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Zbl 1216.46020
Michels, C.
On a formula of le Merdy for the complex interpolation of tensor products.
(English)
[J] Ann. Funct. Anal. AFA 1, No. 2, 92-102, electronic only (2010). ISSN 2008-8752/e

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.
MSC 2000:
*46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological linear spaces
47B06 Riesz operators and eigenvalue distributions
47B10 Operators defined by summability properties
46B28 Normed linear spaces of linear operators, etc.

Keywords: complex interpolation of Banach spaces; injective tensor products; approximation numbers; quasi-Banach operator ideals

Citations: Zbl 0890.47029

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