Cauchy difference operator in \({\mathcal L}^p\)
type: Item
instance of: scholarly article
title: Cauchy difference operator in \({\mathcal L}^p\)
zbMATH Open document ID: 0894.39011
author: Stanisław Siudut
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1998-08-12T00:00:00Z
review text: Let \(E\) be a Banach space, \((G,\cdot,\Sigma,\lambda)\) a complete measurable group with \(\lambda\) (left or right)-invariant, and for \(f:G\to E\) define the Cauchy difference \({\mathcal C}f:G\times G\to E\) by \(({\mathcal C}f)(x,y) := f(xy) - f(x)-f(y)\). Under which conditions on \({\mathcal C}f\) is \(f\) close to an additive function? Under the assumption that \(\lambda(G)=\infty\), \textit{Józef Tabor} [Result. Math. 32, 145-158 (1997)] proved that if \({\mathcal C}f \in {\mathcal L}^p_{\lambda\times\lambda}(G\times G,E)\), then there is a \(K>0\) and an additive \(A:G\to E\) with \(| | f-A| | _p \leq K | | {\mathcal C} f| | _p\). The present paper treats the case \(0<\lambda(G)<\infty\). One of the main results is the same conclusion as above, with \(K=\lambda(G)^{-1/p}\). Another result concerns the case where \(\cdot\) is an arbitrary binary operation on \(G\): If \(f \in L^p_\lambda(G,E)\) and \({\mathcal C}f\) is measurable, then this conclusion again holds, in fact with \(A\equiv 0\). In this case the linear operator \(\mathcal C\) is continuous on an appropriate domain and continuously invertible on its image. This extends some results by \textit{E. V. Shulman} [J. Lond. Math. Soc., II. Ser. 54, 111-120 (1996; Zbl 0861.39020)], but only in the special case of the Cauchy equation on compact groups.
reviewed by: Roland Girgensohn
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1123839
zbMATH Keywords: Cauchy equation
zbMATH Keywords: functional equations
zbMATH Keywords: Cauchy difference
zbMATH Keywords: additive functions
zbMATH Keywords: stability
zbMATH Keywords: compact groups
zbMATH Keywords: Banach space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1123839