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On a formula of le Merdy for the complex interpolation of tensor products. (English) Zbl 1216.46020
Summary: {\it C. Le Merdy} [Proc. Am. Math. Soc. 126, No. 3, 715--719 (1998; Zbl 0890.47029)] proved the following complex interpolation formula for injective tensor products: $[\ell_2\tilde\otimes_\epsilon \ell_1, \ell_2\tilde\otimes_\epsilon \ell_\infty]_{\frac 12}=S_4$. We investigate whether related formulas hold when considering arbitrary $0 < \theta < 1$ instead of $\frac 12$, and give a partially positive answer for $\theta < \frac 12$ and a negative answer for $\theta > \frac 12$. Furthermore, we briefly discuss the more general case when $\ell_2$ is replaced by $\ell_q$, $1<q<2$, and $\ell_1$ and $\ell_\infty$ by $\ell_{p_0}$ and $\ell_{p_1}$, respectively.
46B70Interpolation between normed linear spaces
46M35Abstract interpolation of topological linear spaces
47B06Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers etc.of operators
47B10Operators belonging to operator ideals
46B28Spaces of operators; tensor products; approximation properties
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