Complex interpolation between Hilbert, Banach and operator spaces

Source abstract

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X X satisfying the following property: there is a function ε → Δ X ( ε ) \varepsilon \to \Delta _X(\varepsilon ) tending to zero with ε > 0 \varepsilon >0 such that every operator T : L 2 → L 2 T\colon \ L_2\to L_2 with ‖ T ‖ ≤ ε \|T\|\le \varepsilon that is simultaneously contractive (i.e. of norm ≤ 1 \le 1 ) on L 1 L_1 and on L ∞ L_\infty must be of norm ≤ Δ X ( ε ) \le \Delta _X(\varepsilon ) on L 2 ( X ) L_2(X) . We show that Δ X ( ε ) ∈ O ( ε α ) \Delta _X(\varepsilon ) \in O(\varepsilon ^\alpha ) for some α > 0 \alpha >0 iff X X is isomorphic to a quotient of a subspace of an ultraproduct of θ \theta -Hilbertian spaces for some θ > 0 \theta >0 (see Corollary 6.7), where θ \theta -Hilbertian is meant in a slightly more general sense than in our previous paper (1979). Let B r ( L 2 ( μ ) ) B_{{r}}(L_2(\mu )) be the space of all regular operators on L 2 ( μ ) L_2(\mu ) . We are able to describe the complex interpolation space $$( B r ( L 2 ( μ ) ) , B ( L 2 ( μ ) ) ) θ . (B_{{r}}(L_2(\mu )), B(L_2(\mu )))^\theta .$$ We show that T : L 2 ( μ ) → L 2 ( μ ) T\colon \ L_2(\mu )\to L_2(\mu ) belongs to this space iff T ⊗ i d X T\otimes id_X is bounded on L 2 ( X ) L_2(X) for any θ \theta -Hilbertian space X X . More generally, we are able to describe the spaces $$( B ( ℓ p 0 ) , B ( ℓ p 1 ) ) θ o r ( B ( L p 0 ) , B ( L p 1 ) ) θ (B(\ell _{p_0}), B(\ell _{p_1}))^\theta \ \mathrm {or}\ (B(L_{p_0}), B(L_{p_1}))^\theta$$ for any pair 1 ≤ p 0 , p 1 ≤ ∞ 1\le p_0,p_1\le \infty and 0 > θ > 1 0>\theta >1 . In the same vein, given a locally compact Abelian group G G , let M ( G ) M(G) (resp. P M ( G ) PM(G) ) be the space of complex measures (resp. pseudo-measures) on G G equipped with the usual norm ‖ μ ‖ M ( G ) = | μ | ( G ) \|\mu \|_{M(G)} = |\mu |(G) (resp. \[ ‖ μ ‖ P M ( G ) = sup { | μ ^ ( γ ) | | γ ∈ G ^ } ) . \|\mu \|_{PM(G)} = \sup \{|\hat \mu (\gamma )| \ \big | \ \gamma \in \widehat G\}). \] We describe similarly the interpolation space ( M ( G ) , P M ( G ) ) θ (M(G), PM(G))^\theta . Various extensions and variants of this result will be given, e.g. to Schur multipliers on B ( ℓ 2 ) B(\ell _2) and to operator spaces.