Optimal decompositions for the \(K\)-functional for a couple of Banach lattices
type: Item
instance of: scholarly article
title: Optimal decompositions for the \(K\)-functional for a couple of Banach lattices
zbMATH Open document ID: 1028.46034
DOI: 10.1007/BF02388790
author: Michael Cwikel
author: Uri Keich
published in: Arkiv för Matematik
publication date: 2003-09-30T00:00:00Z
review text: Let \(X_0\) and \(X_1\) be Banach lattices of (equivalence classes of) real valued measurable functions on the same measure space \((\Omega,\Sigma,\mu)\). It is known that \(X_0\) and \(X_1\) form a Banach couple \(\overline{X}=(X_0,X_1)\) in the sense of interpolation theory. Let \(\{f=g_t+h_t\}_{t>0}\) be the optimal decomposition for calculating the exact value of the \(K-\)functional \(K(t,f,\overline{X})\) of an element \(f\) with respect to a couple \(\overline{X}= (X_0,X_1)\) of Banach lattices of measurable functions. In this paper the authors show that this decomposition has a rather simple form in many cases where one of the spaces \(X_0\) and \(X_1\) is either \(L^{\infty}\) or \(L^1\). Many examples are given of couples of lattices \(\overline{X}\) for which \(|g_t|\) increases monotonically a.e. wich respect to \(t\). It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak \(K-\)divisibility constant \(\gamma(\overline{X})\) for the couple. It is also shown that certain couples \(\overline{X}\) do not have this property. These also provide examples of couples of lattices for which \(\gamma(\overline{X})>1\).
Mathematics Subject Classification ID: 46B70
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1987643
zbMATH Keywords: K-functional
zbMATH Keywords: Banach lattices
zbMATH Keywords: exactly monotone couple
zbMATH Keywords: K-divisibility
zbMATH Keywords: interpolation theory
zbMATH Keywords: K-divisibility constant
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
cites work: On the \(K\)-functional of interpolation between \(L^ p\) and Orlicz spaces
cites work: An explicit expression for the \(K_ r\) functionals of interpolation between \(L^ p\) spaces
cites work: Q4112938
cites work: Weak*-sequential closure and the characteristic of subspaces of conjugate Banach spaces
cites work: Monotonicity properties of interpolation spaces
cites work: K-divisibility of the K-functional and Calderón couples
cites work: On the fundamental lemma of interpolation theory
cites work: On the K functional between \(L^ 1\) and \(L^ 2\) and some other K functionals
cites work: Q3748756
cites work: Q4115333
cites work: Spaces \(\Lambda_\alpha\)(X) and interpolation
cites work: A geometrical approach to the \(K\)-divisibility problem
cites work: Interpolation of Operators with Change of Measures
description: scientific article; zbMATH DE number 1987643