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We investigate how compact operators behave under J and K interpolation methods for N spaces and two parameters. First we study those methods: relationship with those already existing in the literature, estimates for the norms of interpolated operators, examples, characterization as Aronszajn‐Gagliardo functors,…. We also describe the relationship between Sparr and Fernandez methods and we derive sharp estimates for the norms of interpolated operators in Fernandez' case. Then we investigate the behaviour of compact operators. We begin with the case when one of the N‐tupIes reduces to a single Banach space, and later we treat the general case by means of the approach developed in [8].

Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for L 2 -approximation of Sobolev functions into estimates for L ∞ -approximation, with precise control of all involved constants. As

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.