We investigate the limit class of interpolation spaces that comes up by the choice θ = 0 in the definition of the real method. These spaces arise naturally interpolating by the J -method associated to the unit square. Their duals coincide with the other extreme spaces obtained by the choice θ = 1. We also study the behavior of compact operators under these two extreme interpolation methods. Moreover, we establish some interpolation formulae for function spaces and for spaces of operators.

K and J spaces and measure of non-compactnessWe establish a formula for the measure of non-compactness of an operator interpolated by the general real method generated by a sequence lattice Γ. The formula is given in terms of the norms of the shift operators in Γ.

Abstract. 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.1. Introduction. Interpolation of compact operators is one of the most active research areas in interpolation theory. Many authors have worked on this subject since the beginning of abstract interpolation theory in the early 1960s. During the last twenty years, new tools have been developed which are intimately related to the type of the interpolation method under consideration. Nowadays, still a lot of work is being done along different lines.Talking only about the real method, it was shown in the joint papers of one of the present authors with Edmunds and Potter [7], with Fernandez [8] and with Peetre [11] that properties of the vector-valued sequence spaces that come up when defining the real interpolation space (A 0 , A 1 ) θ,q are very useful to study the behaviour of compact operators under interpolation. These efforts culminated with Cwikel's [13] proof that if T : A → B withLater, the approach developed in [7,8,11] was used by Cobos, Kühn and Schonbek [9] to give a broad generalization of Cwikel's result, including a function parameter version and even compactness theorems for other interpolation methods. Techniques related to vector-valued sequence spaces have also turned out to be useful to study compactness in the multidimensional case and in the case of infinite families, as can be seen in the papers by Cobos and Peetre [12] and Carro and Peetre [5], respectively.

A procedure is given to reduce the interpolation spaces on an ordered pair generated by the function parameter t θ (1 + |log t|) −b to the classical real interpolation spaces. Applications are given for Lorentz-Zygmund function spaces, Besov spaces of generalized smoothness and Lorentz-Zygmund operator spaces.

No abstract