Strong convergence of functions on Köthe spaces

Abstract:

Let $\Lambda$ be a rearrangement invariant Köthe space over a nondiscrete group G with Haar measure $\mu$. For a function $f \in \Lambda$ and relatively compact 0-neighborhood U in G the function \[ {T_U}f(x) = \frac {1}{{\mu (U)}} \cdot \int _{U + x} {f d\mu } \] is continuous and also belongs to $\Lambda$. The convergence ${T_U}f \to f$ (as $U \to 0$) for the strong Köthe topology on $\Lambda$ is involved in establishing compactness criteria for subsets of a Köthe space. The main result of this paper is a necessary and sufficient condition for convergence ${T_U}f \to f$ in the strong topology on $\Lambda$.