Interpolation of bilinear operators and compactness

MATHEMATICA SCANDINAVICA, 2019

The behavior of bilinear operators acting on the interpolation of Banach spaces in relation to compactness is analyzed, and an one-sided compactness theorem is obtained for bilinear operators interpolated by the ρ interpolation method.

Bulletin of Mathematical Sciences, 2020

This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of Banach spaces generated by abstract methods of interpolation, in the sense of Aronszajn and Gagliardo. To get these results, we prove a key one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an extra approximation property. We give applications to general cases, including Peetre’s method and the general real interpolation methods.

2016

e is compact and show a positive answer under a variety of conditions. For example it suffices that X o be a UMD-space or that X o is reflexive and there is a Banach space so that X 0 = [W,X l~\I for some 0<ot< 1. 1991 Mathematics subject classification: 46M35.

Proceedings of the American Mathematical Society, 2014

We show that multilinear interpolation can be lifted to multilinear operators from spaces generated by the minimal methods to spaces generated by the maximal methods of interpolation defined on a class of couples of compatible p-Banach spaces. We also prove mutlilinear interpolation theorem for operators on Calderón-Lozanovskii spaces between L p-spaces with 0 < p ≤ 1. As an application we obtain interpolation theorems for multilinear operators on quasi-Banach Orlicz spaces. We introduce relevant notation and we recall some definitions. Let (X, •) be a quasinormed space. A quasi-norm induces locally bounded topology. A complete quasi-normed space is called quasi-Banach space. If in addition we have for some 0 < p ≤ 1 x + y p ≤ x p + y p , x, y ∈ X,

Transactions of the American Mathematical Society, 2018

Working in the setting of quasi-Banach couples, we establish a formula for the measure of noncompactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter. c

Arkiv for Matematik, 1989

This originated the question of whether there are generalizations of Theorem L-P to the case Ao~A~ and BomB 1.

Arkiv för matematik, 1997

Archiv der Mathematik, 1990

Acta Applicandae Mathematicae, 1997

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos-Peetre methods of interpolation for finite families.

Studia Mathematica

Let (X j , d j , µ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p κ ≤ ∞ for κ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) × • • • L p m (X m) → L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Stein's complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schrödinger operators on L p is included.