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Let $(X_0,X_1)$ be an admissible pair of complex Banach spaces with $X_0$ continuously embedded in $X_1$. For $0<\theta<1$, let us denote by $X_\theta =(X_0,X_1)_\theta$ the complex interpolation space.

Given a closed subspace $M$ of $X_0$ which is also closed in $X_1$, it looks natural that interpolating the quotients we get $(X_0/M,X_1/M)_\theta = X_\theta/M$.

Is there a suitable reference for this result, maybe under some conditions?

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  • $\begingroup$ Isn't there a counterexample in your book on Tauberian operators? Take an $\ell_2$ subspace $X$ of $L_1$ that is contained in $L_2$ s.t. $L_1/X$ does not embed into $L_1$ and $X$ is complemented in some $L_p$ with $p<2$. I think Pisier proved that such exists (I am traveling and cannot look things up), But you know this material better than I, so maybe I am misremembering again... $\endgroup$
    – Bill Johnson
    Commented Nov 28, 2016 at 19:26
  • $\begingroup$ I cannot see why Pisier's example included in "Tauberian operators" is a counterexample to the previous question. $\endgroup$
    – M.González
    Commented Nov 29, 2016 at 13:05
  • $\begingroup$ I have found that Theorem 4.2 in [S. Janson. Interpolation of subcouples and quotient couples. Ark. Mat. 31 (1993), 307-338] looks like a result more general than my conjecture, but for the real interpolation method. I need to understand this result. $\endgroup$
    – M.González
    Commented Nov 29, 2016 at 13:10

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