Boundedness Properties of Stationary Linear Operators
James D. Baker · SIAM Journal on Mathematical Analysis · 1972
A characterization of boundedness given by R. E. Lane provides a foundation for studying stationary, linear transformations which map the left-continuous quasi-continuous functions on the line into functions on the line. Properties of these operators are identified in terms of the transforms of $J(t) = \chi _{(0,\infty )} $ It is shown that the norm of an operator T is the variation of $TJ$, that there is a Stieltjes integral representation of T with this function as the integrator, and that convergence of a sequence of these operators is equivalent to convergence in variation of the sequence of transforms of J. An operator which is continuous under pointwise convergence is shown to be bounded on a closed interval, and is characterized by the left-continuity of $TJ$.