$ is at most $M_0(m_0\mathrm{/}{m}_1{)}^{-\theta }\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}\le m_0^{1-\theta }{m}_1^{\theta }\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}$, and at $j=1$ it is at most $M_1(m_0\mathrm{/}{m}_1{)}^{1-\theta }\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}\le m_0^{1-\theta }{m}_1^{\theta }\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}$. Taking the infimum over $F$ and then $\delta \downarrow 0$ proves the geometric-mean bound, including either zero endpoint bound. Step 4.1 identifies the target interpolation norm exactly with $L^2(I_{\tau };H_A^{b_{\theta }})$. For the two stated source specializations, [F2] gives the spectral-source identity, and the construction in steps 1.1–4.1 applied to $B,K,I_{\sigma }$ gives the Bochner-source identity; both preserve the exact norm.