International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 497-501
doi:10.1155/S0161171284000533

A property of L−L integral transformations

Abstract

The main result of this paper is the result that the collection of all integral transformations of the form F(x)=∫0∞G(x,y)f(y)dy for all x≥0, where f(y) is defined on [0,∞) and G(x,y) defined on D={(x,y):x≥0,   y≥0} has no identity transformation on L, where L is the space of functions that are Lebesgue integrable on [0,∞) with norm ‖f‖=∫0∞|f(x)|dx. That is to say, there is no G(x,y) defined on D such that for every f∈L, f(x)=∫0∞G(x,y)f(y)dy for almost all x≥0. In addition, this paper gives a theorem that is an improvement of a theorem that is proved by J. B. Tatchell (1953) and Sunonchi and Tsuchikura (1952).