In this paper we use complex interpolation to obtain new inclusion and coincidence theorems for absolutely and multiple summing multilinear mappings between Banach spaces. In particular, we derive optimal coincidence theorems of Bohnenblust-Hille type for multilinear forms on K-convex Banach spaces of cotype 2.

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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:

Based on abstract interpolation, we prove asymptotic formulae for the (F, 2)-summing norm of inclusions id : S n E → S n 2 , where E and F are two Banach sequence spaces. Here, S n E stands for the unitary ideal of operators on the n-dimensional Hilbert space whose singular values belong to E, and S n 2 = S n 2 for the Hilbert-Schmidt operators. Our results are noncommutative analogues of results due to Bennett and Carl, as well as their recent generalizations to Banach sequence spaces. As an application, we give lower and upper estimates for certain s-numbers of the embeddings id : S n E → S n 2 and id : S n 2 → S n E . In the concluding section, we finally consider mixing norms.

Abstract. In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 the identity map id: u → 2 is absolutely (u, 1)-summing, i. e., for every unconditionally summable sequence (xn) in u the scalar sequence ( xn 2 ) is contained in u, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach sequence space E the identity map id : E → 2 is absolutely (E, 1)-summing, i. e., for every unconditionally summable sequence (xn) in E the scalar sequence ( xn 2 ) is contained in E. Various applications are given, e. g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator T on 2 with values in a 2-concave symmetric Banach sequence space E is a multiplier from 2 into E. Furthermore, we prove an asymptotic formula for the k-th approximation number of the identity map id : n 2 → En, where En denotes the linear span of the first n standard unit vectors in E, and apply it to Lorentz and Orlicz sequence spaces.