Evaluation of the Distance to the Space of Multipliers in Integrable Function Spaces

Abstract

In this study we prove the inequality \frac{\alpha}{2}\leq d(T, M)\leq\alpha a for the distance d(T, M) of an operator T \in B (L1 (G)) from the space M of multipliers. Here \sigma sup_{t\in G} \parallel TL_{t} - L_{t} T \parallel and G is a compact Abelian group. Moreover, we prove the same inequality for each operator T \in B (L2 (G)) where G is a locally compact Abelian group.