A complex interpolation formula for tensor products of vector-valued Banach function spaces

Abstract

We prove the complex interpolation formulafor the injective tensor product of vector-valued Banach function spaces X i (E i ) and Y i (F i ) satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalarvalued Banach function spaces.The following theorem for the complex interpolation of injective tensor products of vectorvalued Banach function spaces is proved: holds algebraically and topologically whenever the Banach lattices X 0 , X 1 , Y 0 , Y 1 are 2-concave and the Banach spaces E i and F i satisfy one of the following conditions:

Complex interpolation and summability properties of multilinear operators

“…Recently, in [4,8,18], complex interpolation arguments were used in order to obtain inclusion and coincidence theorems for spaces of absolutely summing and multiple summing mappings involving spaces of cotype 2; the interpolation results were based on the paper [15]. Among other results, we obtain similar results for spaces with cotype greater than 2 as well as for L ∞ -spaces.…”

“…Going back to (4), this is clear if condition (iv) is fulfilled. The corresponding statement under the assumptions in (ii) can be found in [15,Proposition 8], for (iii) in [22,Theorem 1]. To see that it holds under the assumptions in (i), first localize and then simply use the fact that m…”

Summing inclusion maps between symmetric sequence spaces

“…The following lemma partially extends (in the lattice case) results of Pisier and Kouba on the complex interpolation of spaces of operators (see [Kou91], [Pi90] and also [DM00]). Recall that 2 = M ( 2 , 1 ) and ∞ = M ( 2 , 2 ); then the statement below says that, under the given assumption, the interpolation property of the spaces of multipliers (diagonal operators) can be transferred into the corresponding interpolation property of the associated spaces of bounded operators (at least in the finite-dimensional case).…”

On interpolation of bilinear operators

“…To complete the proof, we use the following result (which follows from the proof of Proposition 4 in [10]): if jAF is a non-degenerate function, then for any finitedimensional Banach space N of cotype 2 and any couple ðM 0 ; M 1 Þ of finitedimensional 2-concave Banach lattices with dimðM 0 Þ ¼ dimðM 1 Þ; LðN Ã ; G j ðM 0 ; M 1 ÞÞ+G j ðLðN Ã ; M 0 Þ; LðN Ã ; M 1 ÞÞ with a norm of the continuous inclusion depending only on the 2-concavity constants of M 0 and M 1 and a cotype 2 constant of N:…”