Added the method of complex interpolation.
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Bokuan Li
2026-06-10 23:39:46 -04:00
parent 1c211eac9a
commit 9ec4eea839
5 changed files with 110 additions and 1 deletions

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@@ -104,7 +104,7 @@
\label{definition:interpolation-functor-exponent}
Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_1$ be its categories of compatible couples, $F: \catc_1 \to \catc$ be an interpolation functor, and $\theta \in [0, 1]$, then $F$ is \textbf{of exponent $\theta$} if there exists $C \ge 0$ such that for every $(E_0, E_1), (F_0, F_1) \in \catc_1$ and $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$
\[
\norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; E_1)}^\theta\norm{T}_{L(F_0; F_1)}^{1 - \theta}
\norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; F_0)}^{1 - \theta}\norm{T}_{L(E_1; F_1)}^{\theta}
\]
If $C = 1$, then $F$ is \textbf{of exact exponent $\theta$}.