Uniform \(\lambda\)-property in \(L^1\cap L^\infty\)

Adam Bohonos ; Ryszard Płuciennik

Commentationes Mathematicae, Tome 55 (2015) no. 2, p. 171−181.

Résumé

Here it is proved that the space \(L^{1}\cap L^{\infty }\) equipped with the standard interpolation norm \(\left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}=\max \left\{ \left\Vert \cdot \right\Vert _{L^{1}},\left\Vert \cdot \right\Vert _{L^{\infty }}\right\} \) has the uniform \(\lambda \)-property if and only if \(\mu (T)\leq 1.\) Replacing the standard norm with an equivalent one \(\left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}^{\prime }= \) \(\left\Vert \cdot \right\Vert _{L^{1}}+\left\Vert \cdot \right\Vert _{L^{\infty }}\), a different result is obtained.: \((L^{1}\cap L^{\infty }, \left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}^{\prime } )\) has the uniform \(\lambda \)-property if and only if \(\mu (T)

Détail

DOI : 10.14708/cm.v55i2.1122

Classification : 46E30;46B20, 46B22
Mots-clés : \(\lambda\)-property, uniform \(\lambda\)-property, interpolation spaces, convex series representation property