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A beginner mathmatician
A beginner mathmatician
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Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\theta$ be a closed subspace. Let there be a contractive projection $P:X_\theta\to X_\theta$ for all $0<\theta<1$$0\leq\theta\leq 1$ such that image of $P$ is $E_\theta.$ Is it true that $E_\theta=(E_0,E_1)_\theta$?

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\theta$ be a closed subspace. Let there be a contractive projection $P:X_\theta\to X_\theta$ for all $0<\theta<1$ such that image of $P$ is $E_\theta.$ Is it true that $E_\theta=(E_0,E_1)_\theta$?

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\theta$ be a closed subspace. Let there be a contractive projection $P:X_\theta\to X_\theta$ for all $0\leq\theta\leq 1$ such that image of $P$ is $E_\theta.$ Is it true that $E_\theta=(E_0,E_1)_\theta$?

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A beginner mathmatician
A beginner mathmatician
  • 1.9k
  • 6
  • 11

Complex interpolation of subspaces

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\theta$ be a closed subspace. Let there be a contractive projection $P:X_\theta\to X_\theta$ for all $0<\theta<1$ such that image of $P$ is $E_\theta.$ Is it true that $E_\theta=(E_0,E_1)_\theta$?