Evaluation of the distance to the space of multipliers in integrable function spaces

Abstract

In this study we prove the inequality $frac{alpha}{2}leq d(T, M)leqalpha$ a for the distance d(T, M) of an operator $T in B (L^1 (G))$ from the space M of multipliers. Here $sigma = sup_{tin G} parallel TL_{t} - L_{t} T parallel$ and G is a compact Abelian group. Moreover, we prove the same inequality for each operator $T in B (L^2 (G))$ where G is a locally compact Abelian group.