When $Z$ is an interpolation space between two Banach spaces $X$ and $Y$ (say real / complex method), we have a norm inequality $$ \ x \_Z \le C \ x \_X^\theta \ x \_Y^{1\theta} $$ My question is somehow for the converse : assume that $T$ is a bounded linear operator to $X$, assume $Y$ to be a subspace of $X$ (to make it easy) and we assume the above inequality holds for all $x$ in the range of $T$. Can we then conclude that the image of $T$ sits between any interpolation space with exponent $\theta\pm \varepsilon$? Or even between the real interpolation spaces of exponent $\theta$ with fine indices 1 and $\infty$ respectively ??
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1$\begingroup$ What exactly is meant by "assume $Y$ is a subspace of $X$"? Do you mean $Y$ embeds in $X$? If you use the induced norm on $Y$, then isn't $\\cdot\_X = \\cdot\_Y$? What do you mean by $\x\_Y$ if $x\in \mathrm{Image}(T)$? Are you assuming the image always sit inside $Y$? $\endgroup$ – Willie Wong Jul 25 '16 at 15:07

$\begingroup$ if $Y$ embeds in $X$ then there is no reason that $Y$ with induced norm equals $X$. For example $c_0$ is a subspace (or embeds identically) into $\ell_\infty$ without them being equal. $\endgroup$ – Eric Jul 25 '16 at 22:14

$\begingroup$ For the second question: Yes, I assume $Tx \in Y \subseteq X$. But due to the norm inequality, $Tx$ may always be contained in a larger (interpolation) space, between $Y$ and $X$, and that is precisely the question. $\endgroup$ – Eric Jul 25 '16 at 22:21

$\begingroup$ I misunderstood the first question maybe? The concrete situation I have, $Y$ is the domain of an operator as well, you may give it the graph norm, and then $Y$ embeds indeed into $X$. Is that what you meant? $\endgroup$ – Eric Jul 25 '16 at 22:28

$\begingroup$ So essentially $T$ plays no role in your question. You have as sets $Z\subset Y\subset X$ and norms $\\cdot\_Z$, $\\cdot\_Y$ and $\\cdot\_X$ making them into Banach spaces. You are asking if $\z\_Z \leq C\z\_X^\theta \z\_Y^{1\theta}$, equivalently $Z$ embeds into some interpolation space of $X$ and $Y$, whether $Z$ embeds into "something else". (I am not 100% sure what you mean by the something else in your question.) $\endgroup$ – Willie Wong Jul 26 '16 at 13:13