Exact scale interpolation and endpoint operator transport

On the zero Hilbert space the unique full-domain operator is self-adjoint, and all its powers act on $\{0\}$ by [F4]. Thus its smooth core is $\{0\}$. Its completion at any real scale index is again $\{0\}$: the dense isometric image of the zero space consists only of zero, and a nonzero vector of norm $r>0$ would have a ball of radius $r\mathrm{/}2$ disjoint from that image. For the zero endpoint couple the only strip function is zero, so its interpolation space is $\{0\}$ with its unique norm. This proves the scale identity whenever $H=\{0\}$, and the same reasoning applies to $K$. If either $H=\{0\}$ or $K=\{0\}$, the operator $L$ has zero image and the claimed estimate is $0\le 0$ or $0\le M_0^{1-\theta }{M}_1^{\theta }\mathrm{\parallel }u\mathrm{\parallel }$, respectively. Hence the operator estimate is also established in every such case. For the remaining arguments assume $H,K\mathrm{\ne }\{0\}$.

Suppose $a\le b$, and write $d=(b-a)\mathrm{/}2$. The unitary $J_a$ maps the endpoint couple isometrically and consistently to $(H,D(T))$ for $T=A^d\ge I$ with norm $\mathrm{\parallel }Tv\mathrm{\parallel }$. The Borel calculus gives self-adjointness of $T$ and $T^{\theta }=A^{d\theta }$. Applying this common map and its inverse to strip functions preserves holomorphy, continuity, decay and both endpoint norms. The pivot/domain identity therefore gives $[H_A^a,H_A^b{]}_{\theta }=H_A^{a+2d\theta }$ with identical norms. If $a=b$, use $T=I$. If $a>b$, $F(z)\mapsto F(1-z)$ is an isometry between the admissible classes with swapped endpoints, and evaluates at $1-\theta$, proving the same formula.

Endpoint boundedness makes $L$ bounded between the sum spaces: for each decomposition $u=u_0+u_1$, its target sum norm is at most $M_0\mathrm{\parallel }{u}_0\mathrm{\parallel }+M_1\mathrm{\parallel }{u}_1\mathrm{\parallel }$, so at most $\max (M_0,M_1)\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{s}\mathrm{u}\mathrm{m}}$ after infimizing. Thus composition with $L$ preserves sum-valued holomorphy and continuity; endpoint boundedness preserves boundary continuity and decay. For $\delta >0$ put $m_j=M_j+\delta >0$. If $F(\theta )=u$, the strip function $G(z)=(m_0\mathrm{/}{m}_1{)}^{z-\theta }LF(z)$ is admissible in the target, has $G(\theta )=Lu$, and has boundary norms at most $m_0^{1-\theta }{m}_1^{\theta }\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathcal{F}}$: on boundary zero multiply $M_0$ by $(m_0\mathrm{/}{m}_1{)}^{-\theta }$ and on boundary one multiply $M_1$ by $(m_0\mathrm{/}{m}_1{)}^{1-\theta }$. The scalar factor is bounded on the strip, since the ratio is positive real. Taking the infimum in $F$ and then $\delta \downarrow 0$ gives the asserted norm bound, including $M_j=0$. The target and source norms in this calculation are exactly those in step 1.2.