I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we have the equalities $$ \|A v_0\|_1 \leq \|v_0\|_1$$ and $$ \|A v_0\|_\infty \leq \|v_0\|_\infty $$ for all $v_0 \in V_0.$ Does it follow that $A$ becomes a contraction on $V_0$ for every $\ell_p$-norm, where $1 \leq p \leq \infty?$ (Probably not...?) The answer will change if we transfer the whole question to $\mathbb{C}^n$?
2025-01-13 07:33:47.1736753627
Interpolation of a subspace of codimension one
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