Real interpolation of compact bilinear operators

We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kühn and Schonbek. We work with the assumption that : (A0+ A1)×(B0+ B1) −→ E0+E1 is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderón–Zygmund operators.Ministerio de Economía y Competitividad | Ref. MTM2013-42220-

On interpolation of the measure of noncompactness

We revised the known results on interpolation of the measure of noncompactness and we announce a new approach to establishing the interpolation formula for the real method

On compactness results of Lions-Peetre type for bilinear operators

Let Ā = (A₀ , A₁) , B̄ = (B₀ , B₁) be Banach couples, let E be a Banach space and let T be a bilinear operator such that ||T(a, b)||ᴇ ≤ M[sub]j ||a||ᴀ[sub]j ||b||ʙ[sub]j for a ∈ A₀ ∩ A₁, b ∈ B₀ ∩ B₁, j = 0, 1. If T : A°[sub]j × B°[sub]j −→ E compactly for j = 0 or 1, we show that T may be uniquely extended to a compact bilinear operator from the complex interpolation spaces generated by Ā and B̄ to E. Furthermore, the corresponding result for the real method is given and we also study the case when E is replaced by a couple (E₀, E₁) of Banach function spaces on the same measure space

A compactness result of Janson type for bilinear operators

We establish a compactness interpolation result for bilinear operators of the type proved by Janson for bounded bilinear operators. We also give an application to compactness of convolution operators.Agencia Estatal de Investigación | Ref. MTM2017-84058-

Associate spaces of logarithmic interpolation spaces and generalized Lorentz-Zygmund spaces

We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces

On the interpolation of the measure of non-compactness of bilinear operators with weak assumptions on the boundedness of the operator

We complete the range of the parameters in the interpolation formula established by Mastylo and Silva for the measure of non-compactness of a bilinear operator interpolated by the real method.Agencia Estatal de Investigación | Ref. MTM2017-84058-

On compactness results of Lions-Peetre type for bilinear operators

Let A¯ = (A0, A1) , B¯ = (B0, B1) be Banach couples, let E be a Banach space and let T be a bilinear operator such that kT(a, b)kE ≤ MjkakAj kbkBj for a ∈ A0 ∩ A1, b ∈ B0 ∩ B1, j = 0, 1. If T : A◦ j × B◦ j −→ E compactly for j = 0 or 1, we show that T may be uniquely extended to a compact bilinear operator from the complex interpolation spaces generated by A¯ and B¯ to E. Furthermore, the corresponding result for the real method is given and we also study the case when E is replaced by a couple (E0, E1) of Banach function spaces on the same measure space.Agencia Estatal de Investigación | Ref. MTM2017-84058-

Interpolation of compact bilinear operators among quasi-Banach spaces and applications

We study the interpolation properties of compact bilinear operators by the general real method among quasi-Banach couples. As an application we show that some commutators of Calderón-Zygmund bilinear operators are compact.Agencia Estatal de Investigación | Ref. MTM2017-84058-

Weakly compact bilinear operators among real interpolation spaces

We show a necessary and sufficient condition for weak compactness of bilinear operators interpolated by the real method. This characterization does not hold for interpolated operators by the complex method