### Duality and Lorentz-Marcinkiewicz operator spaces.

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A (real or complex) Banach space E is said to have the unconditionaly property for martingale differences (UMD-property, for short) if E-values martingale differences are unconditional in L(E;[0,1]). The main reason for the interest in this new class of spaces is that the analogues of several classical results on martingales and singular integrals are also true for a Banach space belonging to this class.

The behavior of compactness under real interpolation real is discussed. Classical results due to Krasnoselskii, Lions-Peetre, Persson, and Hayakawa are described, as well as others obtained very recently by Edmunds, Potter, Fernández, and the author.

We study limiting K- and J-methods for arbitrary Banach couples. They are related by duality and they extend the methods already known in the ordered case. We investigate the behaviour of compact operators and we also discuss the representation of the methods by means of the corresponding dual functional. Finally, some examples of limiting function spaces are given.

This paper deals with Besov spaces of logarithmic smoothness ${B}_{p,r}^{0,b}$ formed by periodic functions. We study embeddings of ${B}_{p,r}^{0,b}$ into Lorentz-Zygmund spaces ${L}_{p,q}{\left(logL\right)}_{\beta}$. Our techniques rely on the approximation structure of ${B}_{p,r}^{0,b}$, Nikol’skiĭ type inequalities, extrapolation properties of ${L}_{p,q}{\left(logL\right)}_{\beta}$ and interpolation.

We establish interpolation formulæ for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions.

We investigate the behaviour of bilinear operators under limiting real methods. As an application, we show an interpolation formula for spaces of linear operators. Some results on norm estimates for bounded linear operators are also established.

We prove a reiteration theorem for interpolationmethods defined by means of polygons, and a Wolff theorem for the case when the polygon has 3 or 4 vertices. In particular, we establish a Wolff theorem for Fernandez' spaces, which settles a problem left over in [5].

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space ${\u2102}^{2}$. This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space ${\mathbb{R}}^{2}$. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

We investigate compact operators between approximation spaces, paying special attention to the limit case. Applications are given to embeddings between Besov spaces.

Let Ω be a bounded domain in R and denote by id the restriction operator from the Besov space B (R) into the generalized Lipschitz space Lip(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like e(id) ~ k if α > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.

We review several results on interpolation of Banach algebras and factorization of weakly compact homomorphisms. We also establish a new result on interpolation of multilinear operators.

Given any operator ideal $\mathcal{I}$, there are two natural functionals ${\gamma}_{\mathcal{I}}\left(T\right)$, ${\beta}_{\mathcal{I}}\left(T\right)$ that one can use to show the deviation of the operator $T$ to the closed surjective hull of $\mathcal{I}$ and to the closed injective hull of $\mathcal{I}$, respectively. We describe the behaviour under interpolation of ${\gamma}_{\mathcal{I}}$ and ${\beta}_{\mathcal{I}}$. The results are part of joint works with A. Martínez, A. Manzano and P. Fernández-Martínez.

We investigate the behaviour of the measure of non-compactness of an operator under real interpolation. Our results refer to general Banach couples. An application to the essential spectral radius of interpolated operators is also given.

The paper establishes necessary and sufficient conditions for compactness of operators acting between general K-spaces, general J-spaces and operators acting from a J-space into a K-space. Applications to interpolation of compact operators are also given.

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