The purpose of this note is to announce, without proofs, some results concerning vector valued multilinear operators on a product of C(K) spaces.

A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely continuous...

We study the classes of complex Banach spaces with Valdivia dual unit ball. We give complex analogues of several theorems on real spaces. Further we study relationship of these complex Banach spaces with their real versions and that of real Banach spaces and their complexification. We also formulate several open problems.

We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d{*}_{(}w,1)$ into d(w,1), where $d{*}_{(}w,1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.

For the complex interpolation functors associated with derivatives of analytic functions, the Calderón fundamental inequality is formulated in both additive and multiplicative forms; discretization, reiteration, the Calderón-Lozanovskiĭ construction for Banach lattices, and the Aronszajn-Gagliardo construction concerning minimality and maximality are presented. These more general complex interpolation functors are closely connected with the real and other interpolation functors via function parameters...

We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space $C_E$. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric...

We study the Complex Unconditional Metric Approximation Property for translation invariant spaces $C_{\Lambda }\left(\right)$ of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which $C_{\Lambda }\left(\right)$ has (ℂ-UMAP); though these sets are such that $L_{\Lambda }^{\infty }\left(\right)$ contains functions which are not continuous, we show that there is a linear invariant lifting from these $L_{\Lambda }^{\infty }\left(\right)$ spaces into the Baire class 1 functions.

We show that the Taylor (resp. Bochnak) complexification of the injective (projective) tensor product of any two real Banach spaces is isometrically isomorphic to the injective (projective) tensor product of the Taylor (Bochnak) complexifications of the two spaces.

We give a unified treatment of procedures for complexifying real Banach spaces. These include several approaches used in the past. We obtain best possible results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. These estimates provide generalizations and show sharpness of previously obtained inequalities.

The Banach operator ideal of (q,2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces. A key result in this context is a composition formula for such operators due to H. König, J. R. Retherford and N. Tomczak-Jaegermann. Based on abstract interpolation theory, we prove a variant of this result for (E,2)-summing operators, E a symmetric Banach sequence space.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of positive numbers and $1\le p<\mathrm{\infty }$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}\stackrel{⌃}{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\left|\stackrel{⌃}{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\mathrm{\infty }$ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and ${\sum }_{n=1}^{\mathrm{\infty }}\frac{n^{qj}}{\beta {\left(n\right)}^q}=\mathrm{\infty }$ for some nonnegative integer $j$. We show that if $C_{\phi }$ is compact on $H^p\left(\beta \right)$, then the non-tangential limit of ${\phi }^{\left(j+1\right)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\phi }$ is Fredholm on $H_p\left(\beta \right)$, then $\phi$ must be an automorphism of the open unit disc.

In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.